Chapter 4 Resource Masters · PDF file · 2015-06-19Chapter 4 Mid-Chapter Test .....59 Chapter 4 Vocabulary ... assessment and summative (final) assessment. ... • Form 3 is a free-response

Chapter 4 Resource Masters

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.

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ISBN: 978-0-07-890513-1

MHID: 0-07-890513-3

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13

Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4

Homework Pratice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish Version

Homework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 4 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 4 Test Form 2A and Form 2C.

iii

Contents

Teacher’s Guide to Using the Chapter 4

Resource Masters .........................................iv

Chapter ResourcesStudent-Built Glossary ....................................... 1

Anticipation Guide (English) .............................. 3

Anticipation Guide (Spanish) ............................. 4

Lesson 4-1Classifying Triangles

Study Guide and Intervention ............................ 5

Skills Practice .................................................... 7

Practice .............................................................. 8

Word Problem Practice ..................................... 9

Enrichment ...................................................... 10

Lesson 4-2Angles of Triangles

Study Guide and Intervention .......................... 11

Skills Practice .................................................. 13

Practice ............................................................ 14

Word Problem Practice ................................... 15

Enrichment ...................................................... 16

Cabri Junior. ................................................... 17

Geometer’s Sketchpad Activity ....................... 18

Lesson 4-3Congruent Triangles


Skills Practice .................................................. 21

Practice ............................................................ 22


Enrichment ...................................................... 24

Lesson 4-4Proving Triangles Congruent—SSS, SAS


Skills Practice .................................................. 27

Practice ............................................................ 28


Enrichment ...................................................... 30

Lesson 4-5Proving Triangles Congruent—ASA, AAS


Skills Practice .................................................. 33

Practice ............................................................ 34


Enrichment ...................................................... 36

Lesson 4-6Isosceles and Equilateral Triangles


Skills Practice .................................................. 39

Practice ............................................................ 40


Enrichment ...................................................... 42

Lesson 4-7Congruence Transformations


Skills Practice .................................................. 45

Practice ............................................................ 46


Enrichment ...................................................... 48

Lesson 4-8Triangles and Coordinate Proof


Skills Practice .................................................. 51

Practice ............................................................ 52


Enrichment ...................................................... 54

AssessmentStudent Recording Sheet ................................ 55

Rubric for Extended Response ....................... 56

Chapter 4 Quizzes 1 and 2 ............................. 57

Chapter 4 Quizzes 3 and 4 ............................. 58

Chapter 4 Mid-Chapter Test ............................ 59

Chapter 4 Vocabulary Test ............................. 60

Chapter 4 Test, Form 1 ................................... 61

Chapter 4 Test, Form 2A ................................. 63

Chapter 4 Test, Form 2B ................................. 65

Chapter 4 Test, Form 2C ................................ 67

Chapter 4 Test, Form 2D ................................ 69

Chapter 4 Test, Form 3 ................................... 71

Chapter 4 Extended-Response Test ............... 73

Standardized Test Practice ............................. 74

Answers ........................................... A1–A40

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Teacher’s Guide to Using the

Chapter 4 Resource MastersThe Chapter 4 Resource Masters includes the core materials needed for Chapter 4. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on

the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 4-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator or Spreadsheet

Activities These activities present ways in which technology can be used with the con-cepts in some lessons of this chapter. Use as an alternative approach to some con-cepts or as an integral part of your lesson presentation.

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Assessment OptionsThe assessment masters in the

Chapter 4 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended Response Rubric This master provides information for teachers and stu-dents on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice questions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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Chapter 4 1 Glencoe Geometry

Student-Built Glossary

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.

As you study the chapter, complete each term’s definition or description.

Remember to add the page number where you found the term. Add these pages to

your Geometry Study Notebook to review vocabulary at the end of the chapter.

4

Vocabulary TermFound

on PageDefinition/Description/Example

auxiliary line

base angle

congruent

corresponding parts

congruent polygons

equiangular triangle

equilateral triangle

image

included angle

(continued on the next page)

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Student-Built Glossary (continued)

Vocabulary TermFound

on PageDefinition/Description/Example

included side

isosceles triangle

preimage

reflection

rotation

scalene (SKAY·leen) triangle

transformation

translation

4

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Anticipation Guide

Congruent Triangles

Before you begin Chapter 4

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 4

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

4

Step 1

STEP 1A, D, or NS

StatementSTEP 2A or D

1. For a triangle to be acute, all three angles must be acute.

2. All sides of an equilateral triangle are congruent.

3. An exterior angle is any angle outside of the given triangle.

4. The sum of the measures of the angles in a triangle is 360°.

5. An obtuse triangle is a triangle with three obtuse angles.

6. Triangles that are the same shape and size are congruent.

7. If the angles of one triangle are congruent to the angles of another triangle, then the two triangles are congruent.

8. An isosceles triangle is a triangle with two congruent sides.

9. A triangle can be equiangular and not equilateral.

10. To make the calculations in a coordinate proof easier, place either a vertex or the center of the polygon at the origin.

Step 2

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NOMBRE FECHA PERÍODO

Ejercicios preparatorios

Triá ngulos congruentes

Antes de comenzar el Capítulo 4

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a)).

Después de completar el Capítulo 4

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

Paso 2

Paso 1

PASO 1A, D, o NS

EnunciadoPASO 2A o D

1. Para que un triángulo sea acutángulo, todos sus tres ángulos deben ser agudos.

2. Todos los lados de un triángulo equilátero son iguales.

3. Un ángulo exterior es cualquier ángulo fuera del triángulo dado.

4. La suma de las medidas de los ángulos de un triángulo es 360°.

5. Un triángulo obtusángulo es un triángulo con tres ángulos obtusos.

6. Los triángulos que son iguales en forma y tamaño son congruentes.

7. Si los ángulos de dos triángulos son congruentes, entonces los dos triángulos son congruentes.

8. Un triángulo isósceles es un triángulo con dos lados congruentes.

9. Un triángulo puede ser equiángulo y no equilátero.

10. Para facilitar los cálculos en una demostración por coordenadas, ubica ya sea un vértice o el centro del polígono en el origen.

4

Capítulo 4 4 Geometría de Glencoe

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Less

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4-1


Study Guide and Intervention

Classifying Triangles

Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.

• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.

• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.

• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.

• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.

Classify each triangle.

a.

60°

A

B C

All three angles are congruent, so all three angles have measure 60°.

The triangle is an equiangular triangle.

b.

25°35°

120°

D F

E

The triangle has one angle that is obtuse. It is an obtuse triangle.

c.

90°

60° 30°

G

H J

The triangle has one right angle. It is a right triangle.

Exercises

Classify each triangle as acute, equiangular, obtuse, or right.

1. 67°

90° 23°

K

L M

2.

120°

30° 30°N O

P

3.

60° 60°

60°

Q

R S

4.

65° 65°

50°

U V

T 5.

45°

45°90°X Y

W 6. 60°

28° 92°F D

B

4-1

Example

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Study Guide and Intervention (continued)


Classify Triangles by Sides You can classify a triangle by the number of congruent sides. Equal numbers of hash marks indicate congruent sides.

• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.

• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.

Equilateral triangles can also be considered isosceles.

• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.

Classify each triangle.

a.

L J

H b. N

R P

c.

2312

15X V

T

Two sides are congruent. All three sides are The triangle has no pair

The triangle is an congruent. The triangle of congruent sides. It is

isosceles triangle. is an equilateral triangle. a scalene triangle.

Exercises

Classify each triangle as equilateral, isosceles, or scalene.

1. 2

1

3√G C

A 2. G

K I18

18 18

3.

12 17

19Q O

M

4.

UW

S 5. 8x

32x

32x

B

C

A 6. D E

F

x

x x

Example

4-1

7. ALGEBRA Find x and the length of each

side if △RST is an equilateral triangle.

8. ALGEBRA Find x and the length of

each side if △ABC is isosceles with

AB = BC.

5x - 42x + 2

3x

3y + 24y

3y

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Skills Practice

Classifying Triangles Classify each triangle as acute, equiangular, obtuse, or right.

1.

60° 60°

60° 2. 40°

95° 45°

3.

50° 40°

90°

4. 80°

50°

50°

5.

100°

30°

50° 6. 70°

55° 55°

Classify each triangle as equilateral, isosceles, or scalene.

E CD

AB9

8

8√2 7. △ABE 8. △EDB

9. △EBC 10. △DBC


each side if △ABC is an isosceles

triangle with −−

AB # −−

BC .


side if △FGH is an equilateral triangle.

5x - 83x + 10

4x + 1


each side if △RST is an isosceles

triangle with −−

RS # −−

TS .

3x - 1

9x + 2

6x + 8


side if △DEF is an equilateral triangle.

5x - 212x + 6

7x - 39

4-1

4x - 3 2x + 5

x + 2

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Practice

Classifying Triangles Classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

Classify each triangle in the figure at the right by its angles and sides.

4. △ABD 5. △ABC

6. △EDC 7. △BDC

ALGEBRA For each triangle, find x and the measure of each side.

8. △FGH is an equilateral triangle with FG = x + 5, GH = 3x - 9, and FH = 2x - 2.

9. △LMN is an isosceles triangle, with LM = LN, LM = 3x - 2, LN = 2x + 1, and MN = 5x - 2.

Find the measures of the sides of △KPL and classify each triangle by its sides.

10. K(-3, 2), P(2, 1), L(-2, -3)

11. K(5, -3), P(3, 4), L(-1, 1)

12. K(-2, -6), P(-4, 0), L(3, -1)

13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there?

A CD

E

B

60°

30°

90°

65°30°

85°

40°

100°

40°

4-1

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Word Problem Practice


1. MUSEUMS Paul is standing in front of

a museum exhibition. When he turns his

head 60° to the left, he can see a statue

by Donatello. When he turns his head

60° to the right, he can see a statue by

Della Robbia. The two statues and Paul

form the vertices of a triangle. Classify

this triangle as acute, right, or obtuse.

2. PAPER Marsha cuts a rectangular piece

of paper in half along a diagonal. The

result is two triangles. Classify these

triangles as acute, right, or obtuse.

3. WATERSKIING Kim and Cassandra

are waterskiing. They are holding on to

ropes that are the same length and tied

to the same point on the back of a speed

boat. The boat is going full speed ahead

and the ropes are fully taut.

Kim, Cassandra, and the point where

the ropes are tied on the boat form the

vertices of a triangle. The distance

between Kim and Cassandra is never

equal to the length of the ropes. Classify

the triangle as equilateral, isosceles, or

scalene.

4. BOOKENDS Two bookends are shaped

like right triangles.

The bottom side of each triangle is

exactly half as long as the slanted side

of the triangle. If all the books between

the bookends are removed and they are

pushed together, they will form a single

triangle. Classify the triangle that can

be formed as equilateral, isosceles, or

scalene.

5. DESIGNS Suzanne saw this pattern on

a pentagonal floor tile. She noticed many

different kinds of triangles were created

by the lines on the tile.

a. Identify five triangles that appear to

be acute isosceles triangles.

b. Identify five triangles that appear to

be obtuse isosceles triangles.

4-1

Kim

Cassandra

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J

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Enrichment

Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the text

easier to understand.

Consider three points, A, B, and C on a coordinate grid. The

y-coordinates of A and B are the same. The x-coordinate of B is greater than the

x-coordinate of A. Both coordinates of C are greater than the corresponding

coordinates of B. Is △ABC acute, right, or obtuse?

To answer this question, first draw a sample triangle

that fits the description.

Side −−

AB must be a horizontal segment because the

y-coordinates are the same. Point C must be located

to the right and up from point B.

From the diagram you can see that triangle ABC

must be obtuse.

Exercises

Draw a simple triangle on grid paper to help you.

1. Consider three points, R, S, and 2. Consider three noncollinear points,

T on a coordinate grid. The J, K, and L on a coordinate grid. The

x-coordinates of R and S are the y-coordinates of J and K are the

same. The y-coordinate of T is same. The x-coordinates of K and L

between the y-coordinates of R are the same. Is △JKL acute,

and S. The x-coordinate of T is less right, or obtuse?

than the x-coordinate of R. Is angle

R of △RST acute, right, or

obtuse?

3. Consider three noncollinear points, 4. Consider three points, G, H, and I

D, E, and F on a coordinate grid. on a coordinate grid. Points G and

The x-coordinates of D and E are H are on the positive y-axis, and

opposites. The y-coordinates of D and the y-coordinate of G is twice the

E are the same. The x-coordinate of y-coordinate of H. Point I is on the

F is 0. What kind of triangle must positive x-axis, and the x-coordinate

△DEF be: scalene, isosceles, or of I is greater than the y-coordinate

equilateral? of G. Is △GHI scalene, isosceles, or

equilateral?

BA

C

x

y

O

4-1

Example

Less

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Angles of Triangles

Triangle Angle-Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found.

Triangle Angle Sum

Theorem

The sum of the measures of the angles of a triangle is 180.

In the figure at the right, m∠A + m∠B + m∠C = 180.

CA

B

Find m∠T.

m∠R + m∠S + m∠T = 180 Triangle Angle -

Sum Theorem

25 + 35 + m∠T = 180 Substitution

60 + m∠T = 180 Simplify.

m∠T = 120 Subtract 60 from

each side.

Find the missing

angle measures.

m∠1 + m∠A + m∠B = 180 Triangle Angle - Sum

Theorem

m∠1 + 58 + 90 = 180 Substitution

m∠1 + 148 = 180 Simplify.

m∠1 = 32 Subtract 148 from

each side.

m∠2 = 32 Vertical angles are

congruent.

m∠3 + m∠2 + m∠E = 180 Triangle Angle - Sum

Theorem

m∠3 + 32 + 108 = 180 Substitution

m∠3 + 140 = 180 Simplify.

m∠3 = 40 Subtract 140 from

each side.

58°

90°

108°

1

2 3

E

DAC

B

Exercises

Find the measures of each numbered angle.

1. 2.

3. 4.

5. 6. 20°

152°

DG

A

130°60°

1 2

S

R

TW

V

W T

U

30°

60°

2

1

S

Q R30°

1

90°

62°

1N

M

P

4-2

Example 1

35°

25°R T

S

Example 2

Q

O

NM

P58°

66°

50°

3

21

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Angles of Triangles

Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB.

Exterior Angle

Theorem

The measure of an exterior angle of a triangle is equal to

the sum of the measures of the two remote interior angles.

AC

B

D

1m∠1 = m∠A + m∠B

Find m∠1.

m∠1 = m∠R + m∠S Exterior Angle Theorem

= 60 + 80 Substitution

= 140 Simplify.

RT

S

60°

80°

1

Find x.

m∠PQS = m∠R + m∠S Exterior Angle Theorem

78 = 55 + x Substitution

23 = x Subtract 55 from each

side.

S R

Q

P

55°

78°

x°

Exercises

Find the measures of each numbered angle.

1. 2.

3. 4.

Find each measure.

5. m∠ABC 6. m∠F

E

FGH

58°

x°

x°B

A

DC

95°

2x° 145°

U T

SRV

35° 36°

80°

13

2

POQ

N M60°

60°

3 2

1

BC D

A

25°

35°

12

YZ W

X

65°

50°

1

4-2

Example 1 Example 2

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Skills Practice

Angles of Triangles

Find the measure of each numbered angle.

1. 2.

Find each measure.

3. m∠1

4. m∠2

5. m∠3

Find each measure.

6. m∠1

7. m∠2

8. m∠3

Find each measure.

9. m∠1

10. m∠2

11. m∠3

12. m∠4

13. m∠5

Find each measure.

14. m∠1

15. m∠2 63°

1

2

DA C

B

80°

60°

40°

105°

1 4 5

2

3

150°55°

70°

1 2

3

85°55°

40°

1 2

3

146°1 2

TIGERS80°

73°

1

4-2

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Practice

Angles of Triangles

Find the measure of each numbered angle.

1. 2.

Find each measure.

3. m∠1

4. m∠2

5. m∠3

Find each measure.

6. m∠1

7. m∠4

8. m∠3

9. m∠2

10. m∠5

11. m∠6

Find each measure.

12. m∠1

13. m∠2

14. CONSTRUCTION The diagram shows an

example of the Pratt Truss used in bridge

construction. Use the diagram to find m∠1.

145°1

64°58°1

2A

B

C

D

118°

36°

68°

70°

65°

82°

1

2

3 4

5

6

58°

39°

35°

12

3

40°

1

55°

72°

1

2

4-2

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Angles of Triangles

62˚136˚

1. PATHS Eric walks around a triangular

path. At each corner, he records the

measure of the angle he creates.

He makes one complete circuit around

the path. What is the sum of the

three angle measures that he wrote

down?

2. STANDING Sam, Kendra, and Tony are

standing in such a way that if lines were

drawn to connect the friends they would

form a triangle.

If Sam is looking at Kendra he needs to

turn his head 40° to look at Tony. If

Tony is looking at Sam he needs to turn

his head 50° to look at Kendra. How

many degrees would Kendra have to

turn her head to look at Tony if she is

looking at Sam?

3. TOWERS A lookout tower sits on a

network of struts and posts. Leslie

measured two angles on the tower.

What is the measure of angle 1?

4. ZOOS The zoo lights up the chimpanzee

pen with an overhead light at night. The

cross section of the light beam makes an

isosceles triangle.

The top angle of the triangle is 52 and

the exterior angle is 116. What is the

measure of angle 1?

5. DRAFTING Chloe bought a drafting

table and set it up so that she can draw

comfortably from her stool. Chloe

measured the two angles created by the

legs and the tabletop in case she had to

dismantle the table.

a. Which of the four numbered angles

can Chloe determine by knowing the

two angles formed with the tabletop?

What are their measures?

b. What conclusion can Chloe make

about the unknown angles before she

measures them to find their exact

measurements?

4-2

52˚

116˚1

Kendra

Tony

Sam

29˚

68˚

12

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Enrichment

Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.

In triangle ABC, m∠A is twice m∠B, and m∠C

is 8 more than m∠B. What is the measure of each angle?

Write and solve an equation. Let x = m∠B.

m∠A + m∠B + m∠C = 180

2x + x + (x + 8) = 180

4x + 8 = 180

4x = 172

x = 43

So, m∠A = 2(43) or 86, m∠B = 43, and m∠C = 43 + 8 or 51.

Solve each problem.

1. In triangle DEF, m∠E is three times 2. In triangle RST, m∠T is 5 more than m∠D, and m∠F is 9 less than m∠E. m∠R, and m∠S is 10 less than m∠T.What is the measure of each angle? What is the measure of each angle?

3. In triangle JKL, m∠K is four times 4. In triangle XYZ, m∠Z is 2 more than twice m∠J, and m∠L is five times m∠J. m∠X, and m∠Y is 7 less than twice m∠X. What is the measure of each angle? What is the measure of each angle?

5. In triangle GHI, m∠H is 20 more than 6. In triangle MNO, m∠M is equal to m∠N, m∠G, and m∠G is 8 more than m∠I. and m∠O is 5 more than three times What is the measure of each angle? m∠N. What is the measure of each angle?

7. In triangle STU, m∠U is half m∠T, 8. In triangle PQR, m∠P is equal to and m∠S is 30 more than m∠T. What m∠Q, and m∠R is 24 less than m∠P.is the measure of each angle? What is the measure of each angle?

9. Write your own problems about measures of triangles.

4-2

Example

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Graphing Calculator Activity

Cabri Junior: Angles of Triangles

Cabri Junior can be used to investigate the relationships between the angles in a triangle.

Use Cabri Junior to investigate the sum of the measures of the

angles of a triangle and to investigate the measure of the exterior angle in a

triangle in relation to its remote interior angles.

Step 1 Use Cabri Junior to draw a triangle. • Select F2 Triangle to draw a triangle. • Move the cursor to where you want the first vertex. Then press ENTER . • Repeat this procedure to determine the next two vertices of the triangle.

Step 2 Label each vertex of the triangle. • Select F5 Alph-num.

• Move the cursor to a vertex and press ENTER . Enter A and press ENTER again. • Repeat this procedure to label vertex B and vertex C.

Step 3 Draw an exterior angle of △ABC.

• Select F2 Line to draw a line through −−−

BC .

• Select F2 Point, Point on to draw a point on # $% BC so that C is between B and the new point.

• Select F5 Alph-num to label the point D.

Step 4 Find the measure of the three interior angles of △ABC and the exterior angle, ∠ACD.

• Select F5 Measure, Angle. • To find the measure of ∠ABC, select points A, B, and C

(with the vertex B as the second point selected). • Use this method to find the remaining angle measures.

Exercises

Analyze your drawing.

1. Use F5 Calculate to find the sum of the measures of the three interior angles.

2. Make a conjecture about the sum of the interior angles of a triangle.

3. Change the dimensions of the triangle by moving point A. (Press CLEAR so the pointer becomes a black arrow. Move the pointer close to point A until the arrow becomes transparent and point A is blinking. Press ALPHA to change the arrow to a hand. Then move the point.) Is your conjecture still true?

4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?

5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.

6. Change the dimensions of the triangle by moving point A. Is your conjecture still true?

4-2

Example

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Geometer’s Sketchpad Activity

Angles of Triangles

The Geometer’s Sketchpad can be used to investigate the relationships between the angles in a triangle.

Use The Geometer’s Sketchpad to investigate the sum of the

measures of the angles of a triangle and to investigate the measure of the

exterior angle in a triangle in relation to its remote interior angles.

Step 1 Use The Geometer’s Sketchpad to draw a triangle. • Construct a ray by selecting the Ray tool from the toolbar. First, click where

you want the first point. Then click on a second point to draw the ray. • Next, select the Segment tool from the toolbar. Use the endpoint of the ray as

the first point for the segment and click on a second point to construct the segment.

• Construct another segment joining the second point of the new segment to a point on the ray.

Step 2 Display the labels for each point. Use the pointer tool to select all four points. Display the labels by selecting Show Labels from the Display menu.

Step 3 Find the measures of the three interior angles and the one exterior angle. To find the measure of angle ABC, use the Selection Arrow tool to select points A, B, and C (with the vertex B as the second point selected). Then, under the Measure menu, select Angle. Use this method to find the remaining angle measures.

Exercises

Analyze your drawing.

1. What is the sum of the measures of the three interior angles?

2. Make a conjecture about the sum of the interior angles of a triangle.

3. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?

4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?

5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.

6. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?

m/ABC = 60.00°

m/BAC = 58.19°

m/ACB = 61.81°

m/ACD = 118.19°

m/ABC + m/BAC + m/ACB = 180.00°

m/ABC + m/BAC = 118.19°

B C D

A

4-2

Example

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Less

on

4-3


Congruence and Corresponding Parts Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, △ABC ! △RST.

If △XYZ ! △RST, name the pairs of

congruent angles and congruent sides.

∠X ! ∠R, ∠Y ! ∠S, ∠Z ! ∠T

−−

XY ! −−

RS , −−

XZ ! −−

RT , −−

YZ ! −−

ST

Exercises

Show that the polygons are congruent by identifying all congruent corresponding

parts. Then write a congruence statement.

1.

CA

B

LJ

K 2.

C

D

A

B 3. K

J

L

M

4. F G L K

JE

5. B D

CA

6. R

T

U S

7. Find the value of x.

8. Find the value of y.

Y

XZ

T

SR

AC

B

R

T

S

4-3 Study Guide and Intervention

Congruent Triangles

Third Angles

Theorem

If two angles of one triangle are congruent to two angles

of a second triangle, then the third angles of the triangles are congruent.

Example

2x+ y(2y- 5)°75°

65° 40°

96.6

90.664.3

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Congruent Triangles

4-3

Example

Prove Triangles Congruent Two triangles are congruent if and only if their corresponding parts are congruent. Corresponding parts include corresponding angles and corresponding sides. The phrase “if and only if” means that both the conditional and its converse are true. For triangles, we say, “Corresponding parts of congruent triangles are congruent,” or CPCTC.

Write a two-column proof.

Given: −−

AB " −−−

CB , −−−

AD " −−−

CD , ∠BAD " ∠BCD

−−−

BD bisects ∠ABC

Prove: △ABD " △CBD

Proof:

Statement Reason

1. −−

AB " −−−

CB , −−−

AD " −−−

CD

2. −−−

BD " −−−

BD

3. ∠BAD " ∠BCD

4. ∠ABD " ∠CBD

5. ∠BDA " ∠BDC

6. △ABD " △CBD

1. Given

2. Reflexive Property of congruence

3. Given

4. Definition of angle bisector

5. Third Angles Theorem

6. CPCTC

ExercisesWrite a two-column proof.

1. Given: ∠A " ∠C, ∠D " ∠B, −−−

AD " −−−

CB , −−

AE " −−−

CE ,

−−

AC bisects −−−

BD

Prove: △AED " △CEB

Write a paragraph proof.

2. Given: −−−

BD bisects ∠ABC and ∠ADC,

−−

AB " −−−

CB , −−

AB " −−−

AD , −−−

CB " −−−

DC


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4-3

Show that polygons are congruent by identifying all congruent corresponding

parts. Then write a congruence statement.

1.

L

P

J

S

V

T

2.

In the figure, △ABC " △FDE.



5. PROOF Write a two-column proof.

Given: −−−

AB " −−−

CB , −−−

AD " −−−

CD , ∠ABD " ∠CBD,

∠ADB " ∠CDB


Skills Practice

Congruent Triangles

D

E

G

F

108° 48° y°

(2x - y)°

A F

BC ED

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Practice

Congruent Triangles

Show that the polygons are congruent by indentifying all congruent

corresponding parts. Then write a congruence statement.

1.

D

R

S

CA

B 2. M

N

L

P

Q

Polygon ABCD polygon PQRS.




Given: ∠P ! ∠R, ∠PSQ ! ∠RSQ, −−−

PQ ! −−−

RQ ,

−−

PS ! −−

RS

Prove: △PQS △RQS

6. QUILTING

a. Indicate the triangles that appear to be congruent.

b. Name the congruent angles and congruent sides of a pair of

congruent triangles.

B

A

I

E

FH G

C D

4-3

(2x+ 4)°

(3y- 3)

80°

100°

10

12

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Congruent Triangles

1. PICTURE HANGING Candice hung a picture that was in a triangular frame on her bedroom wall.

Before

A B

C

After

One day, it fell to the floor, but did not break or bend. The figure shows the object before and after the fall. Label the vertices on the frame after the fall according to the “before” frame.

2. SIERPINSKI’S TRIANGLE The figure below is a portion of Sierpinski’s Triangle. The triangle has the property that any triangle made from any combination of edges is equilateral. How many triangles in this portion are congruent to the black triangle at the bottom corner?

3. QUILTING Stefan drew this pattern for a piece of his quilt. It is made up of congruent isosceles right triangles. He drew one triangle and then repeatedly drew it all the way around.

45˚

What are the missing measures of the angles of the triangle?

4. MODELS Dana bought a model airplane kit. When he opened the box, these two congruent triangular pieces of wood fell out of it.

A

B

CQ

R

P

Identify the triangle that is congruent to △ABC.

5. GEOGRAPHY Igor noticed on a map that the triangle whose vertices are the supermarket, the library, and the post office (△SLP) is congruent to the triangle whose vertices are Igor’s home, Jacob’s home, and Ben’s home (△IJB). That is, △SLP ! △IJB.

a. The distance between the supermarket and the post office is 1 mile. Which path along the triangle △IJB is congruent to this?

b. The measure of ∠LPS is 40. Identify the angle that is congruent to this angle in △IJB.

4-3

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Enrichment4-3

Overlapping TrianglesSometimes when triangles overlap each other, they share corresponding parts. When trying to picture the corresponding parts, try redrawing the triangles separately.

In the figure △ABD ! △DCA and m∠BDA = 30°.

Find m∠CAD and m∠CDA.

30°

First, redraw △ABD and △DCA like this.

30°

Now you can see that ∠BDA ! ∠CAD because they are corresponding parts of congruent triangles, so m∠CAD = 30. Using the Triangle Sum Theorem, 180 - (90 + 30) = 60, so m∠CDA = 60.

Exercises

1. In the figure △ABC ! △CDB, m∠ABC = 110, and m∠D = 20. Find m∠ACB and m∠DCB.

2. Write a two column proof.

Given: ∠A and ∠B are right angles, ∠ACD ! ∠BDC,

−−−

AD ! −−−

BC , −−

AC ! −−−

BD

Prove: △ADC ! △CBD

Example

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Proving Triangles Congruent—SSS, SAS

SSS Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle.


Given: −−

AB " −−−

DB and C is the midpoint of −−−

AD .

Prove: △ABC " △DBC B

CDA

Statements Reasons

1. −−

AB " −−−

DB 1. Given

2. C is the midpoint of −−−

AD . 2. Given

3. −−

AC " −−−

DC 3. Midpoint Theorem

4. −−−

BC " −−−

BC 4. Reflexive Property of "

5. △ABC " △DBC 5. SSS Postulate

Exercises


1. B Y

CA XZ

Given: AB "−−XY ,

−−AC "

−−XZ ,

−−−BC "

−−YZ

Prove: △ABC " △XYZ

2. T U

R S

Given: −−

RS " −−−

UT , −−

RT " −−−

US

Prove: △RST " △UTS

4-4

SSS PostulateIf three sides of one triangle are congruent to three sides of a second

triangle, then the triangles are congruent.

Example

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SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.

For each diagram, determine which pairs of triangles can be

proved congruent by the SAS Postulate.

a. A

B C

X

Y Z

b. D H

F E

G

J

c. P Q

S

1

2R

In △ABC, the angle is not The right angles are The included angles, “included” by the sides

−− AB congruent and they are the ∠1 and ∠2, are congruent

and −−

AC . So the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the SAS Postulate. △DEF $ △JGH by the for two parallel lines. SAS Postulate. △PSR $ △RQP by the SAS Postulate.

Exercises

4-4

SAS Postulate

If two sides and the included angle of one triangle are congruent to two

sides and the included angle of another triangle, then the triangles are

congruent.

Example

Write the specified type of proof.

1. Write a two column proof. 2. Write a two-column proof. Given: NP = PM,

−−− NP ⊥

−− PL Given: AB = CD,

−− AB ‖

−−− CD

Prove: △NPL $ △MPL Prove: △ACD $ △CAB

3. Write a paragraph proof.

Given: V is the midpoint of −−

YZ .

V is the midpoint of −−−

WX .

Prove: △XVZ $ △WVY

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-4


Determine whether △ABC ! △KLM. Explain.

1. A(-3, 3), B(-1, 3), C(-3, 1), K(1, 4), L(3, 4), M(1, 6)

2. A(-4, -2), B(-4, 1), C(-1, -1), K(0, -2), L(0, 1), M(4, 1)

PROOF Write the specified type of proof.

3. Write a flow proof.

Given: −−

PR " −−−

DE , −−

PT " −−−

DF

∠R " ∠E, ∠T " ∠F

Prove: △PRT " △DEF

4. Write a two-column proof.

Given: −−

AB " −−−

CB , D is the midpoint of −−

AC .


T

R

P

F

E

D

4-4 Skills Practice


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Practice

Proving Triangles Congruent—SSS, SASDetermine whether △DEF ! △PQR given the coordinates of the vertices. Explain.

1. D(-6, 1), E(1, 2), F(-1, -4), P(0, 5), Q(7, 6), R(5, 0)

2. D(-7, -3), E(-4, -1), F(-2, -5), P(2, -2), Q(5, -4), R(0, -5)


Given: −−

RS " −−

TS

V is the midpoint of −−

RT .

Prove: △RSV " △TSV

Determine which postulate can be used to prove that the triangles are congruent.

If it is not possible to prove congruence, write not possible.

4. 5. 6.

7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths A'B' and AB are equal?

A'B'

A B

C

S

R

V

T

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1. STICKS Tyson had three sticks of

lengths 24 inches, 28 inches, and

30 inches. Is it possible to make two

noncongruent triangles using the same

three sticks? Explain.

2. BAKERY Sonia made a sheet of

baklava. She has markings on her pan

so that she can cut them into large

squares. After she cuts the pastry in

squares, she cuts them diagonally to

form two congruent triangles. Which

postulate could you use to prove the

two triangles congruent?

3. CAKE Carl had a piece of cake in the

shape of an isosceles triangle with

angles 26, 77, and 77. He wanted to

divide it into two equal parts, so he cut

it through the middle of the 26 angle to

the midpoint of the opposite side. He

says that because he is dividing it at the

midpoint of a side, the two pieces are

congruent. Is this enough information?

Explain.

4. TILES Tammy installs bathroom tiles.

Her current job requires tiles that are

equilateral triangles and all the tiles

have to be congruent to each other. She

has a big sack of tiles all in the shape

of equilateral triangles. Although she

knows that all the tiles are equilateral,

she is not sure they are all the same

size. What must she measure on each

tile to be sure they are congruent?

Explain.

5. INVESTIGATION An investigator at a

crime scene found a triangular piece of

torn fabric. The investigator

remembered that one of the suspects

had a triangular hole in their coat.

Perhaps it was a match. Unfortunately,

to avoid tampering with evidence, the

investigator did not want to touch the

fabric and could not fit it to the coat

directly.

a. If the investigator measures all three

side lengths of the fabric and the

hole, can the investigator make a

conclusion about whether or not the

hole could have been filled by the

fabric?

b. If the investigator measures two sides

of the fabric and the included angle

and then measures two sides of the

hole and the included angle can he

determine if it is a match? Explain.

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Enrichment4-4

Congruent Triangles in the Coordinate Plane

If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may be more than one way to do this.

1. Consider △ABD and △CDB whose vertices have coordinates A(0, 0), B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you know about congruent triangles and the coordinate plane to show that △ABD ! △CDB. You may wish to make a sketch to help get you started.

2. Consider △PQR and △KLM whose vertices are the following points.

P(1, 2) Q(3, 6) R(6, 5)

K(-2, 1) L(-6, 3) M(-5, 6)

Briefly describe how you can show that △PQR ! △KLM.

3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.

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Proving Triangles Congruent—ASA, AAS

ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two triangles are congruent.

Write a two column proof.

Given: −−

AB ‖ −−−

CD

∠CBD $ ∠ADB

Prove: △ABD $ △CDB

Statements Reasons

1. −−

AB ‖ −−−

CD

2. ∠CBD $ ∠ADB

3. ∠ABD $ ∠BDC

4. BD = BD

5. △ABD $ △CDB

1. Given

2. Given

3. Alternate Interior Angles Theorem

4. Reflexive Property of congruence

5. ASA

4-5

ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and

the included side of another triangle, then the triangles are congruent.

Example

Exercises


1. Write a two column proof. 2. Write a paragraph proof.

S

V

U

R T

A C

B

E

D

Given: ∠S $ ∠V, Given: −−−

CD bisects −−

AE , −−

AB ‖ −−−

CD

T is the midpoint of −−

SV ∠E $ ∠BCA

Prove: △RTS $ △UTV Prove: △ABC $ △CDE

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AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.

You now have five ways to show that two triangles are congruent.

• definition of triangle congruence • ASA Postulate

• SSS Postulate • AAS Theorem

• SAS Postulate

In the diagram, ∠BCA ! ∠DCA. Which sides

are congruent? Which additional pair of corresponding parts

needs to be congruent for the triangles to be congruent by

the AAS Theorem?

−−

AC " −−

AC by the Reflexive Property of congruence. The congruent angles cannot be ∠1 and ∠2, because

−− AC would be the included side.

If ∠B " ∠D, then △ABC " △ADC by the AAS Theorem.

Exercises


1. Write a two column proof.

Given: −−−

BC ‖ −−

EF

AB = ED

∠C " ∠F

Prove: △ABD " △DEF


Given: ∠S " ∠U; −−

TR bisects ∠STU.

Prove: ∠SRT " ∠URT

S

R T

U

4-5

AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding

two angles and side of a second triangle, then the two triangles are congruent.

Example

D

C1

2A

B

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PROOF Write a flow proof.

1. Given: ∠N ! ∠L

−−

JK ! −−−

MK

Prove: △JKN ! △MKL

2. Given: −−

AB ! −−−

CB

∠A ! ∠C

−−−

DB bisects ∠ABC.

Prove: −−−

AD ! −−−

CD


Given: −−−

DE ‖ −−−

FG

∠E ! ∠G

Prove: △DFG ! △FDE

FG

D E

A C

B

D

N

J

M

K L

4-5 Skills Practice


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Given: S is the midpoint of −−−QT .

−−−

QR ‖ −−−

TU

Prove: △QSR $ △TSU


Given: ∠D $ ∠F

−−−

GE bisects ∠DEF.

Prove: −−−

DG $ −−−

FG

ARCHITECTURE For Exercises 3 and 4, use the following information.

An architect used the window design in the diagram when remodeling

an art studio. −−

AB and −−−

CB each measure 3 feet.

3. Suppose D is the midpoint of −−

AC . Determine whether △ABD $ △CBD.

Justify your answer.

4. Suppose ∠A $ ∠C. Determine whether △ABD $ △CBD. Justify your answer.

D

B

A C

D

G

F

E

U

Q S

R

T

4-5 Practice


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1. DOOR STOPS Two door stops have

cross-sections that are right triangles.

They both have a 20 angle and the

length of the side between the 90 and

20 angles are equal. Are the cross-

sections congruent? Explain.

2. MAPPING Two people decide to take a

walk. One person is in Bombay and the

other is in Milwaukee. They start by

walking straight for 1 kilometer. Then

they both turn right at an angle of 110,

and continue to walk straight again.

After a while, they both turn right

again, but this time at an angle of 120.

They each walk straight for a while in

this new direction until they end up

where they started. Each person walked

in a triangular path at their location.

Are these two triangles congruent?

Explain.

3. CONSTRUCTION The rooftop of

Angelo’s house creates an equilateral

triangle with the attic floor. Angelo

wants to divide his attic into 2 equal

parts. He thinks he should divide it by

placing a wall from the center of the roof

to the floor at a 90 angle. If Angelo does

this, then each section will share a side

and have corresponding 90 angles. What

else must be explained to prove that the

two triangular sections are congruent?

90˚ 90˚

4. LOGIC When Carolyn finished her

musical triangle class, her teacher gave

each student in the class a certificate in

the shape of a golden triangle. Each

student received a different shaped

triangle. Carolyn lost her triangle on her

way home. Later she saw part of a

golden triangle under a grate.

Is enough of the triangle visible to allow

Carolyn to determine that the triangle is

indeed hers? Explain.

5. PARK MAINTENANCE Park officials

need a triangular tarp to cover a field

shaped like an equilateral triangle 200

feet on a side.

a. Suppose you know that a triangular

tarp has two 60 angles and one side

of length 200 feet. Will this tarp cover

the field? Explain.

b. Suppose you know that a triangular

tarp has three 60 angles. Will this

tarp necessarily cover the field?

Explain.

4-5 World Problem Pratice


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Enrichment4-5

What About AAA to Prove Triangle Congruence?You have learned about the SSS and ASA postulates to prove that two triangles are

congruent. Do you think that triangles can be proven congruent by AAA?

Investigate the AAS theorem. AAS represents the conjecture

that two triangles are congruent if they have two pairs of

corresponding angles congruent and corresponding

non-included sides congruent. For example, in the

triangles at the right, ∠A ! ∠Y, ∠B ! ∠X, and −−

AC ! −−

YZ .

1. Refer to the drawing above. What is the measure of the missing angle in the first

triangle? What is the measure of the missing angle in the second triangle?

2. With the information from Exercise 1, what postulate can you now use to prove that the

two triangles are congruent?

Now investigate the AAA conjecture. AAA represents the conjecture that two triangles are

congruent if they have three pairs of corresponding angles congruent. The triangles below

demonstrate AAA.

25˚

25˚

70˚

70˚85˚

85˚

3. Are the triangles above congruent? Explain.

4. Can you draw two triangles that have all three angles congruent that are not congruent?

Can AAA be used to prove that two triangles are congruent?

5. What characteristics do these two triangles drawn by AAA seem to possess?

89˚

89˚

66˚

66˚A

B C

5 in

5 in

X

Y

Z

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Exercises

ALGEBRA Find the value of each variable.

1. R

P

Q2x°

40°

2. S

V

T 3x - 6

2x + 6 3. W

Y Z3x°

4. 2x°

(6x + 6)°

5.

5x°

6.

R S

T

3x°

x°


Given: ∠1 ! ∠2

Prove: −−

AB ! −−−

CB


Isosceles and Equilateral Triangles

Properties of Isosceles Triangles An isosceles triangle has two congruent sides called the legs. The angle formed by the legs is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles.

• If two sides of a triangle are congruent, then the angles opposite

those sides are congruent. (Isosceles Triangle Theorem)

• If two angles of a triangle are congruent, then the sides opposite

those angles are congruent. (Converse of Isosceles Triangle

Theorem)

If −−

AB ! −−−

CB , then ∠A ! ∠C.

If ∠A ! ∠C, then −−

AB ! −−−

CB .

A

B

C

Find x, given −−

BC # −−

BA .

B

A

C (4x + 5)°

(5x - 10)°

BC = BA, so

m∠A = m∠C Isos. Triangle Theorem

5x - 10 = 4x + 5 Substitution

x - 10 = 5 subtract 4x from each side.

x = 15 Add 10 to each side.

Example 1 Example 2 Find x.

RT

S

3x - 13

2x

m∠S = m∠T, so

SR = TR Converse of Isos. △ Thm.

3x - 13 = 2x Substitution

3x = 2x + 13 Add 13 to each side.

x = 13 Subtract 2x from each side.

B

AC

D

E

1 32

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Properties of Equilateral Triangles An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem leads to two corollaries about equilateral triangles.

1. A triangle is equilateral if and only if it is equiangular.

2. Each angle of an equilateral triangle measures 60°.

Prove that if a line is parallel to one side of an

equilateral triangle, then it forms another equilateral triangle.

Proof:

Statements Reasons

1. △ABC is equilateral; −−−

PQ ‖ −−−

BC . 1. Given

2. m∠A = m∠B = m∠C = 60 2. Each ∠ of an equilateral △ measures 60°.

3. ∠1 & ∠B, ∠2 & ∠C 3. If ‖ lines, then corres. ' are &.

4. m∠1 = 60, m∠2 = 60 4. Substitution

5. △APQ is equilateral. 5. If a △ is equiangular, then it is equilateral.

Exercises


1. D

F E6x°

2. G

J H

6x - 5 5x

3.

3x°

4.

4x 40

60°

5. X

Z Y4x - 4

3x + 8 60°

6.

4x°

60°


Given: △ABC is equilateral; ∠1 & ∠2.

Prove: ∠ADB & ∠CDB

A

D

C

B1

2

A

B

P Q

C

1 2

4-6

Example

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Skills Practice


D

C

B

A E

Refer to the figure at the right.

1. If −−

AC " −−−

AD , name two congruent angles.

2. If −−−

BE " −−−

BC , name two congruent angles.

3. If ∠EBA " ∠EAB, name two congruent segments.

4. If ∠CED " ∠CDE, name two congruent segments.

Find each measure.

5. m∠ABC 6. m∠EDF

60°

40°


7.

2x + 4 3x - 10

8.

(2x + 3)°


Given: −−−

CD " −−−

CG

−−−

DE " −−−

GF

Prove: −−−

CE " −−

CF

D

E

F

G

C

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Practice


Lincoln H

awks

E

D B

C

A1

23

4

U

R

TV

S

Refer to the figure at the right.

1. If −−−

RV " −−

RT , name two congruent angles.

2. If −−

RS " −−

SV , name two congruent angles.

3. If ∠SRT " ∠STR, name two congruent segments.

4. If ∠STV " ∠SVT, name two congruent segments.

Find each measure.

5. m∠KML 6. m∠HMG 7. m∠GHM

8. If m∠HJM = 145, find m∠MHJ.

9. If m∠G = 67, find m∠GHM.


Given: −−−

DE ‖ −−−

BC

∠1 " ∠2

Prove: −−

AB " −−

AC

11. SPORTS A pennant for the sports teams at Lincoln High

School is in the shape of an isosceles triangle. If the measure

of the vertex angle is 18, find the measure of each base angle.

4-6

50°

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1

1. TRIANGLES At an art supply store, two

different triangular rulers are available.

One has angles 45°, 45°, and 90°. The

other has angles 30°, 60°, and 90°.

Which triangle is isosceles?

2. RULERS A foldable ruler has two hinges

that divide the ruler into thirds. If the

ends are folded up until they touch,

what kind of triangle results?

3. HEXAGONS Juanita placed one end of

each of 6 black sticks at a common point

and then spaced the other ends evenly

around that point. She connected the

free ends of the sticks with lines. The

result was a regular hexagon. This

construction shows that a regular

hexagon can be made from six congruent

triangles. Classify these triangles.

Explain.

4. PATHS A marble path is constructed

out of several congruent isosceles

triangles. The vertex angles are all 20.

What is the measure of angle 1 in the

figure?

5. BRIDGES Every day, cars drive through

isosceles triangles when they go over the

Leonard Zakim Bridge in Boston. The

ten-lane roadway forms the bases of the

triangles.

a. The angle labeled A in the picture

has a measure of 67. What is the

measure of ∠B?

b. What is the measure of ∠C?

c. Name the two congruent sides.

A

B

C

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Enrichment

Triangle ChallengesSome problems include diagrams. If you are not sure how to solve the problem, begin by using the given information. Find the measures of as many angles as you can, writing each measure on the diagram. This may give you more clues to the solution.

1. Given: BE = BF, ∠BFG ! ∠BEF ! ∠BED, m∠BFE = 82 and ABFG and BCDE each have opposite sides parallel and congruent.

Find m∠ABC.

A

G D

F E

CB

3. Given: m∠UZY = 90, m∠ZWX = 45, △YZU ! △VWX, UVXY is a square (all sides congruent, all angles right angles).

Find m∠WZY.

2. Given: AC = AD, and −−

AB ⊥ −−−

BD , m∠DAC = 44 and −−−

CE bisects ∠ACD.

Find m∠DEC.

A

D C

B

E

4. Given: m∠N = 120, −−−

JN ! −−−

MN , △JNM ! △KLM.

Find m∠JKM.

J

K

L

MN

UV

W

XYZ

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Identify Congruence Transformations A congruence transformation is a transformation where the original figure, or preimage, and the transformed figure, or image, figure are still congruent. The three types of congruence transformations are reflection (or flip), translation (or slide), and rotation (or turn).

Identify the type of congruence transformation shown as a

reflection, translation, or rotation.

x

y

Each vertex and its image are the

same distance from the y-axis.

This is a reflection.

x

y

Each vertex and its image are in

the same position, just two units

down. This is a translation.

x

y

Each vertex and its image are the

same distance from the origin.

The angles formed by each pair of

corresponding points and the

origin are congruent. This is a

rotation.

Exercises

Identify the type of congruence transformation shown as a reflection, translation,

or rotation.

1.

x

y 2.

x

y 3.

x

y

4.

x

y 5.

x

y 6.

x

y


Congruence Transformations

Example

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Verify Congruence You can verify that reflections, translations, and rotations of triangles produce congruent triangles using SSS.

Verify congruence after a transformation.

△WXY with vertices W(3, -7), X(6, -7), Y(6, -2) is a transformation of △RST with vertices R(2, 0), S(5, 0), and T(5, 5). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation.

Graph each figure. Use the Distance Formula to show the sides are congruent and the triangles are congruent by SSS.

RS = 3, ST = 5, TR = √ ####### (5 - 2) 2 + (5 - 0) 2 = √ ## 34

WX = 3, XY = 5, YW= √ ######### (6 - 3) 2 + (-2- (-7)) 2 = √ ## 34

−−

RS % −−−

WX , −−

ST , % −−

XY , −−

TR % −−−

YW

By SSS, △RST % △WXY.

Exercises

COORDINATE GEOMETRY Graph each pair of triangles with the given vertices.

Then identify the transformation, and verify that it is a congruence

transformation.

1. A(−3, 1), B(−1, 1), C(−1, 4) 2. Q(−3, 0), R(−2, 2), S(−1, 0)

D(3, 1), E(1, 1), F(1, 4) T(2, −4), U(3, −2), V(4, −4)

x

y

x

y

4-7 Study Guide and Intervention (continued)


Example

x

y

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4-7


or rotation.

1.

x

y 2.

x

y

COORDINATE GEOMETRY Identify each transformation and verify that it is a

congruence transformation.

3.

x

y 4.

x

y

COORDINATE GEOMETRY Graph each pair of triangles with the given vertices.

Then, identify the transformation, and verify that it is a congruence

transformation.

5. A(1, 3), B(1, 1), C(4, 1); 6. J(−3, 0), K(−2, 4), L(−1, 0);

D(3, -1), E(1, -1), F(1, -4) Q(2, −4), R(3, 0), S(4, −4)

x

y

x

y

4-7 Skills Practice


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or rotation.

1.

x

y 2.

x

y

3. Identify the type of congruence transformation shown as a reflection, translation, or rotation, and verify that it

is a congruence transformation.

4. ∆ABC has vertices A(−4, 2), B(−2, 0), C(−4, -2). ∆DEF has vertices D(4, 2), E(2, 0), F(4, −2). Graph the original figure and its image. Then identify the transformation and verify that it is a congruence transformation.

5. STENCILS Carly is planning on stenciling a pattern of flowers along the ceiling in her bedroom. She wants all of the flowers to look exactly the same. What type of congruence transformation should she use? Why?

4-7 Practice


x

y

x

y

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4-7 Word Problem Practice


1. QUILTING You and your two friends

visit a craft fair and notice a quilt, part

of whose pattern is shown below. Your

first friend says the pattern is repeated

by translation. Your second friend says

the pattern is repeated by rotation. Who

is correct?

2. RECYCLING The international symbol

for recycling is shown below. What type

of congruence transformation does this

illustrate?

3. ANATOMY Carl notices that when he

holds his hands palm up in front of him,

they look the same. What type of

congruence transformation does this

illustrate?

4. STAMPS A sheet of postage stamps

contains 20 stamps in 4 rows of 5

identical stamps each. What type of

congruence transformation does this

illustrate?

5. MOSAICS José cut rectangles out of

tissue paper to create a pattern. He cut

out four congruent blue rectangles and

then cut out four congruent red

rectangles that were slightly smaller to

create the pattern shown below.

a. Which congruence transformation

does this pattern illustrate?

b. If the whole square were rotated 90,

would the transformation still be the

same?

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4-7 Enrichment

Transformations in The Coordinate Plane

The following statement tells one way to relate points to corresponding translated points in the coordinate plane.

(x, y) → (x + 6, y - 3)

This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x + 6, y - 3).” With this transformation, for example, (3, 5) is mapped or moved to (3 + 6, 5 - 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.

1. Does the transformation above appear to be a congruence transformation? Explain your answer.

Draw the transformation image for each figure. Then tell whether the

transformation is or is not a congruence transformation.

2. (x, y) → (x - 4, y) 3. (x, y) → (x + 5, y + 4)

4. (x, y) → (-x , -y) 5. (x, y) → (- 1 −

2 x, y)

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C

X

Z

Y

O

(x, y) # (x + 6, y - 3)

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4-8



Triangles and Coordinate Proof

Position and Label Triangles A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines.

1. Use the origin as a vertex or center of the figure.

2. Place at least one side of the polygon on an axis.

3. Keep the figure in the first quadrant if possible.

4. Use coordinates that make computations as simple as possible.

Position an isosceles triangle on the

coordinate plane so that its sides are a units long and

one side is on the positive x-axis.

Start with R(0, 0). If RT is a, then another vertex is T(a, 0).

For vertex S, the x-coordinate is a −

2 . Use b for the y-coordinate,

so the vertex is S ( a −

2 , b) .

Exercises

Name the missing coordinates of each triangle.

1.

x

y

B(2p, 0)

C(?, q)

A(0, 0)

2.

x

y

S(2a, 0)

T(?, ?)

R(0, 0)

3.

x

y

G(2g, 0)

F(?, b)

E(?, ?)

Position and label each triangle on the coordinate plane.

4. isosceles triangle 5. isosceles right △DEF 6. equilateral triangle △RST with base

−− RS with legs e units long △EQI with vertex

4a units long Q(0, √ $ 3 b) and sides 2b units long

x

y

x

y

x

y

x

y

T(a, 0)R(0, 0)

S(a–2, b)Example

4-8

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Write Coordinate Proofs Coordinate proofs can be used to prove theorems and to verify properties. Many coordinate proofs use the Distance Formula, Slope Formula, or Midpoint Theorem.

Prove that a segment from the vertex

angle of an isosceles triangle to the midpoint of the base

is perpendicular to the base.

First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(-a, 0), and S(0, c). Then U(0, 0) is the midpoint of

−− RT .

Given: Isosceles △RST; U is the midpoint of base −−

RT .

Prove: −−−

SU ⊥ −−

RT

Proof:

U is the midpoint of −−

RT so the coordinates of U are ( -a + a

− 2 ,

0 + 0 −

2 ) = (0, 0). Thus

−−− SU lies on

the y-axis, and △RST was placed so −−

RT lies on the x-axis. The axes are perpendicular, so

−−−

SU ⊥ −−

RT .

Exercise

PROOF Write a coordinate proof for the statement.

Prove that the segments joining the midpoints of the sides of a right triangle form a right triangle.

x

y

T(a, 0)U(0, 0)R(–a, 0)

S(0, c)

4-8 Study Guide and Intervention (continued)


Example

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1. right △FGH with legs 2. isosceles △KLP with 3. isosceles △AND with

a units and b units long base −−

KP 6b units long base −−−

AD 5a units long

x

y

x

y

x

y


4.

x

y

B(2a, 0)

A(0, ?)

C(0, 0)

5.

x

y

Y(2b, 0)

Z(?, ?)

X(0, 0)

6.

x

y

N(3b, 0)

M(?, ?)

O(0, 0)

7.

x

y

Q(?, ?)

R(2a, b)

P(0, 0)

8.

x

y

P(7b, 0)

R(?, ?)

N(0, 0)

9.

x

y

U(a, 0)

T(?, ?)

S(–a, 0)

10. PROOF Write a coordinate proof to prove that in an isosceles right triangle, the segment

from the vertex of the right angle to the midpoint of the hypotenuse is perpendicular to

the hypotenuse.

Given: isosceles right △ABC with ∠ABC the right angle and M the midpoint of −−

AC

Prove: −−−

BM ⊥ −−

AC

4-8 Skills Practice


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1. equilateral △SWY with 2. isosceles △BLP with 3. isosceles right △DGJ

sides 1 − 4 a units long base

−− BL 3b units long with hypotenuse

−− DJ and

legs 2a units long

x

y

x

y

x

y


4.

x

yS(?, ?)

J(0, 0) R( b, 0)1

–3

5.

x

y

C(?, 0)

E(0, ?)

B(–3a, 0)

6.

x

y

P(2b, 0)

M(0, ?)

N(?, 0)

NEIGHBORHOODS For Exercises 7 and 8, use the following information.

Karina lives 6 miles east and 4 miles north of her high school. After school she works part

time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.

7. Proof Write a coordinate proof to prove that Karina’s high school, her home, and the

mall are at the vertices of a right triangle.

Given: △SKM

Prove: △SKM is a right triangle.

8. Find the distance between the mall and Karina’s home.

x

y

S(0, 0)

K(6, 4)

M(–2, 3)

4-8 Practice


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1. SHELVES Martha has a shelf bracket shaped like a right isosceles triangle. She wants to know the length of the hypotenuse relative to the sides. She does not have a ruler, but remembers the Distance Formula. She places the bracket on a coordinate grid with the right angle at the origin. The length of each leg is a. What are the coordinates of the vertices making the two acute angles?

2. FLAGS A flag is shaped like an isosceles triangle. A designer would like to make a drawing of the flag on a coordinate plane. She positions it so that the base of the triangle is on the y-axis with one endpoint located at (0, 0). She locates

the tip of the flag at (a, b−2

).

What are the coordinates of the third vertex?

3. BILLIARDS The figure shows a situation on a billiard table.

(–b, ?) y

x

O

(0, 0)

What are the coordinates of the cue ball before it is hit and the point where the cue ball hits the edge of the table?

4. TENTS The entrance to Matt’s tent makes an isosceles triangle. If placed on a coordinate grid with the base on the x–axis and the left corner at the origin, the right corner would be at (6, 0) and the vertex angle would be at (3, 4). Prove that it is an isosceles triangle.

5. DRAFTING An engineer is designing a roadway. Three roads intersect to form a triangle. The engineer marks two points of the triangle at (-5, 0) and (5, 0) on a coordinate plane.

a. Describe the set of points in the coordinate plane that could not be used as the third vertex of the triangle.

b. Describe the set of points in the coordinate plane that would make the vertex of an isosceles triangle together with the two congruent sides.

c. Describe the set of points in the coordinate plane that would make a right triangle with the other two points if the right angle is located at (-5, 0).



4-8

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Rectangle Paradox

Use grid paper to draw the two rectangles below. Cut them into pieces along the

dashed lines. Then rearrange the pieces to form a new rectangle.

1. Make a sketch of the new rectangle. Label the whole number lengths

2. What is the combined area of the two original rectangles?

3. What is the area of the new rectangle?

4. Make a conjecture as to why there is a discrepancy between the answers to

Exercises 2 and 3.

5. Use a straight edge to precisely draw the four shapes that

make up the larger rectangle. What does you drawing show?

6. How could you use slope to show that you drawing

is accurate?

3

5

3

35

5

4-8 Enrichment

Assessment

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SCORE 4 Student Recording Sheet

Use this recording sheet with pages 316–317 of the Student Edition.

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Multiple Choice

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing

each number or symbol in a box. Then fill in the corresponding circle for that

number or symbol.

8. ———————— (grid in)

9. ———————— (grid in)

10. ———————— (grid in)

11. ————————

12. ————————

13. ———————— (grid in)

14. ————————

Extended Response

Short Response/Gridded Response

Record your answers for Question 15 on the back of this paper.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

8.

10.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9.

13.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

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General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show how

he or she arrived at the answer, the student will be awarded only 1 credit. All extended-

response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all

parts of the question. For example, if a question has three parts, the correct response to

one or two parts of the question that required work to be shown is not considered a fully

correct response.

• Students who use trial and error to solve a problem must show their method. Merely

showing that the answer checks or is correct is not considered a complete response for

full credit.

Exercise 15 Rubric

Score Specific Criteria

4

The points for the verties of △ABC are plotted properly and the

distance formula is appropritely used to find AB = BC = √ #### 4a2+b2 .

Students draw the appropriate conclusion that △ABC is isosceles.

3A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.

4 Rubric for Scoring Extended Response

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SCORE

SCORE

1. Use a protractor to classify the triangle by its angles and sides.

2. MULTIPLE CHOICE What is the best classification of this triangle by its angles and sides?

A acute isosceles C right isosceles

B obtuse isosceles D obtuse equilateral

3. If △ABC is an isosceles triangle, ∠B is the vertex angle, AB = 6x + 3, BC = 8x - 1, and AC = 10x - 10, find the value of x and the length of each side of the triangle.

4. Find the measures of the sides of △ABC with vertices A(1, 5), B(3, -2), and C(-3, 0). Classify the triangle by sides.

Find the measure of each angle in

the figure.

5. m∠1 6. m∠2

7. m∠3 8. m∠4

9. m∠5 10. m∠6

1. Identify the congruent triangles in the figure. K

MLN

2. If △JGO " △RWI, which angle corresponds to ∠I?

3. In the diagram, △ABC " △DEF. Find the values of x and y.

4. In the figure −−−

QR " −−

SR and −−−

PQ " −−

TS . Point R is the midpoint of

−− PT . Determine

which theorem or postulate can be used to show that △QRP " △SRT.

5. What is the relationship between ∠Q and ∠S?

1.

2.

3.

4.

5.

4

4

Chapter 4 Quiz 1(Lessons 4-1 and 4-2)

70°65°

1

2 3

4

5

6 107°

43°


Q

P T

R

S

3x - y

(6y - 4)°

80°

70° 30°

19.7

18.810

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

SCORE

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SCORE

A

K

Z C

1.

2.

3.

4.

5.

For Questions 1 and 2, complete the two-column proof by supplying the missing information for each corresponding location.

Given: ∠Z ! ∠C −−−

AK bisects ∠ZKC.

Prove: △AKZ ! △AKC

Proof:

Statements Reasons

1. ∠Z ! ∠C −−−

AK bisects ∠ZKC. 1. Given

2. ∠ZKA ! ∠CKA 2. (Question 1)

3. −−−

AK ! −−−

AK 3. Reflexive Property

4. △AKZ ! △AKC 4. (Question 2)

For Questions 3 and 4 refer to the figure.

3. Find m∠1 4. Find m∠2

5. The base angle of an isosceles triangle is 30. What is the measure of the vertex angle?

1. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

Position and label the following triangle on a coordinate plane.

2. Right △DJL with hypotenuse −−

DJ ; LJ = 1 − 2 DL and

−−− DL is a

units long.

3. Two of the vertices of an equilateral triangle are (0, -a) and (0, a). The third vertex is on the x-axis to the right of the origin. Name the coordinates of the third vertex.

For Questions 4 and 5, complete the coordinate proof by supplying the missing information for each corresponding location.

Given: △ABC with A(-1, 1), B(5, 1), and C(2, 6).

Prove: △ABC is isosceles.

By the Distance Formula the lengths of the three sides are as follows: (Question 4). Since (Question 5), △ABC is isosceles.

1.

2.

3.

4.

5.

4

4



x

y

SCORE

21

Assessm

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SCORE

1. Use a ruler to measure each side and a protractorto measure each angle. Which best describes the type of triangle?

A acute scalene

B obtuse equilateral

C acute isosceles

D obtuse isosceles

Find the missing angle measures.

2. What is m∠1?

F 50 H 100

G 60 J 105

3. What is m∠2?

A 40 C 60

B 50 D 100

4. If △SJL " △DMT, which segment corresponds to −−

LS ?

F −−−

DT H −−−

MD

G −−−

TD J −−−

MT

50°

50°

60°

2

1

5.

6.

7.

8.

1.

2.

3.

4.

Part II

5. Find the measures of the sides of △ABC with vertices A(1, 3), B(5, -2), and C(0, -4). Classify the triangle by its sides.

6. Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, write not possible.

7. What is the maximum number of acute angles that a triangle can have?

8. Determine whether △DEF " △JKL, given that D(2, 0), E(5, 0), F(5, 5), J(3, -7), K(6, -7), and L(6, -2).

Part I Write the letter for the correct answer in the blank at the right of each question.

4 Chapter 4 Mid-Chapter Test(Lessons 4-1 through 4-4)

SCORE SCORE

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Write whether each sentence is true or false. If false,

replace the underlined word or number to make a true

sentence.

1. A triangle that is equilateral is also called a(n) acute triangle.

2. A(n) obtuse triangle has at least one obtuse angle.

Choose the correct term to complete the sentence.

3. If you slide, flip, or turn a triangle you are performing a (coordinate proof or congruence transformation).

4. An (interior or exterior) angle is formed by one side of a triangle and the extension of another side.

5. The SAS Postulate involves two corresponding sides and the (exterior angle or included angle) they form.

Choose from the terms above to complete each sentence.

6. A(n) organizes a series of statements in logical order, starting with the given statements.

7. A triangle that has one 90° angle is called a(n) .

8. The ASA postulate involves two corresponding angles and

their corresponding .

9. A(n) uses figures in the coordinate plane and algebra to prove geometric concepts.

10. The is formed by the congruent legs of an isosceles triangle.

Define each term in your own words.

11. corollary

12. congruent triangles

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

acute triangle

auxiliary line

base angles

congruence

transformation

congruent triangles

coordinate proof

corollary

corresponding parts

equiangular triangle

equilateral triangle

exterior angle

flow proof

included angle

included side

isosceles triangle

obtuse triangle

reflection

remote interior angles

right triangle

rotation

scalene triangle

translation

vertex angle

4

?

?

?

?

?

Chapter 4 Vocabulary Test SCORE

Assessment

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SCORE

80˚

A

E

CB

D

A C

B

10x - 5

6x + 37.5x

1.

2.

3.

4.

5.

6.

7.

8.

1. Which best describes the type of triangle?

A acute C obtuse

B equiangular D right

2. What is the value of x if △ABC is equilateral?

F -8 H 1 − 2

G - 1 −

8 J 2

Use the figure for Questions 3 and 4.

3. What is m∠2?

A 50 B 70 C 110 D 120

4. What is m∠4?

F 10 G 60 H 100 J 120

5. What are the congruent triangles in the diagram?

A △ABC $ △EBD

B △ABE $ △CBD

C △AEB $ △CBD

D △ABE $ △CDB

6. If △CJW $ △AGS, m∠A = 50, m∠J = 45, and m∠S = 16x + 5, what is the value of x?

F 17.5 H 6

G 11.875 J 5

7. Two triangles are graphed with the following vertices: A(1, 1), B(0, 1), C(0, 0)

and Q(−2, 1), R(−1, 1), S(−1, 0). When a transformation is applied to △ABC,

the result is △QRS. Which best describes the transformation?

A △QRS is a translation of △ABC

B △QRS is a rotation of △ABC

C △QRS is a reflection of △ABC

D △QRS and △ABC are not congruent

8. A triangular-shaped roof of a house has congruent legs. What is the measure of each of the two base angles?

F 25 H 100

G 50 J 120

70°

2 1

3 460°

40°

4 Chapter 4 Test, Form 1

Write the letter for the correct answer in the blank at the right of each question.

C W

J

(16x + 5)°

45°

A S

G

50°

SCORE

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NAME DATE PERIOD


N

K

LMJ

x

yM(a, c)

N(?, ?)L(0, 0)

1 2

M

TL N

Use the proof for Questions 9 and 10.

Given: L is the midpoint of −−−

JM

−−

JK || −−−

NM

Prove: △JKL # △MNL

Proof:

Statements Reasons

1. L is the midpoint of −−−

JM . 1. Given

2. −−

JL # −−−

ML 2. Definition of midpoint

3. −−

JK ‖ −−−

MN 3. Given

4. ∠JKL # ∠MNL 4. Alt. int. & are #.

5. ∠JLK # ∠MLN 5. (Question 9)

6. △JKL # △MNL 6. (Question 10)

9. What is the reason for ∠JLK # ∠MLN?

A definition of midpoint

B corresponding angles

C vertical angles

D alternate interior angles

10. What is the reason for △JKL # △MNL?

F AAS G ASA H SAS J SSS


11. If △LMN is isosceles and T is the midpoint of −−−

LN , which postulate can be used to prove △MLT # △MNT?

A AAA B AAS C SAS D ABC

12. If △MLT # △MNT, what is used to prove ∠1 # ∠2?

F CPCTC

G definition of isosceles triangle

H definition of perpendicular

J definition of angle bisector

13. What are the missing coordinates of this triangle?

A (2a, 2c) C (0, 2a)

B (2a, 0) D (a, 2c)

Bonus Classify the triangle with coordinates A(5, 0), B(0, 5), and C(-5, 0).

4 Chapter 4 Test, Form 1 (continued)

9.

10.

11.

12.

13.

B:

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SCORE


1. What is the length of the sides of this equilateral triangle?

A 42 C 15

B 30 D 12

2. How would △ABC with vertices A(4, 1), B(2, -1), and C(-2, -1) be classified based on the length of its sides?

F equilateral G isosceles H scalene J right


3. What is m∠1?

A 40 B 50 C 70 D 90

4. What is m∠3?

F 40 G 70 H 90 J 110

5. If △DJL " △EGS, which segment in △EGS corresponds to −−−

DL ?

A −−−

EG B −−

ES C −−−

GS D −−−

GE

6. Which triangles are congruent in the figure?

F △KLJ " △MNL

G △JLK " △NLM

H △JKL " △LMN

J △JKL " △MNL

7. Quadrilateral MNQP is made of two congruent triangles. −−−

NP bisects ∠N and ∠P. In the quadrilateral, m∠N = 50 and m∠P = 100. What is the measure of ∠M?

A 25 C 60

B 50 D 105

8. The coordinates of the vertices of △CDE are C(-3, 1), D(-1, 4), and E(-6, 4). A transformation applied to △CDE creates a congruent triangle △SQR. The new coordinates of two vertices are Q(-1, 6) and R(-6, 6). What are the coordinates of S?

F (-3, 3) G (1, 3) H (-1, 1) J (-1, 3)

M

N

Q

P

L

K

NJ

M

70°

50°1

2

3

6x - 3

9x - 123x + 6

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2A SCORE

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D A

NY

A C

B

E F

x

y

B(0, a)

A(?, ?)

x

y

(0, c)

(?, ?) (2b, 0)

9.

10.

11.

12.

13.

B:

9. If △ABC is isosceles with vertex angle ∠B, and

−−AE $

−−FC , which theorem or postulate can be

used to prove △AEB $ △CFB?

A SSS C ASA

B SAS D AAS


Given: −−−

DA || −−−

YN

−−−

DA $ −−−

YN

Prove: ∠NDY $ ∠DNA

Proof:

Statements Reasons

1. −−−

DA ‖ −−−

YN 1. Given

2. ∠ADN $ ∠YND 2. Alt. int. & are $.

3. −−−

DA $ −−−

YN 3. Given

4. −−−

DN $ −−−

DN 4. Reflexive Property

5. △NDY $ △DNA 5. (Question 10)

6. ∠NDY $ ∠DNA 6. (Question 11)

10. What is the reason for statement 5?

F ASA H SAS

G AAS J SSS


A Alt. int. & are $. C Corr. angles are $.

B CPCTC D Isosceles Triangle Theorem

12. What is the classification of a triangle with vertices A(3, 3), B(6, -2), C(0, -2) by the length of its sides?

F isosceles H equilateral

G scalene J right

13. What are the missing coordinates of the triangle?

A (-2b, 0) C (-c, 0)

B (0, 2b) D (0, -c)

Bonus Name the coordinates of points A and C in isosceles right △ABC if point C is in the second quadrant.

4 Chapter 4 Test, Form 2A (continued)

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SCORE


1. What are the lengths of the sides of this equilateral triangle?

A 2.5 C 15

B 5 D 20

2. What is the classification of △ABC with vertices A(0, 0), B(4, 3), and C(4, -3) by its sides?

F equilateral G isosceles H scalene J right


3. What is m∠1?

A 120 B 90 C 60 D 30

4. What is m∠2?

F 120 G 90 H 60 J 30

5. If △TGS " △KEL, which angle in △KEL corresponds to ∠T ?

A ∠L B ∠E C ∠K D ∠A

6. Which triangles are congruent in the figure?

F △HMN " △HGN

G △HMN " △NGH

H △NMH " △NGH

J △MNH " △HGN

7. The rhombus QRST is made of two congruent isosceles triangles. Given m∠QRS = 34, what is the measure of ∠S?

A 56 C 112

B 73 D 146

8. The vertices of △ABC are A(2, 0), B(5, 0), C(2, 6). The vertices of △DEF

are D(-6, -7), E(-3, -7), F(-6, -1) so that △ABC " △DEF. Which best describes the congruence transformation?

F flip G rotation H reflection J translation

H G

NM

120°1

2

7x - 15

3x + 54x

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2B

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G A

N

E I

A DF E

CB

(5x + 60)° (2x + 51)°

(30 - 10x)°(43 - 2x)°

x

y

(?, ?)

(a, 0)(-a, 0)

9.

10.

11.

12.

13.

B:

9. If −−AF !

−−−DE ,

−−AB !

−−FC and

−−AB ||

−−FC , which theorem

or postulate can be used to prove △ABE ! △FCD?

A AAS C SAS

B ASA D SSS


Given: −−−

EG ! −−

IA ∠EGA ! ∠IAG

Prove: ∠GEN ! ∠AIN

Proof:

Statements Reasons

1. −−−

EG ! −−

IA 1. Given

2. ∠EGA ! ∠IAG 2. Given

3. −−−

GA ! −−−

GA 3. Reflexive Property

4. △EGA ! △IAG 4. (Question 10)

5. ∠GEN ! ∠AIN 5. (Question 11)


F SSS G ASA H SAS J AAS


A Alt. int. % are !. C Corr. angles are !.

B Same side interior angles D CPCTC

12. What is the classification of a triangle with vertices A(-3, -1), B(-2, 2), C(3, 1) by its sides?

F scalene H equilateral

G isosceles J right

13. What are the missing coordinates of the triangle?

A (a, 0) C (c, 0)

B (b, 0) D (∅, a −

2 )

Bonus Find the value of x.

4 Chapter 4 Test, Form 2B (continued)

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SCORE

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, and AC if △ABC is equilateral.

3. Find the measure of the sides of the triangle if the vertices of △EFG are E(-3, 3), F(1, -1), and G(-3, -5). Then classify the triangle by its sides.

Use the figure for Question 4-6. Find the measure of

each angle.

4. m∠1

5. m∠2

6. m∠3

7. Identify the congruent triangles and name their corresponding congruent angles.

8. Identify the transformation and verify that it is a congruence transformation.

x

y

A

D F

G

B

C

110° 2

1

3

A C

B

8x

7x + 310x - 6

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2C

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9. A rectangular piece of paper measuring 8 inches by 16 inches

is folded in half to form a square. It is then folded again along

its diagonal to form a triangle. Find the measures of the

angles of the triangle.

10. △KLM is an isosceles triangle

and ∠1 " ∠2. Name the theorem

that could be used to determine

∠LKP " ∠LMN. Then name the

postulate that could be used to

prove △LKP " △LMN. Choose

from SSS, SAS, ASA, and AAS.

11. Find m∠1.


13. Position and label isosceles △ABC with base −−

AB , b units long,

on the coordinate plane.

14. −−

CP joins point C in isosceles right

△ABC to the midpoint P, of −−

AB .

Name the coordinates of P. Then

determine the relationship

between −−

AB and −−

CP .

Bonus Without finding any other angles or sides congruent,

determine which pair of triangles can be proved to be

congruent by the HL Theorem.

A

B

C D

E

F X

Y

Z M

N

O

x

y

A(0, b)

B(b, 0)C(0, 0)

P

(10x + 20)°15x°

(18x - 12)°

40°

1

P

1 2

NMK

L

4 Chapter 4 Test, Form 2C (continued)

9.

10.

11.

12.

13.

14.

B:

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SCORE

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, AC if △ABC is isosceles with AB = BC.

3. Find the measure of the sides of the triangle if the vertices of △EFG are E(1, 4), F(5, 1), and G(2, -3). Then classify the triangle by its sides.

Use the figure for Questions 4-6. Find the measure of

each angle.

4. m∠1

5. m∠2

6. m∠3

7. Identify the congruent triangles and name their corresponding congruent angles.

8. Identify the transformation and verify that it is a congruence transformation.

9. A square piece of paper measuring 8 inches on each side is folded in half forming a triangle. It is then folded in half again to form another triangle. Find the angles of the resulting triangle.

DB

EC

FA

70°1 32

80°

A C

B

2x + 20

9x - 5

5x + 5

4 Chapter 4 Test, Form 2D

x

y

1.

2.

3.

4.

5.

6.

7.

8.

9.

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10. △ABC is an isosceles triangle with −−−

BD ⊥ −−

AC . Name the theorem that

could be used to determine

∠A % ∠C. Then name the

postulate that could be used to

prove △BDA % △BDC. Choose

from SSS, SAS, ASA, and AAS.

11. Find m∠1.


13. Position and label equilateral △KLM with side lengths

3a units long on the coordinate plane.

14. −−−

MN joins the midpoint of −−

AB and the

midpoint of −−

AC in △ABC. Find the

coordinates of M and N.

Bonus Without finding any other angles or sides congruent,

determine which pair of triangles can be proved to be

congruent by the LL Theorem.

A

B

C D

E

F X

Y

Z M

N

O

x

y

B(a, 0)

C(0, b)

M(?, ?)

N(?, ?)

A(0, 0)

(6x + 4)°

(18x - 8)°

80°1

B D

C

A

10.

11.

12.

13.

14.

B:

4 Chapter 4 Test, Form 2D (continued)

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SCORE

1. If △ABC is isosceles, ∠B is the vertex angle, AB = 20x - 2, BC = 12x + 30, and AC = 25x, find x and the length of each side of the triangle.

2. Find the length of the sides of the triangle with vertices A(0, 4), B(5, 4), and C(-3, -2). Classify the triangle by its sides and angles.

Use the figure to answer Questions 3–5.


4. Find m∠1, if m∠1 = 4x + 10.

5. Find m∠2.

6. The vertices of △ABC are A(-1, 1), B(4, 2), and C(1, 5). The vertices of △DEF are D(-1, -1), E(4, -2), and F(1, -5) so that △ABC " △DEF. Identify the congruence transformation.

7. Determine whether △GHI " △JKL, given G(1, 2), H(5, 4), I(3, 6) and J(-4, -5), K(0, -3), L(-2, -1). Explain.

8. In the figure, −−

AC " −−−

FD , −−

AB || −−−

DE ,

and −−

AC || −−−

FD . Determine which postulate can be used to prove △ABC " △DEF. Choose from SSS, SAS, ASA, and AAS.

D

E

A

B

F

C

(8x - 30)°(3x - 10)°2

1

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 3

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For Questions 9 and 10, complete this two-column proof.

Given: △ABC is an isosceles triangle with base −−

AC . D is the midpoint of

−− AC .

Prove: −−−

BD bisects ∠ABC

Proof:

Statements Reasons

1. △ABC is isosceles with base

−− AC .

1. Given

2. −−

AB $ −−−

CB 2. Def. of isosceles triangle.

3. ∠A $ ∠C 3. (Question 9)

4. D is the midpoint of −−

AC . 4. Given

5. −−−

AD $ −−−

CD 5. Midpoint Theorem

6. △ABD $ △CBD 6. (Question 10)

7. ∠1 $ ∠2 7. CPCTC

8. −−−

BD bisects ∠ABC 8. Def. of angle bisector

11. Find x.

12. Position and label isosceles △ABC with base −−

AB ,

(a + b) units long, on a coordinate plane

13. A wood column is supported by four cablesthat form four triangles. Determine which postulate can be used to show that all the triangles are congruent.

Bonus In the figure, △ABC is isosceles, △ADC is equilateral, △AEC is isosceles, and the measures of ∠9, ∠1, and ∠3 are all equal. Find the measures of the nine numbered angles.

987

1

65

23

4

B

C

DE

A

(15x + 15)°(21x - 3)°

(17x + 9)°

21

A C

B

D

9.

10.

11.

12.

13.

B:

4 Chapter 4 Test, Form 3 (continued)

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SCORE

Demonstrate your knowledge by giving a clear, concise solution to

each problem. Be sure to include all relevant drawings and justify

your answers. You may show your solution in more than one way

or investigate beyond the requirements of the problem.

1.

a. Classify the triangle by its angles and sides.

b. Show the steps needed to solve for x.

2. a. Describe how to determine whether a triangle with coordinates A(1, 4), B(1, -1), and C(-4, 4) is an equilateral triangle.

b. Is the triangle equilateral? Explain.

3. Explain how to find m∠1 and m∠2 in the figure.

4.

a. Determine which theorem or postulate can be used to prove that the triangles are congruent.

b. List their corresponding congruent angles and sides.

5. Write a two-column proof.

Given: −−

AB || −−−

DE , −−−

AD bisects −−−

BE . Prove: △ABC $ △DEC using the ASA postulate.

A

C

B

E D

J

DL

E

G

S

C

BD E

A 58°

40° 62°

1 2

(9x + 4)°

(20x - 10)°

4 Chapter 4 Extended-Response Test

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4 Standardized Test Practice(Chapters 1– 4)

A CB

D

?

U V

XW

1. If m∠1 = 5x - 4 and m∠2 = 52 - 9y, which values for x and y would make ∠1 and ∠2 complementary? (Lesson 1-5)

A x = 2, y = 12 C x = 12, y = 2

B x = 27, y = 1 − 3 D x = 1 −

3 , y = 27

2. Which is not a polygon? (Lesson 1-6)

F G H J

3. Complete the statement so that its conditional and its converse are true.

If ∠1 # ∠2, then ∠1 and ∠2 . (Lesson 2-3)

A are supplementary. C are complementary.

B have the same measure. D are consecutive interior angles.

4. Complete this proof. (Lesson 2-7)

Given: −−−

UV # −−−

VW −−−

VW # −−−

WX Prove: UV = WX

Proof:

Statements Reasons

1. −−−

UV # −−−

VW ; −−−

VW # −−−

WX 1. Given

2. UV = VW; VW = WX 2.

3. UV = WX 3. Transitive Property

F Definition of congruent segments

G Substitution Property

H Segment Addition Postulate

J Symmetric Property

5. Which equation has a slope of 1 − 3 and a y-intercept of -2? (Lesson 3-4)

A y = 1 − 3 x + 2 C y = 1 −

3 x - 2

B y = 2x - 1 − 3 D y = -2x + 1 −

3

6. Classify △DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1)

F acute G equiangular H obtuse J right

7. Which postulate or theorem can be used to prove △ABD # △CBD? (Lesson 4-4)

A SAS C SSS

B ASA D AAS

?

SCORE

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

A B C D

A B C D

Assessment

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SCORE 4 Standardized Test Practice (continued)

8. If the volume of a cylinder with a height of 3 feet is 75π cubic feet, find the surface area of the cylinder in square feet. (Lesson 1-7)

F 25π G 50π H 80π J 30π

9. Let C be the midpoint of the line segment AB having endpoints A(2, -7) and B(6, -3). Find the length of

−−− CB . (Lesson 1-3)

A 2 √ $ 2 units B √ $$ 53 units C √ $$ 20 units D √ $$ 10 units

10. Find x so that p || q. (Lesson 3-5)

F 11.50 H 20.5

G 20 J 25

11. If ∠1 is supplementary to ∠2 and ∠3 is complementary to ∠2, find m∠3 if m∠1 is 145. (Lesson 2-8)

A 35 C 55

B 45 D 90

12. Let △ABC be an isosceles triangle with △ABC ' △PQR. If m∠B = 154, find m∠R. (Lesson 4-3)

F 154 G 126 H 26 J 13

13. In proving the congruence of two triangles, which postulate involves an included angle? (Lessons 4-4 and 4-5)

A SSS B SAS C ASA D AAS

14. △PQR is an isosceles triangle with base −−−

QR . If m∠P = 6x + 40 and m∠Q = x - 10, find x. (Lesson 4-6)

F 20 G 25 H 30 J 100

qp

(2x -

55)°

(8x + 30)°

8.

9.

10.

11.

12.

13.

14. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

15. If −−

JK || −−− LM , then ∠4 must

be supplementary to

∠ . (Lesson 3-5)

16. Find PR if △PQR is isosceles, ∠Q is the vertex angle, PQ = 4x - 8, QR = x + 7, and PR = 6x - 12. (Lesson 4-1)

?

6

7

4

5H

L

M

J

K

15.

16. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the

answer in a column box and then shading in the appropriate circle

that corresponds to that entry.

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17. The perimeter of a regular pentagon is 14.5 feet. Find the new

perimeter if the length of each side of the pentagon is doubled.

(Lesson 1-6)

18. Make a conjecture about the next number in the sequence

5, 7, 11, 17, 25. . .. (Lesson 2-1)

19. Find m∠PQR. (Lesson 4-2)

20. If PQ = QS, QS = SR, and

m∠R = 20, find m∠PSQ.

(Lesson 4-6)

21. Name the corresponding congruent angles and sides for

△PQR " △HGB. (Lesson 4-3)

22. Refer to the figure at the right to

answer the questions below.

a. Name the segment that represents

the distance from F to # %& AD . (Lesson 3-6)

b. Classify △ADC. (Lesson 4-1)

c. Find m∠ACD. (Lesson 4-2)

50°30°

85°ED

A CB

F

P RS

Q

63° 125° 10°P S

Q

R

T

17.

18.

19.

20.

21.

22a.

22b.

22c.

4 Standardized Test Practice (continued)

Part 3: Short Response

Instructions: Write your answers in the space provided.

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 4-1

Ch

ap

ter

4

5

Gle

nc

oe

Ge

om

etr

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y T

ria

ng

les

by

An

gle

s O

ne w

ay t

o c

lass

ify a

tri

an

gle

is

by t

he m

easu

res

of

its

an

gle

s.

• If

all

thre

e o

f th

e a

ng

les o

f a

tria

ng

le a

re a

cu

te a

ng

les,

the

n t

he

tria

ng

le is a

n a

cu

te t

ria

ng

le.

• If

all

thre

e a

ng

les o

f a

n a

cu

te t

ria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

n e

qu

ian

gu

lar

tria

ng

le.

• If

on

e o

f th

e a

ng

les o

f a

tria

ng

le is a

n o

btu

se

an

gle

, th

en

th

e t

ria

ng

le is a

n o

btu

se

tri

an

gle

.

• If

on

e o

f th

e a

ng

les o

f a

tria

ng

le is a

rig

ht

an

gle

, th

en

th

e t

ria

ng

le is a

rig

ht

tria

ng

le.

C

lassif

y e

ach

tria

ng

le.

a

.

60°A

BC

All

th

ree a

ngle

s are

con

gru

en

t, s

o a

ll t

hre

e a

ngle

s h

ave m

easu

re 6

0°.

Th

e t

rian

gle

is

an

equ

ian

gu

lar

tria

ngle

.

b.

25°

35°

120°

DF

E

Th

e t

rian

gle

has

on

e a

ngle

th

at

is o

btu

se.

It i

s an

obtu

se t

rian

gle

.

c.

90°

60°

30°

G

HJ

Th

e t

rian

gle

has

on

e r

igh

t an

gle

. It

is

a r

igh

t tr

ian

gle

.

Exerc

ises

Cla

ssif

y e

ach

tria

ng

le a

s a

cu

te,

eq

uia

ng

ula

r,

ob

tuse,

or r

igh

t.

1

. 67°

90°

23°

K LM

2.

120°

30°

30°

NO

P

3

.

60°

60°

60°

Q

RS

r

igh

t o

btu

se

eq

uia

ng

ula

r

4

.

65°

65°

50°

UV

T

5.

45°

45°

90°

XY

W

6.

60°

28°

92°

FDB

a

cu

te

rig

ht

ob

tuse

4-1 Exam

ple

Chapter Resources


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

3

Gle

nc

oe

Ge

om

etr

y

An

tici

pati

on

Gu

ide

Co

ng

ruen

t Tri

an

gle

s

B

efo

re y

ou

beg

in C

ha

pte

r 4

•

Read

each

sta

tem

en

t.

•

Deci

de w

heth

er

you

Agre

e (

A)

or

Dis

agre

e (

D)

wit

h t

he s

tate

men

t.

•

Wri

te A

or

D i

n t

he f

irst

colu

mn

OR

if

you

are

not

sure

wh

eth

er

you

agre

e o

r d

isagre

e,

wri

te N

S (

Not

Su

re).

A

fter y

ou

co

mp

lete

Ch

ap

ter 4

•

Rere

ad

each

sta

tem

en

t an

d c

om

ple

te t

he l

ast

colu

mn

by e

nte

rin

g a

n A

or

a D

.

•

Did

an

y o

f you

r op

inio

ns

abou

t th

e s

tate

men

ts c

han

ge f

rom

th

e f

irst

colu

mn

?

•

For

those

sta

tem

en

ts t

hat

you

mark

wit

h a

D,

use

a p

iece

of

pap

er

to w

rite

an

exam

ple

of

wh

y y

ou

dis

agre

e.

4 Ste

p 1

ST

EP

1A

, D

, o

r N

SS

tate

men

tS

TE

P 2

A o

r D

1

. F

or

a t

rian

gle

to b

e a

cute

, all

th

ree a

ngle

s m

ust

be

acu

te.

A

2

. A

ll s

ides

of

an

equ

ilate

ral

tria

ngle

are

con

gru

en

t.A

3

. A

n e

xte

rior

an

gle

is

an

y a

ngle

ou

tsid

e o

f th

e g

iven

tr

ian

gle

.D

4

. T

he s

um

of

the m

easu

res

of

the a

ngle

s in

a t

rian

gle

is

360°.

D

5

. A

n o

btu

se t

rian

gle

is

a t

rian

gle

wit

h t

hre

e o

btu

se

an

gle

s.D

6

. T

rian

gle

s th

at

are

th

e s

am

e s

hap

e a

nd

siz

e a

re

con

gru

en

t.A

7

. If

th

e a

ngle

s of

on

e t

rian

gle

are

con

gru

en

t to

th

e

an

gle

s of

an

oth

er

tria

ngle

, th

en

th

e t

wo t

rian

gle

s are

co

ngru

en

t.D

8

. A

n i

sosc

ele

s tr

ian

gle

is

a t

rian

gle

wit

h t

wo c

on

gru

en

t si

des.

A

9

. A

tri

an

gle

can

be e

qu

ian

gu

lar

an

d n

ot

equ

ilate

ral.

D

10

. T

o m

ak

e t

he c

alc

ula

tion

s in

a c

oord

inate

pro

of

easi

er,

p

lace

eit

her

a v

ert

ex o

r th

e c

en

ter

of

the p

oly

gon

at

the o

rigin

.A

Ste

p 2

Answers (Anticipation Guide and Lesson 4-1)

Chapter 4 A1 Glencoe Geometry

Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

Lesson 4-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

7

Gle

nc

oe

Ge

om

etr

y

Sk

ills

Pra

ctic

e

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y e

ach

tria

ng

le a

s a

cu

te,

eq

uia

ng

ula

r,

ob

tuse,

or r

igh

t.

1

.

60°

60°

60°

2

. 40

° 95°

45°

3.

50°

40°

90°

e

qu

ian

gu

lar

ob

tuse

rig

ht

4

. 80

°50

°

50°

5

.

100°

30°

50°

6.

70°

55°

55°

a

cu

te

ob

tuse

acu

te

Cla

ssif

y e

ach

tria

ng

le a

s e

qu

ila

tera

l, i

so

scele

s,

or s

ca

len

e.

EC

D

AB

9

88√2

7

. △

AB

E

8.

△E

DB

s

cale

ne

iso

scele

s

9

. △

EB

C

10

. △

DB

C

s

cale

ne

eq

uala

tera

l

11

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

AB

C i

s an

iso

scele

s

tria

ngle

wit

h −−

A

B #

−−

BC

.

12

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

FG

H i

s an

equ

ilate

ral

tria

ngle

.

5x-8

3x+10

4x+1

13

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

RS

T i

s an

iso

scele

s

tria

ngle

wit

h −

−

RS

# −

−

TS .

3x-1

9x+2

6x+8

14

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

DE

F i

s an

equ

ilate

ral

tria

ngle

.

5x-21

2x+6

7x-39

4-1 x

= 4

; A

B =

BC

= 1

3;

AC

= 6

x =

9;

FG

= G

H =

FH

= 3

7

x =

2;

RS =

ST =

20;

RT

= 5

x =

9;

DE =

EF =

DF =

24

4x-3

2x+5

x+2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

6

Gle

nc

oe

Ge

om

etr

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y T

ria

ng

les

by

Sid

es

You

can

cla

ssif

y a

tri

an

gle

by t

he n

um

ber

of

con

gru

en

t si

des.

Equ

al

nu

mbers

of

hash

mark

s in

dic

ate

con

gru

en

t si

des.

• If

all

thre

e s

ide

s o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

n e

qu

ila

tera

l tr

ian

gle

.

• If

at

lea

st

two s

ide

s o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

n i

so

sc

ele

s t

ria

ng

le.

Eq

uila

tera

l tr

ian

gle

s c

an

als

o b

e c

on

sid

ere

d iso

sce

les.

• If

no

tw

o s

ide

s o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

sc

ale

ne

tri

an

gle

.

C

lassif

y e

ach

tria

ng

le.

a

.

LJ

H

b.

N

RP

c.

2312

15X

V

T

T

wo s

ides

are

con

gru

en

t.

A

ll t

hre

e s

ides

are

T

he t

rian

gle

has

no p

air

Th

e t

rian

gle

is

an

co

ngru

en

t. T

he t

rian

gle

of

con

gru

en

t si

des.

It

is

isosc

ele

s tr

ian

gle

. is

an

equ

ilate

ral

tria

ngle

. a s

cale

ne t

rian

gle

.

Exerc

ises

Cla

ssif

y e

ach

tria

ng

le a

s e

qu

ila

tera

l, i

so

scele

s,

or s

ca

len

e.

1

. 2

1

3√

G

CA

2.

G

KI

18

1818

3

.

1217

19Q

O

M

s

cale

ne

eq

uilate

ral

scale

ne

4

.

UW

S

5.

8x

32x

32x

B C

A

6.

DE

Fx

xx

i

so

scele

s

iso

scele

s

eq

uilate

ral

Exam

ple

4-1

7

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

RS

T i

s an

equ

ilate

ral

tria

ngle

.

8

. A

LG

EB

RA

F

ind

x a

nd

th

e l

en

gth

of

each

sid

e i

f △

AB

C i

s is

osc

ele

s w

ith

AB

= B

C.

5x-4

2x+2

3x

3y+2

4y

3y

2;

62;

AB

= 8

; B

C =

8,

AC

= 6

Answers (Lesson 4-1)


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

8

Gle

nc

oe

Ge

om

etr

y

Practi

ce

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y e

ach

tria

ng

le a

s acute

, equiangular, obtuse,

or right.

1

.

2.

3.

Cla

ssif

y e

ach

tria

ng

le i

n t

he

fig

ure

at

the

rig

ht

by

its

an

gle

s a

nd

sid

es.

4

. △

AB

D

5.

△A

BC

eq

uia

ng

ula

r; e

qu

ilate

ral

rig

ht;

scale

ne

6

. △

ED

C

7.

△B

DC

ri

gh

t; s

cale

ne

ob

tuse;

iso

scele

s

ALG

EB

RA

F

or e

ach

tria

ng

le,

fin

d x

an

d t

he

me

asu

re

of

ea

ch

sid

e.

8

. △

FG

H i

s an

equ

ilate

ral

tria

ngle

wit

h F

G =

x +

5,

GH

= 3

x -

9,

an

d F

H =

2x -

2.

9.

△L

MN

is

an

iso

scele

s tr

ian

gle

, w

ith

LM

= L

N,

LM

= 3

x -

2,

LN

= 2

x +

1,

an

d

MN

= 5

x -

2.

Fin

d t

he

me

asu

re

s o

f th

e s

ide

s o

f △KPL

an

d c

lassif

y e

ach

tria

ng

le b

y i

ts s

ide

s.

10

. K

(-3,

2),

P(2

, 1),

L(-

2,

-3)

11

. K

(5,

-3),

P(3

, 4),

L(-

1,

1)

12

. K

(-2,

-6),

P(-

4,

0),

L(3

, -

1)

13

. D

ES

IGN

D

ian

a e

nte

red

th

e d

esi

gn

at

the r

igh

t in

a l

ogo c

on

test

sp

on

sore

d b

y a

wil

dli

fe e

nvir

on

men

tal

gro

up

. U

se a

pro

tract

or.

H

ow

man

y r

igh

t an

gle

s are

th

ere

?

AC

DE

B

60°

30°

90°

65°

30°

85°

40°

100°

40°

4-1 x

= 7

; FG

= 1

2,

GH

= 1

2,

FH

= 1

2

x =

3;

LM

= 7

, LN

= 7

, M

N =

13

KP

= √

##

26 ,

PL

= 4

√ #

2 ,

LK

= √

##

26 ;

iso

scele

s

KP

= √

##

53 ,

PL

= 5

, LK

= 2

√ #

#

13 ; s

cale

ne

KP

= 2

√ #

#

10 ,

PL

= 5

√ #

2 ,

LK

= 5

√ #

2 ;

iso

scele

s

5

o

btu

se

acu

te

rig

ht

Lesson 4-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

9

Gle

nc

oe

Ge

om

etr

y

Wo

rd

Pro

ble

m P

racti

ce

Cla

ssif

yin

g T

rian

gle

s

1

. M

US

EU

MS

Pau

l is

sta

nd

ing i

n f

ron

t of

a m

use

um

exh

ibit

ion

. W

hen

he t

urn

s h

is

head

60

° to

th

e l

eft

, h

e c

an

see a

sta

tue

by D

on

ate

llo.

Wh

en

he t

urn

s h

is h

ead

60

° to

th

e r

igh

t, h

e c

an

see a

sta

tue b

y

Dell

a R

obbia

. T

he t

wo s

tatu

es

an

d P

au

l fo

rm t

he v

ert

ices

of

a t

rian

gle

. C

lass

ify

this

tri

an

gle

as

acu

te,

righ

t, o

r obtu

se.

2

. P

AP

ER

M

ars

ha c

uts

a r

ect

an

gu

lar

pie

ce

of

pap

er

in h

alf

alo

ng a

dia

gon

al.

Th

e

resu

lt i

s tw

o t

rian

gle

s. C

lass

ify t

hese

tr

ian

gle

s as

acu

te,

righ

t, o

r obtu

se.

3. W

AT

ER

SK

IIN

G K

im a

nd

Cass

an

dra

are

wate

rsk

iin

g.

Th

ey a

re h

old

ing o

n t

o

rop

es

that

are

th

e s

am

e l

en

gth

an

d t

ied

to

th

e s

am

e p

oin

t on

th

e b

ack

of

a s

peed

boat.

Th

e b

oat

is g

oin

g f

ull

sp

eed

ah

ead

an

d t

he r

op

es

are

fu

lly t

au

t.

K

im,

Cass

an

dra

, an

d t

he p

oin

t w

here

th

e r

op

es

are

tie

d o

n t

he b

oat

form

th

e

vert

ices

of

a t

rian

gle

. T

he d

ista

nce

betw

een

Kim

an

d C

ass

an

dra

is

never

equ

al

to t

he l

en

gth

of

the r

op

es.

Cla

ssif

y

the t

rian

gle

as

equ

ilate

ral,

iso

scele

s, o

r sc

ale

ne.

4

. B

OO

KE

ND

S

Tw

o b

ook

en

ds

are

sh

ap

ed

li

ke r

igh

t tr

ian

gle

s.

Th

e b

ott

om

sid

e o

f each

tri

an

gle

is

exact

ly h

alf

as

lon

g a

s th

e s

lan

ted

sid

e

of

the t

rian

gle

. If

all

th

e b

ook

s betw

een

th

e b

ook

en

ds

are

rem

oved

an

d t

hey a

re

pu

shed

togeth

er,

th

ey w

ill

form

a s

ingle

tr

ian

gle

. C

lass

ify t

he t

rian

gle

th

at

can

be f

orm

ed

as

equ

ilate

ral,

iso

scele

s, o

r sc

ale

ne.

5

. D

ES

IGN

S

Su

zan

ne s

aw

th

is p

att

ern

on

a p

en

tagon

al

floor

tile

. S

he n

oti

ced

man

y

dif

fere

nt

kin

ds

of

tria

ngle

s w

ere

cre

ate

d

by t

he l

ines

on

th

e t

ile.

a.

Iden

tify

fiv

e t

rian

gle

s th

at

ap

pear

to

be a

cute

iso

scele

s tr

ian

gle

s.

b.

Iden

tify

fiv

e t

rian

gle

s th

at

ap

pear

to

be o

btu

se i

sosc

ele

s tr

ian

gle

s.

4-1

Kim

Cassandra

CA

GH

IF

JB

ED

Th

ey a

re b

oth

rig

ht

tria

ng

les.

ob

tuse

eq

uilate

ral

Sam

ple

an

sw

ers

: △

EB

D,

△C

AE

, △

AD

C,

△B

GH

, △

CH

I, △

DIJ

, △

EJF,

△A

GF

Sam

ple

an

sw

ers

: △

AFE

, △

EJD

, △

CID

, △

BH

C,

△B

GA

, △

AH

D,

△C

GE

, △

AC

J

iso

scele

s



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

10

G

len

co

e G

eo

me

try

En

rich

men

t

Read

ing

Math

em

ati

cs

Wh

en

you

read

geom

etr

y,

you

may n

eed

to d

raw

a d

iagra

m t

o m

ak

e t

he t

ext

easi

er

to u

nd

ers

tan

d.

C

on

sid

er t

hre

e p

oin

ts, A

, B

, a

nd

C o

n a

co

ord

ina

te g

rid

. T

he

y-c

oo

rd

ina

tes o

f A

an

d B

are

th

e s

am

e.

Th

e x

-co

ord

ina

te o

f B

is g

re

ate

r t

ha

n t

he

x-c

oo

rd

ina

te o

f A

. B

oth

co

ord

ina

tes o

f C

are

gre

ate

r t

ha

n t

he

co

rre

sp

on

din

g

co

ord

ina

tes o

f B

. Is

△ABC

acu

te,

rig

ht,

or o

btu

se

?

To a

nsw

er

this

qu

est

ion

, fi

rst

dra

w a

sam

ple

tri

an

gle

that

fits

th

e d

esc

rip

tion

.

Sid

e −

−

AB

mu

st b

e a

hori

zon

tal

segm

en

t beca

use

th

e

y-c

oord

inate

s are

th

e s

am

e.

Poin

t C

mu

st b

e l

oca

ted

to t

he r

igh

t an

d u

p f

rom

poin

t B

.

Fro

m t

he d

iagra

m y

ou

can

see t

hat

tria

ngle

AB

C

mu

st b

e o

btu

se.

Exerc

ises

Dra

w a

sim

ple

tria

ng

le o

n g

rid

pa

pe

r t

o h

elp

yo

u.

1

. C

on

sid

er

thre

e p

oin

ts,

R,

S,

an

d

2. C

on

sid

er

thre

e n

on

coll

inear

poin

ts,

T o

n a

coord

inate

gri

d.

Th

e

J

, K

, an

d L

on

a c

oord

inate

gri

d.

Th

e

x-c

oord

inate

s of

R a

nd

S a

re t

he

y-c

oord

inate

s of

J a

nd

K a

re t

he

sam

e.

Th

e y

-coord

inate

of

T i

s sa

me.

Th

e x

-coord

inate

s of

K a

nd

L

betw

een

th

e y

-coord

inate

s of

R

are

th

e s

am

e.

Is △

JK

L a

cute

,

an

d S

. T

he x

-coord

inate

of

T i

s le

ss

ri

gh

t, o

r obtu

se?

than

th

e x

-coord

inate

of

R.

Is a

ngle

R o

f △

RS

T a

cute

, ri

gh

t, o

r

obtu

se?

3

. C

on

sid

er

thre

e n

on

coll

inear

poin

ts,

4. C

on

sid

er

thre

e p

oin

ts,

G,

H,

an

d I

D,

E,

an

d F

on

a c

oord

inate

gri

d.

on

a c

oord

inate

gri

d.

Poin

ts G

an

d

Th

e x

-coord

inate

s of

D a

nd

E a

re

H

are

on

th

e p

osi

tive y

-axis

, an

d

op

posi

tes.

Th

e y

-coord

inate

s of

D a

nd

th

e y

-coord

inate

of

G i

s tw

ice t

he

E a

re t

he s

am

e.

Th

e x

-coord

inate

of

y-c

oord

inate

of

H.

Poin

t I

is o

n t

he

F i

s 0.

Wh

at

kin

d o

f tr

ian

gle

mu

st

p

osi

tive x

-axis

, an

d t

he x

-coord

inate

△D

EF

be:

scale

ne,

isosc

ele

s, o

r of

I is

gre

ate

r th

an

th

e y

-coord

inate

equ

ilate

ral?

of

G.

Is △

GH

I sc

ale

ne,

isosc

ele

s, o

r

equ

ilate

ral?

BA

C

x

y

O

4-1 Exam

ple

iso

scele

s

rig

ht

scale

ne

acu

te

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

11

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

An

gle

s o

f Tri

an

gle

s

Tri

an

gle

An

gle

-Su

m T

he

ore

m

If t

he m

easu

res

of

two a

ngle

s of

a t

rian

gle

are

k

now

n,

the m

easu

re o

f th

e t

hir

d a

ngle

can

alw

ays

be f

ou

nd

.

Tri

an

gle

An

gle

Su

m

Th

eo

rem

Th

e s

um

of

the

me

asu

res o

f th

e a

ng

les o

f a

tria

ng

le is 1

80

.

In t

he

fig

ure

at

the

rig

ht,

m∠

A +

m∠

B +

m∠

C =

18

0.

CA

B

F

ind

m∠T

.

m∠

R +

m∠

S +

m∠

T =

180

Triangle

Angle

-

Sum

Theore

m

25 +

35 +

m∠

T =

180

Substitu

tion

60 +

m∠

T =

180

Sim

plif

y.

m

∠T

= 1

20

Subtr

act

60 f

rom

each s

ide.

F

ind

th

e m

issin

g

an

gle

me

asu

re

s.

m∠

1 +

m∠

A +

m∠

B =

180

Triangle

Angle

- S

um

T

heore

m

m∠

1 +

58 +

90 =

180

Substitu

tion

m∠

1 +

148 =

180

Sim

plif

y.

m∠

1 =

32

Subtr

act

148 f

rom

each s

ide.

m∠

2 =

32

Vert

ical angle

s a

re

congru

ent.

m∠

3 +

m∠

2 +

m∠

E =

180

Triangle

Angle

- S

um

T

heore

m

m∠

3 +

32 +

108 =

180

Substitu

tion

m∠

3 +

140 =

180

Sim

plif

y.

m∠

3 =

40

Subtr

act

140 f

rom

each s

ide.

58°90°

108°

1

23

E

DA

C

B

Exerc

ises

Fin

d t

he

me

asu

re

s o

f e

ach

nu

mb

ere

d a

ng

le.

1

.

2.

3

.

4.

5

.

6.

20°

152°

DG

A

130°

60°1

2

S

R

TW

V WT

U

30°

60°

2

1

S

QR

30°

1

90°

62°

1NM

P

4-2

Exam

ple

1

35°

25°

RT

S

Exam

ple

2

m∠

1 =

28

m∠

1 =

120

m∠

1 =

30,

m∠

2 =

60

m∠

1 =

56,

m∠

2 =

56,

m∠

3 =

74

m∠

1 =

30,

m∠

2 =

60

m∠

1 =

8

Q

O

NM P

58°

66°

50°

3

21

Answers (Lesson 4-1 and Lesson 4-2)


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

12

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

An

gle

s o

f Tri

an

gle

s

Ex

teri

or

An

gle

Th

eo

rem

A

t each

vert

ex o

f a t

rian

gle

, th

e a

ngle

form

ed

by o

ne s

ide

an

d a

n e

xte

nsi

on

of

the o

ther

sid

e i

s ca

lled

an

ex

terio

r a

ng

le o

f th

e t

rian

gle

. F

or

each

exte

rior

an

gle

of

a t

rian

gle

, th

e r

em

ote

in

terio

r a

ng

les a

re t

he i

nte

rior

an

gle

s th

at

are

not

ad

jace

nt

to t

hat

exte

rior

an

gle

. In

th

e d

iagra

m b

elo

w, ∠

B a

nd

∠A

are

th

e r

em

ote

in

teri

or

an

gle

s fo

r exte

rior ∠

DC

B.

Ex

teri

or

An

gle

Th

eo

rem

Th

e m

ea

su

re o

f a

n e

xte

rio

r a

ng

le o

f a

tria

ng

le is e

qu

al to

the

su

m o

f th

e m

ea

su

res o

f th

e t

wo

re

mo

te in

terio

r a

ng

les.

AC

B

D

1m∠

1 =

m∠

A +

m∠

B

F

ind

m∠

1.

m∠

1 =

m∠

R +

m∠

S

Exte

rior

Angle

Theore

m

= 6

0 +

80

Substitu

tion

= 1

40

Sim

plif

y.

RT

S

60°80°

1

F

ind

x.

m∠

PQ

S =

m∠

R +

m∠

S

Exte

rior

Angle

Theore

m

78 =

55 +

x

Substitu

tion

23 =

x

Subtr

act

55 f

rom

each

sid

e.

SR

Q

P

55°

78°

x°

Exerc

ises

Fin

d t

he

me

asu

re

s o

f e

ach

nu

mb

ere

d a

ng

le.

1

.

2.

3

.

4.

Fin

d e

ach

me

asu

re

.

5

. m∠

AB

C

6. m∠

F

E

FG

H58°

x°

x°

B

A

DC

95°

2x°

145°

UT

SR

V

35°

36°

80°

13

2

PO

Q

NM

60°

60°

32

1

BC

D

A

25°

35°

12

YZ

W

X

65°50°

1

4-2

Exam

ple

1Exam

ple

2

m∠

1 =

115

m∠

1 =

60,

m∠

2 =

120

m∠

1 =

60,

m∠

2 =

60,

m∠

3 =

12

0m∠

1 =

109,

m∠

2 =

29,

m∠

3 =

71

50

29

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

13

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

An

gle

s o

f Tri

an

gle

s

Fin

d t

he

me

asu

re

of

ea

ch

nu

mb

ere

d a

ng

le.

1

.

2.

Fin

d e

ach

me

asu

re

.

3

. m∠

1

4

. m∠

2

5

. m∠

3

Fin

d e

ach

me

asu

re

.

6

. m∠

1

7

. m∠

2

8

. m∠

3

Fin

d e

ach

me

asu

re

.

9

. m∠

1

10

. m∠

2

11

. m∠

3

12

. m∠

4

13

. m∠

5

Fin

d e

ach

me

asu

re

.

14

. m∠

1

15

. m∠

263°

1

2

DA

C

B

80°

60°

40°

105°

14

5

2

3

150°

55°

70°

12

3

85°

55°

40°

12

3

146°

12

TIG

ER

S80°

73°

1

4-2

m∠

1 =

27

m∠

1 =

m∠

2 = 1

7

55

55

70

125

55

95

140

40

65

75

115

27

27



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

14

G

len

co

e G

eo

me

try

Practi

ce

An

gle

s o

f Tri

an

gle

s

Fin

d t

he

me

asu

re

of

ea

ch

nu

mb

ere

d a

ng

le.

1

.

2.

Fin

d e

ach

me

asu

re

.

3

. m∠

1

97

4

. m∠

2

83

5

. m∠

3

62

Fin

d e

ach

me

asu

re

.

6

. m∠

1

104

7

. m∠

4

45

8

. m∠

3

65

9

. m∠

2

79

10

. m∠

5

73

11

. m∠

6

147

Fin

d e

ach

me

asu

re

.

12

. m∠

1

26

13

. m∠

2

32

14

. C

ON

ST

RU

CT

ION

T

he d

iagra

m s

how

s an

exam

ple

of

the P

ratt

Tru

ss u

sed

in

bri

dge

con

stru

ctio

n.

Use

th

e d

iagra

m t

o f

ind

m∠

1.

145°

1

64°

58°

1

2AB

C

D

118°

36°

68°

70° 65° 82°

1

2

34

5

6

58°

39°

35°

12

3

40°

1

55°

72°

1

2

4-2 5

5

m

∠1 =

18,

m∠

1 =

90

m∠

1 =

85

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

15

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

An

gle

s o

f Tri

an

gle

s

62˚

136˚

1

. P

AT

HS

E

ric

walk

s aro

un

d a

tri

an

gu

lar

path

. A

t each

corn

er,

he r

eco

rds

the

measu

re o

f th

e a

ngle

he c

reate

s.

He m

ak

es

on

e c

om

ple

te c

ircu

it a

rou

nd

the p

ath

. W

hat

is t

he s

um

of

the

thre

e a

ngle

measu

res

that

he w

rote

dow

n?

2

. S

TA

ND

ING

S

am

, K

en

dra

, an

d T

on

y a

re

stan

din

g i

n s

uch

a w

ay t

hat

if l

ines

were

dra

wn

to c

on

nect

th

e f

rien

ds

they w

ou

ld

form

a t

rian

gle

.

If S

am

is

look

ing a

t K

en

dra

he n

eed

s to

turn

his

head

40°

to l

ook

at

Ton

y.

If

Ton

y i

s lo

ok

ing a

t S

am

he n

eed

s to

tu

rn

his

head

50°

to l

ook

at

Ken

dra

. H

ow

man

y d

egre

es

wou

ld K

en

dra

have t

o

turn

her

head

to l

ook

at

Ton

y i

f sh

e i

s

look

ing a

t S

am

?

3

. T

OW

ER

S

A l

ook

ou

t to

wer

sits

on

a

netw

ork

of

stru

ts a

nd

post

s. L

esl

ie

measu

red

tw

o a

ngle

s on

th

e t

ow

er.

W

hat

is t

he m

easu

re o

f an

gle

1?

4

. Z

OO

S T

he z

oo l

igh

ts u

p t

he c

him

pan

zee

pen

wit

h a

n o

verh

ead

lig

ht

at

nig

ht.

Th

e

cross

sect

ion

of

the l

igh

t beam

mak

es

an

isosc

ele

s tr

ian

gle

.

Th

e t

op

an

gle

of

the t

rian

gle

is

52 a

nd

the e

xte

rior

an

gle

is

116.

Wh

at

is t

he

measu

re o

f an

gle

1?

5

. D

RA

FT

ING

C

hlo

e b

ou

gh

t a d

raft

ing

table

an

d s

et

it u

p s

o t

hat

she c

an

dra

w

com

fort

ably

fro

m h

er

stool.

Ch

loe

measu

red

th

e t

wo a

ngle

s cr

eate

d b

y t

he

legs

an

d t

he t

able

top

in

case

sh

e h

ad

to

dis

man

tle t

he t

able

.

a.

Wh

ich

of

the f

ou

r n

um

bere

d a

ngle

s

can

Ch

loe d

ete

rmin

e b

y k

now

ing t

he

two a

ngle

s fo

rmed

wit

h t

he t

able

top

?

Wh

at

are

th

eir

measu

res?

b.

Wh

at

con

clu

sion

can

Ch

loe m

ak

e

abou

t th

e u

nk

now

n a

ngle

s befo

re s

he

measu

res

them

to f

ind

th

eir

exact

measu

rem

en

ts?

4-2

52˚

116˚

1

Kendra

Tony

Sam

29˚

68˚

12

34

180

90

98

64

∠1 a

nd

∠2;

m∠

1 =

97,

m∠

2 =

83

Th

e s

um

of

the

me

as

ure

s o

f a

ng

le

3 a

nd

4 i

s e

qu

al

to t

he

me

as

ure

of

an

gle

1;

m∠

3 +

m∠

4 =

97



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

16

G

len

co

e G

eo

me

try

En

rich

men

t

Fin

din

g A

ng

le M

easu

res i

n T

rian

gle

sY

ou

can

use

alg

ebra

to s

olv

e p

roble

ms

involv

ing t

rian

gle

s.

In

tria

ng

le ABC

, m∠A

is t

wic

e m∠B

, a

nd

m∠C

is 8

mo

re

th

an

m∠B

. W

ha

t is

th

e m

ea

su

re

of

ea

ch

an

gle

?

Wri

te a

nd

solv

e a

n e

qu

ati

on

. L

et

x =

m∠

B.

m∠

A +

m∠

B +

m∠

C =

180

2

x +

x +

(x +

8)

= 1

80

4x +

8 =

180

4x =

172

x =

43

So,

m∠

A =

2(4

3)

or

86,

m∠

B =

43,

an

d m

∠C

= 4

3 +

8 o

r 51.

So

lve

ea

ch

pro

ble

m.

1. In

tri

an

gle

DE

F,

m∠

E i

s th

ree t

imes

2. In

tri

an

gle

RS

T,

m∠

T i

s 5 m

ore

th

an

m

∠D

, an

d m

∠F

is

9 l

ess

th

an

m∠

E.

m

∠R

, an

d m

∠S

is

10 l

ess

th

an

m∠

T.

Wh

at

is t

he m

easu

re o

f each

an

gle

? W

hat

is t

he m

easu

re o

f each

an

gle

?

3

. In

tri

an

gle

JK

L,

m∠

K i

s fo

ur

tim

es

4. In

tri

an

gle

XY

Z,

m∠

Z i

s 2 m

ore

th

an

tw

ice

m∠

J,

an

d m

∠L

is

five t

imes

m∠

J.

m

∠X

, an

d m

∠Y

is

7 l

ess

th

an

tw

ice m

∠X

.

Wh

at

is t

he m

easu

re o

f each

an

gle

? W

hat

is t

he m

easu

re o

f each

an

gle

?

5. In

tri

an

gle

GH

I, m

∠H

is

20 m

ore

th

an

6. In

tri

an

gle

MN

O,

m∠

M i

s equ

al

to m

∠N

, m

∠G

, an

d m

∠G

is

8 m

ore

th

an

m∠

I.

an

d m

∠O

is

5 m

ore

th

an

th

ree t

imes

W

hat

is t

he m

easu

re o

f each

an

gle

?

m

∠N

. W

hat

is t

he m

easu

re o

f each

an

gle

?

7. In

tri

an

gle

ST

U,

m∠

U i

s h

alf

m∠

T,

8. In

tri

an

gle

PQ

R,

m∠

P i

s equ

al

to

an

d m

∠S

is

30 m

ore

th

an

m∠

T.

Wh

at

m

∠Q

, an

d m

∠R

is

24 l

ess

th

an

m∠

P.

is t

he m

easu

re o

f each

an

gle

? W

hat

is t

he m

easu

re o

f each

an

gle

?

9

. W

rite

you

r ow

n p

roble

ms

abou

t m

easu

res

of

tria

ngle

s.

4-2 Exam

ple

m∠

D =

27,

m∠

E =

81,

m∠

F =

72

m∠

R =

60,

m∠

S =

55,

m∠

T =

65

m∠

J =

18,

m∠

K =

72,

m∠

L =

90

m∠

X =

37,

m∠

Y =

67,

m∠

Z =

76

m∠

G =

56,

m∠

H =

76,

m∠

I =

48

m∠

M =

m∠

N =

35,

m∠

O =

110

m∠

S =

90,

m∠

T =

60,

m∠

U =

30

m∠

P =

m∠

Q =

68,

m∠

R =

44

See s

tud

en

ts’

wo

rk.

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

17

G

len

co

e G

eo

me

try

Gra

ph

ing C

alc

ula

tor

Act

ivit

y

Cab

ri J

un

ior:

An

gle

s o

f Tri

an

gle

s

Cabri

Ju

nio

r ca

n b

e u

sed

to i

nvest

igate

th

e r

ela

tion

ship

s betw

een

th

e a

ngle

s in

a t

rian

gle

.

U

se

Ca

bri

Ju

nio

r t

o i

nv

esti

ga

te t

he

su

m o

f th

e m

ea

su

re

s o

f th

e

an

gle

s o

f a

tria

ng

le a

nd

to

in

ve

sti

ga

te t

he

me

asu

re

of

the

ex

terio

r a

ng

le i

n a

tria

ng

le i

n r

ela

tio

n t

o i

ts r

em

ote

in

terio

r a

ng

les.

Ste

p 1

U

se C

abri

Ju

nio

r to

dra

w a

tri

an

gle

.

• S

ele

ct F

2 T

ria

ng

le t

o d

raw

a t

rian

gle

.

• M

ove t

he c

urs

or

to w

here

you

wan

t th

e f

irst

vert

ex.

Th

en

pre

ss

EN

TE

R.

•

R

ep

eat

this

pro

ced

ure

to d

ete

rmin

e t

he n

ext

two v

ert

ices

of

the t

rian

gle

.

Ste

p 2

L

abel

each

vert

ex o

f th

e t

rian

gle

.

• S

ele

ct F

5 A

lph

-nu

m.

•

M

ove t

he c

urs

or

to a

vert

ex a

nd

pre

ss

EN

TE

R.

En

ter

A a

nd

pre

ss

EN

TE

R a

gain

.

• R

ep

eat

this

pro

ced

ure

to l

abel

vert

ex B

an

d v

ert

ex C

.

Ste

p 3

D

raw

an

exte

rior

an

gle

of

△A

BC

.

•

S

ele

ct F

2 L

ine

to d

raw

a l

ine t

hro

ugh

−−−

BC

.

•

S

ele

ct F

2 P

oin

t, P

oin

t o

n t

o d

raw

a p

oin

t on

$ %&

BC

so t

hat

C i

s betw

een

B a

nd

th

e n

ew

poin

t.

• S

ele

ct F

5 A

lph

-nu

m t

o l

abel

the p

oin

t D

.

Ste

p 4

F

ind

th

e m

easu

re o

f th

e t

hre

e i

nte

rior

an

gle

s of

△A

BC

an

d t

he

exte

rior

an

gle

, ∠

AC

D.

•

S

ele

ct F

5 M

ea

su

re

, A

ng

le.

•

T

o f

ind

th

e m

easu

re o

f ∠

AB

C,

sele

ct p

oin

ts A

, B

, an

d C

(wit

h t

he v

ert

ex B

as

the s

eco

nd

poin

t se

lect

ed

).

• U

se t

his

meth

od

to f

ind

th

e r

em

ain

ing a

ngle

measu

res.

Exerc

ises

An

aly

ze

yo

ur d

ra

win

g.

1

. U

se F

5 C

alc

ula

te t

o f

ind

th

e s

um

of

the m

easu

res

of

the t

hre

e i

nte

rior

an

gle

s.

2

. M

ak

e a

con

ject

ure

abou

t th

e s

um

of

the i

nte

rior

an

gle

s of

a t

rian

gle

.

3

. C

han

ge t

he d

imen

sion

s of

the t

rian

gle

by m

ovin

g p

oin

t A

. (P

ress

CLE

AR

so t

he p

oin

ter

beco

mes

a b

lack

arr

ow

. M

ove t

he p

oin

ter

close

to p

oin

t A

un

til

the a

rrow

beco

mes

tran

spare

nt

an

d p

oin

t A

is

bli

nk

ing.

Pre

ss

ALP

HA

to c

han

ge t

he a

rrow

to a

han

d.

Th

en

m

ove t

he p

oin

t.)

Is y

ou

r co

nje

ctu

re s

till

tru

e?

4

. W

hat

is t

he m

easu

re o

f ∠

AC

D?

Wh

at

is t

he s

um

of

the m

easu

res

of

the t

wo r

em

ote

in

teri

or

an

gle

s (∠

AB

C a

nd

∠B

AC

)?

5

. M

ak

e a

con

ject

ure

abou

t th

e r

ela

tion

ship

betw

een

th

e m

easu

re o

f an

exte

rior

an

gle

of

a

tria

ngle

an

d i

ts t

wo r

em

ote

in

teri

or

an

gle

s.

6

. C

han

ge t

he d

imen

sion

s of

the t

rian

gle

by m

ovin

g p

oin

t A

. Is

you

r co

nje

ctu

re s

till

tru

e?

4-2 Exam

ple

yes

Th

e s

um

of

the m

easu

res o

f th

e i

nte

rio

r an

gle

s o

f a t

rian

gle

is 1

80°

yes

See s

tud

en

ts’ w

ork

Th

e m

easu

re o

f an

exte

rio

r an

gle

is

eq

ual to

th

e s

um

of

the m

easu

res o

f it

s t

wo

rem

ote

in

teri

or

an

gle

s.

180°



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

18

G

len

co

e G

eo

me

try

Geo

mete

r’s

Sk

etc

hp

ad

Act

ivit

y

An

gle

s o

f Tri

an

gle

s

Th

e G

eom

ete

r’s

Sk

etc

hp

ad

can

be u

sed

to i

nvest

igate

th

e r

ela

tion

ship

s betw

een

th

e a

ngle

s in

a t

rian

gle

.

U

se

Th

e G

eo

me

ter’s

Sk

etc

hp

ad

to

in

ve

sti

ga

te t

he

su

m o

f th

e

me

asu

re

s o

f th

e a

ng

les o

f a

tria

ng

le a

nd

to

in

ve

sti

ga

te t

he

me

asu

re

of

the

ex

terio

r a

ng

le i

n a

tria

ng

le i

n r

ela

tio

n t

o i

ts r

em

ote

in

terio

r a

ng

les.

Ste

p 1

U

se T

he G

eom

ete

r’s

Sk

etc

hp

ad

to d

raw

a t

rian

gle

.

•

Con

stru

ct a

ray b

y s

ele

ctin

g t

he R

ay t

ool

from

th

e t

oolb

ar.

Fir

st,

clic

k w

here

you

wan

t th

e f

irst

poin

t. T

hen

cli

ck o

n a

seco

nd

poin

t to

dra

w t

he r

ay.

•

N

ext,

sele

ct t

he S

egm

en

t to

ol

from

th

e t

oolb

ar.

Use

th

e e

nd

poin

t of

the r

ay a

s th

e f

irst

poin

t fo

r th

e s

egm

en

t an

d c

lick

on

a s

eco

nd

poin

t to

con

stru

ct t

he

segm

en

t.

•

Con

stru

ct a

noth

er

segm

en

t jo

inin

g t

he s

eco

nd

poin

t of

the n

ew

segm

en

t to

a

poin

t on

th

e r

ay.

Ste

p 2

D

isp

lay t

he l

abels

for

each

poin

t.

Use

th

e p

oin

ter

tool

to s

ele

ct a

ll

fou

r p

oin

ts.

Dis

pla

y t

he l

abels

by s

ele

ctin

g S

ho

w L

ab

els

fro

m

the D

isp

lay

men

u.

Ste

p 3

F

ind

th

e m

easu

res

of

the t

hre

e

inte

rior

an

gle

s an

d t

he o

ne e

xte

rior

an

gle

. T

o f

ind

th

e m

easu

re o

f an

gle

A

BC

, u

se t

he S

ele

ctio

n A

rrow

tool

to s

ele

ct p

oin

ts A

, B

, an

d C

(w

ith

th

e v

ert

ex B

as

the s

eco

nd

poin

t se

lect

ed

). T

hen

, u

nd

er

the M

ea

su

re

m

en

u,

sele

ct A

ng

le.

Use

th

is m

eth

od

to

fin

d t

he r

em

ain

ing a

ngle

measu

res.

Exerc

ises

An

aly

ze

yo

ur d

ra

win

g.

1

. W

hat

is t

he s

um

of

the m

easu

res

of

the t

hre

e i

nte

rior

an

gle

s?

2

. M

ak

e a

con

ject

ure

abou

t th

e s

um

of

the i

nte

rior

an

gle

s of

a t

rian

gle

.

3

. C

han

ge t

he d

imen

sion

s of

the t

rian

gle

by s

ele

ctin

g p

oin

t A

wit

h t

he

poin

ter

tool

an

d m

ovin

g i

t. I

s you

r co

nje

ctu

re s

till

tru

e?

4

. W

hat

is t

he m

easu

re o

f ∠

AC

D?

Wh

at

is t

he s

um

of

the m

easu

res

of

the

two r

em

ote

in

teri

or

an

gle

s (∠

AB

C a

nd

∠B

AC

)?

5

. M

ak

e a

con

ject

ure

abou

t th

e r

ela

tion

ship

betw

een

th

e m

easu

re o

f an

exte

rior

an

gle

of

a t

rian

gle

an

d i

ts t

wo r

em

ote

in

teri

or

an

gle

s.

6

. C

han

ge t

he d

imen

sion

s of

the t

rian

gle

by s

ele

ctin

g p

oin

t A

wit

h t

he

poin

ter

tool

an

d m

ovin

g i

t. I

s you

r co

nje

ctu

re s

till

tru

e?

m/

AB

C =

60.00°

m/

BA

C =

58.19°

m/

AC

B =

61.81°

m/

AC

D =

118.19°

m/

AB

C +

m/

BA

C +

m/

AC

B

= 180.00°

m/

AB

C +

m/

BA

C =

118.19°

BC

D

A

4-2 Exam

ple

180

°

yes

See s

tud

en

ts’

wo

rk.

Th

e s

um

of

the m

easu

res o

f th

e i

nte

rio

r an

gle

s o

f a t

rian

gle

is 1

80

°.

Th

e m

easu

res o

f an

exte

rio

r an

gle

is e

qu

al to

th

e s

um

of

the m

easu

re o

f it

s t

wo

rem

ote

in

teri

or

an

gle

s.

yes


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Ch

ap

ter

4

19

G

len

co

e G

eo

me

try

Co

ng

rue

nce

an

d C

orr

esp

on

din

g P

art

s T

rian

gle

s th

at

have t

he s

am

e s

ize a

nd

sam

e s

hap

e a

re

co

ng

ru

en

t tr

ian

gle

s.

Tw

o t

rian

gle

s are

con

gru

en

t if

an

d

on

ly i

f all

th

ree p

air

s of

corr

esp

on

din

g a

ngle

s are

con

gru

en

t an

d a

ll t

hre

e p

air

s of

corr

esp

on

din

g s

ides

are

con

gru

en

t. I

n

the f

igu

re,

△A

BC

" △

RS

T.

If

△XYZ

# △

RST

, n

am

e t

he

pa

irs o

f

co

ng

ru

en

t a

ng

les a

nd

co

ng

ru

en

t sid

es.

∠X

" ∠

R,

∠Y

" ∠

S,

∠Z

" ∠

T

−−

XY

" −

−

RS

, −

−

XZ

" −

−

RT

, −

−

YZ

" −

−

ST

Exerc

ises

Sh

ow

th

at

the

po

lyg

on

s a

re

co

ng

ru

en

t b

y i

de

nti

fyin

g a

ll c

on

gru

en

t co

rre

sp

on

din

g

pa

rts

. T

he

n w

rit

e a

co

ng

ru

en

ce

sta

tem

en

t.

1

.

CA

B

LJ

K

2.

C

D

A

B

3.

K J

L M

4

. F

GL

K JE

5

. B

D

CA

6

. R TU

S

△A

BC

# △

DE

F.

7

. F

ind

th

e v

alu

e o

f x.

8

. F

ind

th

e v

alu

e o

f y.

Y

XZ

T

SR

AC

B

R

TS

4-3

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Co

ng

ruen

t Tri

an

gle

s

Th

ird

An

gle

s

Th

eo

rem

If two a

ngles o

f one triangle a

re c

ongru

ent to

two a

ngles

of a s

econd triangle, th

en the third a

ngles o

f th

e triangles a

re c

ongru

ent.

Exam

ple

2x

+y

(2y

-5)°

75°

65°

40°

96.6

90.6

64.3

∠A

# ∠

J;

∠B

# ∠

K;

∠A

# ∠

D;

∠A

BC

# ∠

BC

D

∠

J #

∠L

; ∠

JK

M #

∠LM

K;

∠C

# ∠

L; −

−

AB

# −

−

JK

;

∠

AC

B #

∠C

BD

; −

−

AC

# −

−

BD

∠K

MJ #

∠M

KL; −

−

KJ #

−−

LM

−

−

BC

# −

−

KL ; −

−

AC

# −

−

JL

−

−

AB

# −

−

CD

−

−

KL #

−−

JM

△A

BC

# △

JK

L

△

AB

C #

△D

CB

△JK

M #

△LM

K

27.8

35

∠E

# ∠

J;

∠F #

∠K

;

∠

A #

∠D

;

∠

R #

∠T

;

∠G

# ∠

L; −

−

EF #

−−

JK

;

∠

AB

C #

∠D

CB

;

∠

RS

U #

∠T

SU

;

−

−

EG

# −

−

JL ; −

−

FG

# −

−

KL

∠

AC

B #

∠D

BC

;

∠

RU

S #

∠T

US

;

△FG

E #

△K

LJ

−

−

AB

# −

−

DC

; −

−

AC

# −

−

DB

;

−

−

RU

# −

−

TU

; −

−

RS #

−−

TS ;

−

−

BC

# −

−

CB

; △

AB

C #

△D

CB

−

−

SU

# −

−

SU

; △

RS

U #

△TS

U



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

20

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Co

ng

ruen

t Tri

an

gle

s

4-3 Exam

ple

Pro

ve

Tri

an

gle

s C

on

gru

en

t T

wo t

rian

gle

s are

con

gru

en

t if

an

d o

nly

if

their

co

rresp

on

din

g p

art

s are

con

gru

en

t. C

orr

esp

on

din

g p

art

s in

clu

de c

orr

esp

on

din

g a

ngle

s an

d

corr

esp

on

din

g s

ides.

Th

e p

hra

se “

if a

nd

on

ly i

f” m

ean

s th

at

both

th

e c

on

dit

ion

al

an

d i

ts

con

vers

e a

re t

rue.

For

tria

ngle

s, w

e s

ay,

“Corr

esp

on

din

g p

art

s of

con

gru

en

t tr

ian

gle

s are

co

ngru

en

t,”

or

CP

CT

C.

W

rit

e a

tw

o-c

olu

mn

pro

of.

G

ive

n:

−−

AB

" −

−−

CB

, −

−−

AD

" −

−−

CD

, ∠

BA

D "

∠B

CD

−

−−

BD

bis

ect

s ∠

AB

C

P

ro

ve

: △

AB

D "

△C

BD

P

ro

of:

Sta

tem

en

tR

ea

so

n

1.

−−

AB

" −

−−

CB

, −

−−

AD

" −

−−

CD

2.

−−

−

BD

" −

−−

BD

3.

∠B

AD

" ∠

BC

D

4.

∠A

BD

" ∠

CB

D

5.

∠B

DA

" ∠

BD

C

6.

△A

BD

" △

CB

D

1.

Giv

en

2.

Refl

exiv

e P

rop

ert

y o

f co

ngru

en

ce

3.

Giv

en

4.

Defi

nit

ion

of

an

gle

bis

ect

or

5.

Th

ird

An

gle

s T

heore

m

6.

CP

CT

C

Exerc

ises

Writ

e a

tw

o-c

olu

mn

pro

of.

1

. G

ive

n:

∠A

" ∠

C,

∠D

" ∠

B,

−−

−

AD

" −

−−

CB

, −

−

AE

" −

−−

CE

, −

−

AC

bis

ect

s −

−−

BD

Pro

ve

: △

AE

D "

△C

EB

Pro

of:

Sta

tem

en

tsR

easo

ns

1. ∠

A

∠C

, ∠

D

∠B

2. ∠

AE

D

∠C

EB

3.

−−

AD

−

−

CB

, −

−

AE

−

−

CE

4.

−−

DE

−

−

BE

5.

△A

ED

△

CE

B

1.

Giv

en

2.

Vert

ical

an

gle

s

3.

Giv

en

4.

Defi

nit

ion

of

seg

men

t b

isecto

r

5.

CP

CT

C

Writ

e a

pa

ra

gra

ph

pro

of.

2

. G

ive

n:

−−

−

BD

bis

ect

s ∠

AB

C a

nd ∠

AD

C,

−−

AB

" −

−−

CB

, −

−

AB

" −

−−

AD

, −

−−

CB

"

−−

−

DC

P

ro

ve

: △

AB

D "

△C

BD

We a

re g

iven

−−

BD

bis

ects

∠A

BC

an

d ∠

AD

C.

Th

ere

fore

∠

AB

D

∠C

BD

an

d ∠

AD

B

∠C

DB

by t

he d

efi

nit

ion

o

f an

gle

bis

ecto

rs.

By t

he T

hir

d A

ng

le T

heo

rem

, w

e

fin

d t

hat

∠A

∠

C.

We a

re g

iven

th

at

−−

AB

−

−

CB

, −

−

AB

−

−

AD

, an

d −

−

CB

−

−

DC

. U

sin

g t

he s

ub

sti

tuti

on

pro

pert

y,

we c

an

dete

rmin

e t

hat

−−

AD

−

−

CD

. F

inally,

−−

BD

−

−

BD

usin

g t

he R

efl

exiv

e P

rop

ert

y o

f co

ng

ruen

ce.

Th

ere

fore

△A

BD

△

CB

D b

y C

PC

TC

.


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Ch

ap

ter

4

21

G

len

co

e G

eo

me

try

4-3

Sh

ow

th

at

po

lyg

on

s a

re

co

ng

ru

en

t b

y i

de

nti

fyin

g a

ll c

on

gru

en

t co

rre

sp

on

din

g

pa

rts

. T

he

n w

rit

e a

co

ng

ru

en

ce

sta

tem

en

t.

1

.

L

P

J

S

V

T

2.

In t

he

fig

ure

, △

AB

C

△F

DE

.

3

. F

ind

th

e v

alu

e o

f x.

36

4

. F

ind

th

e v

alu

e o

f y.

48

5

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

G

ive

n:

−−

−

AB

" −

−−

CB

, −

−−

AD

" −

−−

CD

, ∠

AB

D "

∠C

BD

,∠

AD

B "

∠C

DB

Pro

ve

: △

AB

D

△C

BD

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

−−

AB

−

−

CB

, −

−

AD

−

−

CD

1.

Giv

en

2.

∠A

BD

∠

CB

D,

∠A

DB

∠

CD

B

2.

Giv

en

3.

∠A

∠

C

3.

Th

ird

An

gle

Th

eo

rem

4.

△A

BD

△

CB

D

4.

CP

CT

C

Sk

ills

Pra

ctic

e

Co

ng

ruen

t Tri

an

gle

s

D

E G

F

108°

48°

y°

(2x

-y)°

AF

BC

ED

∠J

∠T;

∠P

∠

V;

∠E

DF

∠FD

G;

∠E

∠

G;

∠L

∠S

; −

−

JP

−

−

TV

∠

EFD

∠

GFD

; −

−

DE

−

−

DG

;

−−

PL

−−

VS

; −

−

JL

−−

TS

−

−

GF

−−

EF ;

△D

EF

△D

GF

△JP

L

△TV

S



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

22

G

len

co

e G

eo

me

try

Practi

ce

Co

ng

ruen

t Tri

an

gle

s

Sh

ow

th

at

the

po

lyg

on

s a

re

co

ng

ru

en

t b

y i

nd

en

tify

ing

all

co

ng

ru

en

t

co

rre

sp

on

din

g p

arts

. T

he

n w

rit

e a

co

ng

ru

en

ce

sta

tem

en

t.

1

.

D

R

S

CA

B

2.

M

N

L

P

Q

Po

lyg

on

ABCD

p

oly

go

n PQRS

.

3

. F

ind

th

e v

alu

e o

f x.

4

. F

ind

th

e v

alu

e o

f y.

5. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

Giv

en

: ∠

P !

∠R

, ∠

PS

Q !

∠R

SQ

, −

−−

PQ

! −

−−

RQ

, −

−

PS

! −

−

RS

Pro

ve

: △

PQ

S

△R

QS

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

∠P

∠

R,

∠P

SQ

∠

RS

Q

1.

Giv

en

2.

∠P

QS

∠

RQ

S

2.

Th

ird

An

gle

Th

eo

rem

3.

−−

PQ

−

−

RQ

, −

−

PS

−

−

RS

3.

Giv

en

4.

−−

QS

−

−

QS

4.

Refl

exiv

e P

rop

ert

y

5.

△P

QS

△

RQ

S

5.

CP

CT

C

6

. Q

UIL

TIN

G

a.

Ind

icate

th

e t

rian

gle

s th

at

ap

pear

to b

e c

on

gru

en

t.

b.

Nam

e t

he c

on

gru

en

t an

gle

s an

d c

on

gru

en

t si

des

of

a p

air

of

con

gru

en

t tr

ian

gle

s.

B

A I

E FH

G

CD

4-3

(2x

+4)° (3

y-

3)

80°

100°

10

12

∠A

∠

D;

∠B

∠

R

∠L

∠Q

; ∠

M

∠P

∠C

∠

S;

−−

AB

−

−

RD

∠

MN

L

∠P

NQ

;

−−

BC

−

−

RS

; −

−

AC

−

−

SD

−

−

LM

−

−

PQ

; −

−

MN

−

−

NP

△A

BC

△

DR

S

−−

LM

−

−

NQ

△LM

N

△Q

PN

△A

BI

△

EB

F,

△C

BD

△

HB

G

S

am

ple

an

sw

er:

∠A

∠

E,

∠A

BI

∠

EB

F,

∠I

∠

F;

−−

AB

−

−

EB

, −

−

BI

−

−

BF ,

−−

AI

−

−

EF

x =

48

y =

5


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Ch

ap

ter

4

23

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Co

ng

ruen

t Tri

an

gle

s

1

. P

ICT

UR

E H

AN

GIN

G C

an

dic

e h

un

g a

p

ictu

re t

hat

was

in a

tri

an

gu

lar

fram

e

on

her

bed

room

wall

.

BeforeA

B

C

A

B

C

After

On

e d

ay,

it f

ell

to t

he f

loor,

bu

t d

id n

ot

bre

ak

or

ben

d.

Th

e f

igu

re s

how

s th

e

obje

ct b

efo

re a

nd

aft

er

the f

all

. L

abel

the v

ert

ices

on

th

e f

ram

e a

fter

the f

all

acc

ord

ing t

o t

he “

befo

re”

fram

e.

2

. S

IER

PIN

SK

I’S

TR

IAN

GLE

T

he f

igu

re

belo

w i

s a p

ort

ion

of

Sie

rpin

ski’s

Tri

an

gle

. T

he t

rian

gle

has

the p

rop

ert

y

that

an

y t

rian

gle

mad

e f

rom

an

y

com

bin

ati

on

of

ed

ges

is e

qu

ilate

ral.

H

ow

man

y t

rian

gle

s in

th

is p

ort

ion

are

co

ngru

en

t to

th

e b

lack

tri

an

gle

at

the

bott

om

corn

er?

3

. Q

UIL

TIN

G S

tefa

n d

rew

th

is p

att

ern

for

a p

iece

of

his

qu

ilt.

It

is m

ad

e u

p o

f co

ngru

en

t is

osc

ele

s ri

gh

t tr

ian

gle

s. H

e

dre

w o

ne t

rian

gle

an

d t

hen

rep

eate

dly

d

rew

it

all

th

e w

ay a

rou

nd

.

45˚

Wh

at

are

th

e m

issi

ng m

easu

res

of

the

an

gle

s of

the t

rian

gle

?

4

. M

OD

ELS

D

an

a b

ou

gh

t a m

od

el

air

pla

ne

kit

. W

hen

he o

pen

ed

th

e b

ox,

these

tw

o

con

gru

en

t tr

ian

gu

lar

pie

ces

of

wood

fell

ou

t of

it.

A

B

CQ

R

P

Id

en

tify

th

e t

rian

gle

th

at

is c

on

gru

en

t to

△

AB

C.

5

. G

EO

GR

AP

HY

Ig

or

noti

ced

on

a m

ap

th

at

the t

rian

gle

wh

ose

vert

ices

are

th

e

sup

erm

ark

et,

th

e l

ibra

ry,

an

d t

he p

ost

off

ice (

△S

LP

) is

con

gru

en

t to

th

e

tria

ngle

wh

ose

vert

ices

are

Igor’

s h

om

e,

Jaco

b’s

hom

e,

an

d B

en

’s h

om

e (

△IJ

B).

T

hat

is,

△S

LP

! △

IJB

.

a.

Th

e d

ista

nce

betw

een

th

e

sup

erm

ark

et

an

d t

he p

ost

off

ice i

s 1 m

ile.

Wh

ich

path

alo

ng t

he t

rian

gle

△

IJB

is

con

gru

en

t to

th

is?

b.

Th

e m

easu

re o

f ∠

LP

S i

s 40.

Iden

tify

th

e a

ngle

th

at

is c

on

gru

en

t to

th

is

an

gle

in

△IJ

B.

4-3 1

1

45 a

nd

90

△Q

RP

the p

ath

betw

een

Ig

or’

s

ho

me a

nd

Ben

’s h

om

e

∠JB

I



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

24

G

len

co

e G

eo

me

try

En

rich

men

t4-3

Overl

ap

pin

g T

rian

gle

sS

om

eti

mes

wh

en

tri

an

gle

s overl

ap

each

oth

er,

th

ey s

hare

corr

esp

on

din

g p

art

s. W

hen

try

ing

to p

ictu

re t

he c

orr

esp

on

din

g p

art

s, t

ry r

ed

raw

ing t

he t

rian

gle

s se

para

tely

.

In t

he f

igu

re △

AB

D !

△D

CA

an

d m

∠B

DA

= 3

0°.

Fin

d m

∠C

AD

an

d m

∠C

DA

.

30°

Fir

st,

red

raw

△A

BD

an

d △

DC

A l

ike t

his

.

30°

Now

you

can

see t

hat

∠B

DA

! ∠

CA

D b

eca

use

th

ey a

re c

orr

esp

on

din

g p

art

s of

con

gru

en

t tr

ian

gle

s, s

o m

∠C

AD

= 3

0.

Usi

ng t

he T

rian

gle

Su

m T

heore

m,

180 -

(90 +

30)

= 6

0,

so

m∠

CD

A =

60

.

Exerc

ises

1

. In

th

e f

igu

re △

AB

C !

△C

DB

, m

∠A

BC

= 1

10,

an

d

m∠

D =

20.

Fin

d m

∠A

CB

an

d m

∠D

CB

.

2

. W

rite

a t

wo c

olu

mn

pro

of.

Giv

en

: ∠

A a

nd

∠B

are

rig

ht

an

gle

s, ∠

AC

D !

∠B

DC

,

−−

−

AD

! −

−−

BC

, −

−

AC

! −

−−

BD

Pro

ve

: △

AD

C !

△C

BD

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

∠A

an

d ∠

B a

re r

igh

t an

gle

s

1.

Giv

en

2.

∠A

" ∠

B

2.

All r

t #

are

"

3.

∠A

CD

" ∠

BD

C

3.

Giv

en

4.

∠A

DC

" ∠

BC

D

4.

Th

ird

An

gle

Th

eo

rem

5.

−−

AD

" −

−

BC

, −

−

AC

" −

−

BD

5.

Giv

en

6.

−−

DC

" −

−

CD

6.

Refl

exiv

e p

rop

ert

y

7.

△A

DC

" △

CB

D

7.

CP

CT

C

Exam

ple

m∠

AC

B =

m∠

DB

C =

50


NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Ch

ap

ter

4

25

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

SS

S P

ostu

late

Y

ou

kn

ow

th

at

two t

rian

gle

s are

con

gru

en

t if

corr

esp

on

din

g s

ides

are

co

ngru

en

t an

d c

orr

esp

on

din

g a

ngle

s are

con

gru

en

t. T

he S

ide-S

ide-S

ide (

SS

S)

Post

ula

te l

ets

you

sh

ow

th

at

two t

rian

gle

s are

con

gru

en

t if

you

kn

ow

on

ly t

hat

the s

ides

of

on

e t

rian

gle

are

con

gru

en

t to

th

e s

ides

of

the s

eco

nd

tri

an

gle

.

W

rit

e a

tw

o-c

olu

mn

pro

of.

Giv

en

: −

−

AB

! −

−−

DB

an

d C

is

the m

idp

oin

t of

−−

−

AD

.

Pro

ve

: △

AB

C !

△D

BC

B C

DA

Sta

tem

en

ts

Re

aso

ns

1.

−−

AB

! −

−−

DB

1

. G

iven

2.

C i

s th

e m

idp

oin

t of

−−

−

AD

. 2

. G

iven

3.

−−

AC

! −

−−

DC

3

. M

idp

oin

t T

heore

m

4.

−−

−

BC

! −

−−

BC

4

. R

efl

exiv

e P

rop

ert

y o

f !

5.

△A

BC

! △

DB

C

5.

SS

S P

ost

ula

te

Exerc

ises

Writ

e a

tw

o-c

olu

mn

pro

of.

1

. B

Y

CA

XZ

Giv

en

:A

B!

−−

XY

, −

−A

C !

−−

XZ

, −

−−

BC

!−

−Y

Z

Pro

ve

:△

AB

C!

△X

YZ

2

. T

U

RS

Giv

en

: −

−

RS

! −

−−

UT

, −

−

RT

! −

−−

US

Pro

ve

: △

RS

T !

△U

TS

4-4 SS

S P

os

tula

teIf

th

ree

sid

es o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

th

ree

sid

es o

f a

se

co

nd

tria

ng

le,

the

n t

he

tria

ng

les a

re c

on

gru

en

t.

Exam

ple

Sta

tem

en

ts

Reaso

ns

Sta

tem

en

ts

Reaso

ns

1.

−−

AB

" −

−

XY

1.

Giv

en

−

−

AC

" −

−

XZ

−

−

BC

" −

−

YZ

2.

△A

BC

" △

XY

Z

2.

SS

S P

ost.

1.

−−

RS

" −

−

UT

1.

Giv

en

−

−

RT "

−−

US

2.

−−

ST "

−−

TS

2

. R

efl

. P

rop

.

3.

△R

ST "

△U

TS

3.

SS

S P

ost.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

26

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

SA

S P

ostu

late

A

noth

er

way t

o s

how

th

at

two t

rian

gle

s are

con

gru

en

t is

to u

se t

he

Sid

e-A

ngle

-Sid

e (

SA

S)

Post

ula

te.

F

or e

ach

dia

gra

m,

de

term

ine

wh

ich

pa

irs o

f tr

ian

gle

s c

an

be

pro

ve

d c

on

gru

en

t b

y t

he

SA

S P

ostu

late

.

a

. A B

C

X YZ

b

. D

H

FE

G

J

c.

PQ

S

1

2R

In

△A

BC

, th

e a

ngle

is

not

T

he r

igh

t an

gle

s are

T

he i

ncl

ud

ed

an

gle

s,

“in

clu

ded

” by t

he s

ides

−−

AB

co

ngru

en

t an

d t

hey a

re t

he

∠

1 a

nd

∠2,

are

con

gru

en

t an

d −

−

AC

. S

o t

he t

rian

gle

s

in

clu

ded

an

gle

s fo

r th

e

beca

use

th

ey a

re

can

not

be p

roved

con

gru

en

t

co

ngru

en

t si

des.

alt

ern

ate

in

teri

or

an

gle

s by t

he S

AS

Post

ula

te.

△

DE

F $

△J

GH

by t

he

fo

r tw

o p

ara

llel

lin

es.

S

AS

Post

ula

te.

△

PS

R $

△R

QP

by t

he

S

AS

Post

ula

te.

Exerc

ises

4-4

SA

S P

os

tula

te

If t

wo

sid

es a

nd

th

e in

clu

de

d a

ng

le o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

tw

o

sid

es a

nd

th

e in

clu

de

d a

ng

le o

f a

no

the

r tr

ian

gle

, th

en

th

e t

ria

ng

les a

re

co

ng

rue

nt.

Exam

ple

Writ

e t

he

sp

ecif

ied

ty

pe

of

pro

of.

1

. W

rite

a t

wo c

olu

mn

pro

of.

2

. W

rite

a t

wo-c

olu

mn

pro

of.

Giv

en

: N

P =

PM

, −−

− N

P ⊥

−−

PL

G

ive

n:

AB

= C

D,

−−

AB

‖ −

−−

CD

Pro

ve

: △

NP

L $

△M

PL

P

ro

ve

: △

AC

D $

△C

AB

Pro

of:

Pro

of:

3

. W

rite

a p

ara

gra

ph

pro

of.

Giv

en

: V

is

the m

idp

oin

t of

−−

YZ

.

V

is

the m

idp

oin

t of

−−

−

WX

.

Pro

ve

: △

XV

Z $

△W

VY

S

ince V

is t

he m

idp

oin

t o

f −

−

YZ a

nd

th

e m

idp

oin

t o

f −

−

WX ,

by t

he d

efi

nit

ion

of

mid

po

int,

Y

V =

VZ

an

d W

V =

VX

. S

ince ∠

YV

W a

nd

∠X

VZ

are

vert

ical

an

gle

s,

by t

he V

ert

ical

An

gle

Th

eo

rem

th

e a

ng

les a

re c

on

gru

en

t. T

here

fore

, b

y S

AS

, △

XV

Z $

△W

VY

Sta

tem

en

tsR

easo

ns

1.

NP

= P

M, −

−

NP

⊥ −

−

PL

2.

∠M

PL i

s a

rig

ht

an

gle

∠N

PL i

s a

rig

ht

an

gle

3.

∠M

PL $

∠N

PL

4.

PL =

PL

5.

△N

PL $

△M

PL

1.

Giv

en

2.

perp

en

dic

ula

r lin

es

form

rig

ht

an

gle

s

3.

all r

igh

t an

gle

s a

re

co

ng

ruen

t

4.

Refl

exiv

e P

rop

ert

y

of

co

ng

ruen

ce

5.

SA

S

Sta

tem

en

tsR

easo

ns

1.

AB

= C

D, −

−

AB

‖ −

−

CD

2.

∠B

AC

$ ∠

DC

A

3.

AC

= A

C

4.

△A

CD

$ △

CA

B

1.

Giv

en

2.

Alt

ern

ate

In

teri

or

An

gle

s T

heo

rem

3.

Re

fle

xiv

e P

rop

ert

y

of

co

ng

rue

nc

e

4.

SA

S


NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Ch

ap

ter

4

27

G

len

co

e G

eo

me

try

De

term

ine

wh

eth

er △

AB

C $

△K

LM

. E

xp

lain

.

1

. A

(-3,

3),

B(-

1,

3),

C(-

3,

1),

K(1

, 4),

L(3

, 4),

M(1

, 6)

2

. A

(-4,

-2),

B(-

4,

1),

C(-

1,

-1),

K(0

, -

2),

L(0

, 1),

M(4

, 1)

PR

OO

F W

rit

e t

he

sp

ecif

ied

ty

pe

of

pro

of.

3

. W

rite

a f

low

pro

of.

G

ive

n:

−−

PR

$ −

−−

DE

, −

−

PT

$ −−

− D

F

∠R

$ ∠

E,

∠T

$ ∠

F

P

ro

ve

: △

PR

T $

△D

EF

PR

$DE

Given

PT

$DF

Given

∠R

$∠E

Given

∠P

$∠D

Thir

d A

ng

le

Theo

rem

△PRT

$△DEF

SA

S

∠T

$∠F

Given

4

. W

rite

a t

wo-c

olu

mn

pro

of.

Giv

en

: −

−

AB

$ −−

− C

B ,

D i

s th

e m

idp

oin

t of

−−

AC

.

Pro

ve

: △

AB

D $

△C

BD

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

−−

AB

$ −

−

CB

, D

is t

he m

idp

oin

t o

f −

−

AC

1.

Giv

en

2.

−−

AD

$ −

−

DC

2.

Defi

nit

ion

of

Mid

po

int

3.

−−

BD

$ −

−

BD

3.

Refl

exiv

e P

rop

ert

y o

f C

on

gru

en

ce

4.

△A

BD

$ △

CB

D

4.

SS

S

T

R

P

F

E

D

4-4

Sk

ills

Pra

ctic

e

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

AB

= 2

, K

L =

2,

BC

= 2

√ ( 2

, LM

= 2

√ ( 2

, A

C =

2,

KM

= 2

.

Th

e c

orr

esp

on

din

g s

ides h

ave t

he s

am

e m

easu

re a

nd

are

co

ng

ruen

t,

so

△A

BC

$ △

KLM

by S

SS

.

AB

= 3

, K

L =

3,

BC

= √

((

13 ,

LM

= 4

, A

C =

√ (

(

10 ,

KM

= 5

.

Th

e c

orr

esp

on

din

g s

ides a

re n

ot

co

ng

ruen

t, s

o △

AB

C i

s n

ot

co

ng

ruen

t to

△K

LM

.

Pro

of:



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

28

G

len

co

e G

eo

me

try

Practi

ce

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

De

term

ine

wh

eth

er △

DE

F !

△P

QR

giv

en

th

e c

oo

rd

ina

tes o

f th

e v

erti

ce

s.

Ex

pla

in.

1

. D

(-6,

1),

E(1

, 2),

F(-

1,

-4),

P(0

, 5),

Q(7

, 6),

R(5

, 0)

2

. D

(-7,

-3),

E(-

4,

-1),

F(-

2,

-5),

P(2

, -

2),

Q(5

, -

4),

R(0

, -

5)

3

. W

rite

a f

low

pro

of.

Giv

en

: −

−

RS

" −

−

TS

V i

s th

e m

idp

oin

t of

−−

RT

.

Pro

ve

: △

RS

V "

△T

SV

De

term

ine

wh

ich

po

stu

late

ca

n b

e u

se

d t

o p

ro

ve

th

at

the

tria

ng

les a

re

co

ng

ru

en

t.

If i

t is

no

t p

ossib

le t

o p

ro

ve

co

ng

ru

en

ce

, w

rit

e n

ot

po

ssib

le.

4

.

5.

6.

7

. IN

DIR

EC

T M

EA

SU

RE

ME

NT

T

o m

easu

re t

he w

idth

of

a s

ink

hole

on

his

pro

pert

y,

Harm

on

mark

ed

off

con

gru

en

t tr

ian

gle

s as

show

n i

n t

he

dia

gra

m.

How

does

he k

now

th

at

the l

en

gth

s A'B' an

d A

B a

re e

qu

al?

A'

B'

AB

C

S

R V T

4-4 D

E =

5 √

$

2 ,

PQ

= 5

√ $

2 ,

EF =

2 √

$$

10 ,

QR

= 2

√ $

$

10 ,

DF =

5 √

$

2 ,

PR

= 5

√ $

2 .

△D

EF

! △

PQ

R b

y S

SS

sin

ce c

orr

esp

on

din

g s

ides h

ave t

he s

am

e

measu

re a

nd

are

co

ng

ruen

t.

DE

= √

$$

13 ,

PQ

= √

$$

13 ,

EF =

2 √

$

5 ,

QR

= √

$$

26 ,

DF =

√ $

$

29 ,

PR

= √

$$

13 .

Co

rresp

on

din

g s

ides a

re n

ot

co

ng

ruen

t, s

o △

DE

F i

s n

ot

co

ng

ruen

t to

△P

QR

.

Sin

ce ∠

AC

B a

nd

∠A

'CB

' are

vert

ical

an

gle

s,

they a

re c

on

gru

en

t. I

n t

he

fig

ure

, −

−

AC

! −

−−

A′C

an

d −

−

BC

! −

−−

B'C

. S

o △

AB

C !

△A

'B'C

by S

AS

. B

y C

PC

TC

, th

e l

en

gth

s A

'B' a

nd

AB

are

eq

ual.

Pro

of:

n

ot

po

ssib

le

SA

S o

r S

SS

S

SS

RS TS

Given

SV SV

Refl

exiv

e

Pro

per

ty

RV VT

Defi

nitio

n

of

mid

po

int

V is

the

mid

po

int

of RT.

Given

∆RSV

∆TSV

SS

S


NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Ch

ap

ter

4

29

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

1

. S

TIC

KS

T

yso

n h

ad

th

ree s

tick

s of

len

gth

s 24 i

nch

es,

28 i

nch

es,

an

d

30 i

nch

es.

Is

it p

oss

ible

to m

ak

e t

wo

non

con

gru

en

t tr

ian

gle

s u

sin

g t

he s

am

e

thre

e s

tick

s? E

xp

lain

.

2

. B

AK

ER

Y S

on

ia m

ad

e a

sh

eet

of

bak

lava.

Sh

e h

as

mark

ings

on

her

pan

so

th

at

she c

an

cu

t th

em

in

to l

arg

e

squ

are

s. A

fter

she c

uts

th

e p

ast

ry i

n

squ

are

s, s

he c

uts

th

em

dia

gon

all

y t

o

form

tw

o c

on

gru

en

t tr

ian

gle

s. W

hic

h

post

ula

te c

ou

ld y

ou

use

to p

rove t

he

two t

rian

gle

s co

ngru

en

t?

3

. C

AK

E C

arl

had

a p

iece

of

cak

e i

n t

he

shap

e o

f an

iso

scele

s tr

ian

gle

wit

h

an

gle

s 26,

77,

an

d 7

7.

He w

an

ted

to

div

ide i

t in

to t

wo e

qu

al

part

s, s

o h

e c

ut

it t

hro

ugh

th

e m

idd

le o

f th

e 2

6 a

ngle

to

the m

idp

oin

t of

the o

pp

osi

te s

ide.

He

says

that

beca

use

he i

s d

ivid

ing i

t at

the

mid

poin

t of

a s

ide,

the t

wo p

iece

s are

co

ngru

en

t. I

s th

is e

nou

gh

in

form

ati

on

? E

xp

lain

.

4

. T

ILE

S T

am

my i

nst

all

s bath

room

til

es.

H

er

curr

en

t jo

b r

equ

ires

tile

s th

at

are

equ

ilate

ral

tria

ngle

s an

d a

ll t

he t

iles

have t

o b

e c

on

gru

en

t to

each

oth

er.

Sh

e

has

a b

ig s

ack

of

tile

s all

in

th

e s

hap

e

of

equ

ilate

ral

tria

ngle

s. A

lth

ou

gh

sh

e

kn

ow

s th

at

all

th

e t

iles

are

equ

ilate

ral,

sh

e i

s n

ot

sure

th

ey a

re a

ll t

he s

am

e

size.

Wh

at

mu

st s

he m

easu

re o

n e

ach

ti

le t

o b

e s

ure

th

ey a

re c

on

gru

en

t?

Exp

lain

.

5. IN

VE

ST

IGA

TIO

NA

n i

nvest

igato

r at

a

crim

e s

cen

e f

ou

nd

a t

rian

gu

lar

pie

ce o

f to

rn f

abri

c. T

he i

nvest

igato

r re

mem

bere

d t

hat

on

e o

f th

e s

usp

ect

s h

ad

a t

rian

gu

lar

hole

in

th

eir

coat.

P

erh

ap

s it

was

a m

atc

h.

Un

fort

un

ate

ly,

to a

void

tam

peri

ng w

ith

evid

en

ce,

the

invest

igato

r d

id n

ot

wan

t to

tou

ch t

he

fabri

c an

d c

ou

ld n

ot

fit

it t

o t

he c

oat

dir

ect

ly.

a.

If t

he i

nvest

igato

r m

easu

res

all

th

ree

sid

e l

en

gth

s of

the f

abri

c an

d t

he

hole

, ca

n t

he i

nvest

igato

r m

ak

e a

co

ncl

usi

on

abou

t w

heth

er

or

not

the

hole

cou

ld h

ave b

een

fil

led

by t

he

fabri

c?

b.

If t

he i

nvest

igato

r m

easu

res

two s

ides

of

the f

abri

c an

d t

he i

ncl

ud

ed

an

gle

an

d t

hen

measu

res

two s

ides

of

the

hole

an

d t

he i

ncl

ud

ed

an

gle

can

he

dete

rmin

e i

f it

is

a m

atc

h?

Exp

lain

.

4-4 N

o.

Th

e s

ticks d

o n

ot

ch

an

ge

siz

e,

so

an

y a

rran

gem

en

t w

ill

yie

ld a

tri

an

gle

co

ng

ruen

t to

th

e

on

e b

efo

re.

eit

her

SS

S o

r S

AS

Sam

ple

an

sw

er:

No

. T

hat

is

on

ly e

xp

lain

ing

on

e s

ide t

hat

is

co

ng

ruen

t o

n t

he t

rian

gle

s.

He

need

s t

o s

ho

w t

wo

mo

re s

ides

of

the t

rian

gle

are

co

ng

ruen

t to

pro

ve t

he p

ieces a

re c

on

gru

en

t.

yes

Sh

e j

ust

need

s t

o m

easu

re o

ne

sid

e o

f each

tile b

ecau

se t

hey

are

all e

qu

ilate

ral

tria

ng

les.

On

ly i

f th

e t

wo

sid

es a

nd

in

clu

ded

an

gle

of

on

e t

rian

gle

are

th

e c

orr

esp

on

din

g s

ides

an

d i

nclu

ded

an

gle

of

the

oth

er

tria

ng

le.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

30

G

len

co

e G

eo

me

try

En

rich

men

t4-4

Co

ng

ruen

t Tri

an

gle

s i

n t

he C

oo

rdin

ate

Pla

ne

If y

ou

kn

ow

th

e c

oord

inate

s of

the v

ert

ices

of

two t

rian

gle

s in

th

e c

oord

inate

p

lan

e,

you

can

oft

en

deci

de w

heth

er

the t

wo t

rian

gle

s are

con

gru

en

t. T

here

m

ay b

e m

ore

th

an

on

e w

ay t

o d

o t

his

.

1

. C

on

sid

er

△A

BD

an

d △

CD

B w

hose

vert

ices

have c

oord

inate

s A

(0,

0),

B

(2,

5),

C(9

, 5),

an

d D

(7,

0).

Bri

efl

y d

esc

ribe h

ow

you

can

use

wh

at

you

k

now

abou

t co

ngru

en

t tr

ian

gle

s an

d t

he c

oord

inate

pla

ne t

o s

how

th

at

△A

BD

! △

CD

B.

You

may w

ish

to m

ak

e a

sk

etc

h t

o h

elp

get

you

sta

rted

.

2

. C

on

sid

er

△P

QR

an

d △

KL

M w

hose

vert

ices

are

th

e f

oll

ow

ing p

oin

ts.

P

(1,

2)

Q

(3,

6)

R

(6,

5)

K

(-2,

1)

L

(-6,

3)

M

(-5,

6)

B

riefl

y d

esc

ribe h

ow

you

can

sh

ow

th

at

△P

QR

! △

KL

M.

3

. If

you

kn

ow

th

e c

oord

inate

s of

all

th

e v

ert

ices

of

two t

rian

gle

s, i

s it

a

lwa

ys

poss

ible

to t

ell

wh

eth

er

the t

rian

gle

s are

con

gru

en

t? E

xp

lain

.

Sam

ple

an

sw

er:

Sh

ow

th

at

the s

lop

es o

f −

−

AD

an

d −

−

BC

are

eq

ual. C

on

clu

de

that

−−

BC

||

−−

AD

. U

se t

he a

ng

le r

ela

tio

nsh

ips f

or

para

llel

lin

es a

nd

a

tran

svers

al

to c

on

clu

de t

hat

∠C

BD

is c

on

gru

en

t to

∠B

DA

. C

ou

nt

the

len

gth

s o

f th

e h

ori

zo

nta

l seg

men

ts t

o c

on

clu

de t

hat

BC

= A

D.

Usin

g

these f

acts

an

d t

he f

act

that

−−

BD

is a

co

mm

on

sid

e f

or

the t

rian

gle

s t

o

co

nclu

de t

hat

△A

BD

$ △

CD

B b

y S

AS

.

Use t

he D

ista

nce F

orm

ula

to

fin

d t

he l

en

gth

s o

f th

e s

ides o

f b

oth

tr

ian

gle

s.

Co

nclu

de t

hat

△P

QR

$ △

KLM

by S

SS

.

Yes;

yo

u c

an

use t

he D

ista

nce F

orm

ula

an

d S

SS

.


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Ch

ap

ter

4

31

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

AS

A P

ostu

late

T

he A

ngle

-Sid

e-A

ngle

(A

SA

) P

ost

ula

te l

ets

you

sh

ow

th

at

two t

rian

gle

s are

con

gru

en

t.

W

rit

e a

tw

o c

olu

mn

pro

of.

G

ive

n:

−−

AB

‖ −

−−

CD

∠C

BD

! ∠

AD

B

P

ro

ve

: △

AB

D !

△C

DB

Sta

tem

en

tsR

ea

so

ns

1.

−−

AB

‖ −

−−

CD

2.

∠C

BD

! ∠

AD

B

3.

∠A

BD

! ∠

BD

C

4.

BD

= B

D

5.

△A

BD

! △

CD

B

1.

Giv

en

2.

Giv

en

3.

Alt

ern

ate

In

teri

or

An

gle

s T

heore

m

4.

Refl

exiv

e P

rop

ert

y o

f co

ngru

en

ce

5.

AS

A

4-5 AS

A P

os

tula

teIf

tw

o a

ng

les a

nd

th

e in

clu

de

d s

ide

of

on

e t

ria

ng

le a

re c

on

gru

en

t to

tw

o a

ng

les a

nd

the

in

clu

de

d s

ide

of

an

oth

er

tria

ng

le,

the

n t

he

tria

ng

les a

re c

on

gru

en

t.

Exam

ple

Exerc

ises

PR

OO

F W

rit

e t

he

sp

ecif

ied

ty

pe

of

pro

of.

1

. W

rite

a t

wo c

olu

mn

pro

of.

2

. W

rite

a p

ara

gra

ph

pro

of.

S

V

U

RT

AC

B

E

D

Giv

en

: ∠

S !

∠V

,

Giv

en

: −

−−

CD

bis

ect

s −

−

AE

, −

−

AB

‖ −

−−

CD

T

is

the m

idp

oin

t of

−−

SV

∠

E !

∠B

CA

Pro

ve

: △

RT

S !

△U

TV

P

ro

ve

: △

AB

C !

△C

DE

Pro

of:

We k

no

w t

hat

∠E

$ ∠

BC

A

an

d −−

− C

D b

isects

−−

AE

. S

ince −−

− C

D

bis

ects

−−

AE

, b

y t

he d

efi

nit

ion

of

bis

ecto

r, A

C =

CE

. W

e a

re a

lso

g

iven

th

at

−−−

AB

‖ −−

− C

D ,

fro

m t

his

we

can

dete

rmin

e t

hat

∠A

is

co

ng

ruen

t to

∠D

CE

by t

he

Alt

ern

ate

In

teri

or

An

gle

T

heo

rem

. F

rom

th

is w

e k

no

w

that

∠A

BC

$ ∠

CD

E b

y t

he

AS

A T

heo

rem

.

Sta

tem

en

tsR

easo

ns

1.

∠S

$ ∠

V

T i

s t

he

mid

po

int

of

−−

SV

2.

ST

= T

V

3.

∠R

TS

$ ∠

VTU

4.

△R

TS

$ △

UTV

1.

Giv

en

2.

defi

nit

ion

of

Mid

po

int

3.

Vert

ical

An

gle

th

eo

rem

.

4.

AS

A



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

32

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

AA

S T

he

ore

m

An

oth

er

way t

o s

how

th

at

two t

rian

gle

s are

con

gru

en

t is

th

e A

ngle

-A

ngle

-Sid

e (

AA

S)

Th

eore

m.

You

now

have f

ive w

ays

to s

how

th

at

two t

rian

gle

s are

con

gru

en

t.

• d

efi

nit

ion

of

tria

ngle

con

gru

en

ce

• A

SA

Post

ula

te

• S

SS

Post

ula

te

• A

AS

Th

eore

m

• S

AS

Post

ula

te

In

th

e d

iag

ra

m,

∠BCA

! ∠

DCA

. W

hic

h s

ide

s

are

co

ng

ru

en

t? W

hic

h a

dd

itio

na

l p

air

of

co

rre

sp

on

din

g p

arts

ne

ed

s t

o b

e c

on

gru

en

t fo

r t

he

tria

ng

les t

o b

e c

on

gru

en

t b

y

the

AA

S T

he

ore

m?

−−

AC

" −

−

AC

by t

he R

efl

exiv

e P

rop

ert

y o

f co

ngru

en

ce.

Th

e c

on

gru

en

t an

gle

s ca

nn

ot

be ∠

1 a

nd

∠2,

beca

use

−−

AC

wou

ld b

e t

he i

ncl

ud

ed

sid

e.

If ∠

B "

∠D

, th

en

△A

BC

" △

AD

C b

y t

he A

AS

Th

eore

m.

Exerc

ises

PR

OO

F W

rit

e t

he

sp

ecif

ied

ty

pe

of

pro

of.

1

. W

rite

a t

wo c

olu

mn

pro

of.

G

ive

n:

−−−

BC

‖ −

−

EF

A

B =

ED

∠

C "

∠F

P

ro

ve

: △

AB

D "

△D

EF

Sta

tem

en

tsR

easo

ns

1.

−−

BC

‖ −

−

EF

A

B =

ED

∠

C !

∠F

2.

∠A

BD

! ∠

DE

F

3.

△A

BD

! △

DE

F

1.

Giv

en

2.

Co

rresp

on

din

g A

ng

les T

heo

rem

3.

AA

S

2

. W

rite

a f

low

pro

of.

G

ive

n:

∠S

" ∠

U;

−−

TR

bis

ect

s ∠

ST

U.

P

ro

ve

: ∠

SR

T "

∠U

RT

P

roo

f:

S

RT

U

4-5 AA

S T

he

ore

mIf

tw

o a

ng

les a

nd

a n

on

inclu

de

d s

ide

of

on

e t

ria

ng

le a

re c

on

gru

en

t to

th

e c

orr

esp

on

din

g

two

an

gle

s a

nd

sid

e o

f a

se

co

nd

tria

ng

le,

the

n t

he

tw

o t

ria

ng

les a

re c

on

gru

en

t.

Exam

ple

D

C1 2

A

B

Given

Given

RT

RT

Refl

. P

rop. of

Def

.of

∠ b

isec

tor

TR

bis

ects ∠

ST

U.

SR

T

U

RT

∠ST

R

∠U

TR

AA

S

∠SR

T

∠U

RT

CP

CTC

∠S

∠U


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Ch

ap

ter

4

33

G

len

co

e G

eo

me

try

PR

OO

F W

rit

e a

flo

w p

ro

of.

1

. G

ive

n:

∠N

" ∠

L

−

−

JK

" −

−−

MK

P

ro

ve

: △

JK

N "

△M

KL

2

. G

ive

n:

−−

AB

" −−

− C

B

∠

A "

∠C

−

−−

DB

bis

ect

s ∠

AB

C.

P

ro

ve

: −−

− A

D "

−−

−

CD

3

. W

rite

a p

ara

gra

ph

pro

of.

G

ive

n:

−−

−

DE

‖ −−

− F

G

∠E

" ∠

G

P

ro

ve

: △

DF

G "

△F

DE

FG

DE

AC

B D

N

J

M

KL

4-5

Sk

ills

Pra

ctic

e

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

Pro

of:

P

roo

f: S

ince i

t is

giv

en

th

at

−−

DE

‖ −

−

FG

, it

fo

llo

ws t

hat

∠E

DF !

∠G

FD

,

b

ecau

se a

lt.

int.

& o

f p

ara

llel

lin

es a

re !

. It

is g

iven

th

at

∠E

! ∠

G.

By t

he

Refl

exiv

e

P

rop

ert

y,

−−

DF !

−−

FD

. S

o △

DFG

! △

FD

E b

y A

AS

.

∠N

"∠

L

Given

JK

"M

K

Given

∠JK

N"

∠M

KL

Ver

tica

l & a

re '

.

△JK

N"

△M

KL

AA

S

Pro

of:

∠A

"∠

C

Given

AB

"C

B

Given

CP

CT

C

AD

'C

D

DB

bis

ects ∠

AB

C.

Given

△A

BD

"△

CB

D

AS

A

∠A

BD

"∠

CB

D

Def

. o

f ∠

bis

ecto

r



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

34

G

len

co

e G

eo

me

try

PR

OO

F W

rit

e t

he

sp

ecif

ied

ty

pe

of

pro

of.

1

. W

rite

a f

low

pro

of.

G

ive

n:

S i

s th

e m

idp

oin

t of

−−−

QT

.

−−

−

QR

‖ −

−−

TU

P

ro

ve

: △

QS

R $

△T

SU

2

. W

rite

a p

ara

gra

ph

pro

of.

G

ive

n:

∠D

$ ∠

F

−−

−

GE

bis

ect

s ∠

DE

F.

P

ro

ve

: −

−−

DG

$ −−

− F

G

AR

CH

ITE

CT

UR

E

Fo

r E

xe

rcis

es 3

an

d 4

, u

se

th

e f

oll

ow

ing

in

form

ati

on

.

An

arc

hit

ect

use

d t

he w

ind

ow

desi

gn

in

th

e d

iagra

m w

hen

rem

od

eli

ng

an

art

stu

dio

. −

−

AB

an

d −−

− C

B e

ach

measu

re 3

feet.

3

. S

up

pose

D i

s th

e m

idp

oin

t of

−−

AC

. D

ete

rmin

e w

heth

er

△A

BD

$ △

CB

D.

Ju

stif

y y

ou

r an

swer.

4

. S

up

pose

∠A

$ ∠

C.

Dete

rmin

e w

heth

er

△A

BD

$ △

CB

D.

Ju

stif

y y

ou

r an

swer.

DB

AC

D

G

F

E

U

QS

R

T

4-5

Practi

ce

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

Pro

of:

Pro

of:

Sin

ce i

t is

giv

en

th

at

−−

GE

bis

ects

∠D

EF,

∠D

EG

# ∠

FE

G b

y t

he

defi

nit

ion

of

an

an

gle

bis

ecto

r. I

t is

giv

en

th

at

∠D

# ∠

F.

By t

he

Refl

exiv

e P

rop

ert

y,

−−

GE

# −

−

GE

. S

o △

DE

G #

△FE

G b

y A

AS

. T

here

fore

−−

DG

# −

−

FG

by C

PC

TC

.

Sin

ce D

is t

he m

idp

oin

t o

f −

−

AC

, −

−

AD

# −

−

CD

by t

he M

idp

oin

t T

heo

rem

.

−−

AB

# −

−

CB

by t

he d

efi

nit

ion

of

co

ng

ruen

t seg

men

ts.

By t

he R

efl

exiv

e

Pro

pert

y −

−

BD

# −

−

BD

. S

o △

AB

D #

△C

BD

by S

SS

.

We a

re g

iven

−−

AB

# −

−

CB

an

d ∠

A #

∠C

. −

−

BD

# −

−

BD

by t

he R

efl

exiv

e

Pro

pert

y.

Sin

ce S

SA

can

no

t b

e u

sed

to

pro

ve t

hat

tria

ng

les a

re

co

ng

ruen

t, w

e c

an

no

t say w

heth

er

△A

BD

# △

CB

D.

∠Q

∠T

Given

QR

||TU

Given

Mid

poin

t Theo

rem

Alt. In

t. &

are

.

QS TS

S is

the

mid

po

int

of QT

.

QSR

TSU

AS

A

∠QSR

∠TSU

Ver

tica

l & a

re

.


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Ch

ap

ter

4

35

G

len

co

e G

eo

me

try

1

. D

OO

R S

TO

PS

T

wo d

oor

stop

s h

ave

cross

-sect

ion

s th

at

are

rig

ht

tria

ngle

s.

Th

ey b

oth

have a

20 a

ngle

an

d t

he

len

gth

of

the s

ide b

etw

een

th

e 9

0 a

nd

20 a

ngle

s are

equ

al.

Are

th

e c

ross

-

sect

ion

s co

ngru

en

t? E

xp

lain

.

2

. M

AP

PIN

G T

wo p

eop

le d

eci

de t

o t

ak

e a

walk

. O

ne p

ers

on

is

in B

om

bay a

nd

th

e

oth

er

is i

n M

ilw

au

kee.

Th

ey s

tart

by

walk

ing s

traig

ht

for

1 k

ilom

ete

r. T

hen

they b

oth

tu

rn r

igh

t at

an

an

gle

of

110,

an

d c

on

tin

ue t

o w

alk

str

aig

ht

again

.

Aft

er

a w

hil

e,

they b

oth

tu

rn r

igh

t

again

, bu

t th

is t

ime a

t an

an

gle

of

120.

Th

ey e

ach

walk

str

aig

ht

for

a w

hil

e i

n

this

new

dir

ect

ion

un

til

they e

nd

up

wh

ere

th

ey s

tart

ed

. E

ach

pers

on

walk

ed

in a

tri

an

gu

lar

path

at

their

loca

tion

.

Are

th

ese

tw

o t

rian

gle

s co

ngru

en

t?

Exp

lain

.

3

. C

ON

ST

RU

CT

ION

T

he r

ooft

op

of

An

gelo

’s h

ou

se c

reate

s an

equ

ilate

ral

tria

ngle

wit

h t

he a

ttic

flo

or.

An

gelo

wan

ts t

o d

ivid

e h

is a

ttic

in

to 2

equ

al

part

s. H

e t

hin

ks

he s

hou

ld d

ivid

e i

t by

pla

cin

g a

wall

fro

m t

he c

en

ter

of

the r

oof

to t

he f

loor

at

a 9

0 a

ngle

. If

An

gelo

does

this

, th

en

each

sect

ion

wil

l sh

are

a s

ide

an

d h

ave c

orr

esp

on

din

g 9

0 a

ngle

s. W

hat

els

e m

ust

be e

xp

lain

ed

to p

rove t

hat

the

two t

rian

gu

lar

sect

ion

s are

con

gru

en

t?

90˚

90˚

4

. LO

GIC

W

hen

Caro

lyn

fin

ish

ed

her

mu

sica

l tr

ian

gle

cla

ss,

her

teach

er

gave

each

stu

den

t in

th

e c

lass

a c

ert

ific

ate

in

the s

hap

e o

f a g

old

en

tri

an

gle

. E

ach

stu

den

t re

ceiv

ed

a d

iffe

ren

t sh

ap

ed

tria

ngle

. C

aro

lyn

lost

her

tria

ngle

on

her

way h

om

e.

Late

r sh

e s

aw

part

of

a

gold

en

tri

an

gle

un

der

a g

rate

.

Is e

nou

gh

of

the t

rian

gle

vis

ible

to a

llow

Caro

lyn

to d

ete

rmin

e t

hat

the t

rian

gle

is

ind

eed

hers

? E

xp

lain

.

5

. P

AR

K M

AIN

TE

NA

NC

E P

ark

off

icia

ls

need

a t

rian

gu

lar

tarp

to c

over

a f

ield

shap

ed

lik

e a

n e

qu

ilate

ral

tria

ngle

200

feet

on

a s

ide.

a.

Su

pp

ose

you

kn

ow

th

at

a t

rian

gu

lar

tarp

has

two 6

0 a

ngle

s an

d o

ne s

ide

of

len

gth

200 f

eet.

Wil

l th

is t

arp

cover

the f

ield

? E

xp

lain

.

b.

Su

pp

ose

you

kn

ow

th

at

a t

rian

gu

lar

tarp

has

thre

e 6

0 a

ngle

s. W

ill

this

tarp

nece

ssari

ly c

over

the f

ield

?

Exp

lain

.

4-5

Wo

rld

Pro

ble

m P

rati

ce

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

yes,

by A

SA

Sam

ple

an

sw

er:

Becau

se t

he

ori

gin

al

tria

ng

le w

as e

qu

ilate

ral,

on

e m

ore

pair

of

an

gle

s i

s

co

ng

ruen

t. T

he t

rian

gle

s a

re

co

ng

ruen

t b

y A

AS

.

No

; th

ere

are

eq

uia

ng

ula

r

tria

ng

les o

f all s

izes.

yes,

by A

AS

Yes,

if s

he k

no

ws t

he m

easu

res

of

her

lost

tria

ng

le.

Sin

ce t

wo

an

gle

s a

nd

th

e i

nclu

ded

sid

e

betw

een

th

em

can

be s

een

, sh

e

can

dete

rmin

e i

f it

is c

on

gru

en

t

by t

he A

SA

Th

eo

rem

.

Yes,

becau

se t

wo

kn

ow

n

an

gle

s a

re 6

0,

the t

hir

d m

ust

be a

lso

. A

SA

or

AA

S c

an

be

used

to

sh

ow

th

at

it w

ill

co

ver.



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

4

36

G

len

co

e G

eo

me

try

En

rich

men

t4-5

Wh

at

Ab

ou

t A

AA

to

Pro

ve T

rian

gle

Co

ng

ruen

ce?

You

have l

earn

ed

abou

t th

e S

SS

an

d A

SA

post

ula

tes

to p

rove t

hat

two t

rian

gle

s are

con

gru

en

t. D

o y

ou

th

ink

th

at

tria

ngle

s ca

n b

e p

roven

con

gru

en

t by A

AA

?

Invest

igate

th

e A

AS

th

eore

m.

AA

S r

ep

rese

nts

th

e c

on

ject

ure

that

two t

rian

gle

s are

con

gru

en

t if

th

ey h

ave t

wo p

air

s of

corr

esp

on

din

g a

ngle

s co

ngru

en

t an

d c

orr

esp

on

din

g

non

-in

clu

ded

sid

es

con

gru

en

t. F

or

exam

ple

, in

th

e

tria

ngle

s at

the r

igh

t, ∠

A !

∠Y

, ∠

B !

∠X

, an

d −

−

AC

! −

−

YZ

.

1

. R

efe

r to

th

e d

raw

ing a

bove.

Wh

at

is t

he m

easu

re o

f th

e m

issi

ng a

ngle

in

th

e f

irst

tria

ngle

? W

hat

is t

he m

easu

re o

f th

e m

issi

ng a

ngle

in

th

e s

eco

nd

tri

an

gle

?

2

. W

ith

th

e i

nfo

rmati

on

fro

m E

xerc

ise 1

, w

hat

post

ula

te c

an

you

now

use

to p

rove t

hat

the

two t

rian

gle

s are

con

gru

en

t?

Now

in

vest

igate

th

e A

AA

con

ject

ure

. A

AA

rep

rese

nts

th

e c

on

ject

ure

th

at

two t

rian

gle

s are

con

gru

en

t if

th

ey h

ave t

hre

e p

air

s of

corr

esp

on

din

g a

ngle

s co

ngru

en

t. T

he t

rian

gle

s belo

w

dem

on

stra

te A

AA

.

25˚

25˚

70˚

70˚

85˚

85˚

3

. A

re t

he t

rian

gle

s above c

on

gru

en

t? E

xp

lain

.

4

. C

an

you

dra

w t

wo t

rian

gle

s th

at

have a

ll t

hre

e a

ngle

s co

ngru

en

t th

at

are

not

con

gru

en

t?

Can

AA

A b

e u

sed

to p

rove t

hat

two t

rian

gle

s are

con

gru

en

t?

5

. W

hat

chara

cteri

stic

s d

o t

hese

tw

o t

rian

gle

s d

raw

n b

y A

AA

seem

to p

oss

ess

?

89˚

89˚

66˚

66˚

A

BC

5 in

5 in

X

Y

Z

25;

25

AS

A

No

, th

e s

ides h

ave d

iffe

ren

t le

ng

ths.

yes;

no

Th

ey h

ave t

he s

am

e s

hap

e,

bu

t d

iffe

ren

t siz

es.

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Ch

ap

ter

4

37

G

len

co

e G

eo

me

try

Exerc

ises

ALG

EB

RA

F

ind

th

e v

alu

e o

f e

ach

va

ria

ble

.

1

.

R

P

Q2

x°

40

°

2.

S

V

T3

x-

6

2x

+ 6

3.

W

YZ

3x°

4

.

2x°

(6x

+6)°

5.

5x°

6.

RS

T 3x°

x°

7

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

G

ive

n:

∠1 !

∠2

P

ro

ve

: −

−

AB

! −

−−

CB

Pro

of:

Sta

tem

en

ts

Reaso

ns

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Pro

pe

rtie

s o

f Is

osc

ele

s T

ria

ng

les

An

iso

sce

les t

ria

ng

le h

as

two c

on

gru

en

t si

des

call

ed

th

e l

eg

s.

Th

e a

ngle

form

ed

by t

he l

egs

is c

all

ed

th

e v

erte

x a

ng

le.

Th

e o

ther

two

an

gle

s are

call

ed

ba

se

an

gle

s.

You

can

pro

ve a

th

eore

m a

nd

its

con

vers

e a

bou

t is

osc

ele

s tr

ian

gle

s.

•

If

tw

o s

ides

of

a t

rian

gle

are

con

gru

en

t, t

hen

th

e a

ngle

s op

posi

te

those

sid

es

are

con

gru

en

t. (

Iso

sce

les T

ria

ng

le T

he

ore

m)

•

If

tw

o a

ngle

s of

a t

rian

gle

are

con

gru

en

t, t

hen

th

e s

ides

op

posi

te

those

an

gle

s are

con

gru

en

t. (

Co

nv

erse

of

Iso

sce

les T

ria

ng

le

Th

eo

re

m)

If −

−

AB

! −

−−

CB

, th

en ∠

A !

∠C

.

If ∠

A !

∠C

, th

en −

−

AB

! −

−−

CB

.

A

B

C

F

ind

x,

giv

en

−−

BC

# −

−

BA

.

B

AC( 4

x+

5) °

( 5x

- 1

0) °

BC

= B

A,

so

m∠

A =

m∠

C

Isos.

Triangle

Theore

m

5x -

10 =

4x +

5

Substitu

tion

x -

10 =

5

subtr

act

4x f

rom

each s

ide.

x =

15

Add 1

0 t

o e

ach s

ide.

Exam

ple

1Exam

ple

2

Fin

d x

.

RT

S 3x

- 1

3

2x

m∠

S =

m∠

T,

so

SR

= T

R

Convers

e o

f Is

os. △

Thm

.

3x -

13 =

2x

Substitu

tion

3x =

2x +

13

A

dd 1

3 t

o e

ach s

ide.

x =

13

Subtr

act

2x f

rom

each s

ide.

B

AC

D

E

13

2

4-6 1

. ∠

1 #

∠2

1. G

iven

2.

∠2 #

∠3

2. V

ert

ical

an

gle

s a

re c

on

gru

en

t.3.

∠1 #

∠3

3. T

ran

sit

ive P

rop

ert

y o

f #

4.

−−

AB

# −

−

CB

4. C

on

vers

e o

f Is

os.

Tri

an

gle

Th

m.

12

936

35

12

15



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

38

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Pro

pe

rtie

s o

f E

qu

ila

tera

l T

ria

ng

les

An

eq

uil

ate

ra

l tr

ian

gle

has

thre

e c

on

gru

en

t si

des.

Th

e I

sosc

ele

s T

rian

gle

Th

eore

m l

ead

s to

tw

o c

oro

llari

es

abou

t equ

ilate

ral

tria

ngle

s.

1.

A t

ria

ng

le is e

qu

ilate

ral if a

nd

on

ly if

it is e

qu

ian

gu

lar.

2.

Ea

ch

an

gle

of

an

eq

uila

tera

l tr

ian

gle

me

asu

res 6

0°.

P

ro

ve

th

at

if a

lin

e i

s p

ara

lle

l to

on

e s

ide

of

an

eq

uil

ate

ra

l tr

ian

gle

, th

en

it

form

s a

no

the

r e

qu

ila

tera

l tr

ian

gle

.

Pro

of:

Sta

tem

en

ts

Re

aso

ns

1.

△A

BC

is

equ

ilate

ral;

−−−

PQ

‖ −−

− B

C .

1.

Giv

en

2.

m∠

A =

m∠

B =

m∠

C =

60

2.

Each

∠ o

f an

equ

ilate

ral

△ m

easu

res

60

°.

3.

∠1 &

∠B

, ∠

2 &

∠C

3.

If ‖

lin

es,

th

en

corr

es.

' a

re &

.

4.

m∠

1 =

60,

m∠

2 =

60

4.

Su

bst

itu

tion

5.

△A

PQ

is

equ

ilate

ral.

5

. If

a △

is

equ

ian

gu

lar,

th

en

it

is e

qu

ilate

ral.

Exerc

ises

ALG

EB

RA

F

ind

th

e v

alu

e o

f e

ach

va

ria

ble

.

1

. D

FE

6x°

2.

G

JH

6x

- 5

5x

3.

3x°

4

.

4x

40

60

°

5.

X

ZY

4x

- 4

3x

+ 8

60

°

6.

4x°

60°

7

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

G

ive

n:

△A

BC

is

equ

ilate

ral;

∠1 &

∠2.

P

ro

ve

: ∠

AD

B &

∠C

DB

P

roo

f:

S

tate

men

ts

Reaso

ns

A D C

B1 2

A

B

PQ

C

12

4-6 Exam

ple

1

. △

AB

C i

s e

qu

ilate

ral.

1.

Giv

en

2

. −

−

AB

$ −

−

CB

; ∠

A $

∠C

2.

An

eq

uilate

ral

△ h

as $

sid

es a

nd

$ a

ng

les.

3

. ∠

1 $

∠2

3.

Giv

en

4

. △

AB

D $

△C

BD

4.

AS

A P

ostu

late

5

. ∠

AD

B $

∠C

DB

5.

CP

CT

C

10

12

15

10

520

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Ch

ap

ter

4

39

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

D

C

B

AE

Re

fer t

o t

he

fig

ure

at

the

rig

ht.

1

. If

−−

AC

& −−

− A

D ,

nam

e t

wo c

on

gru

en

t an

gle

s.

2

. If

−−−

BE

& −−

− B

C ,

nam

e t

wo c

on

gru

en

t an

gle

s.

3

. If

∠E

BA

& ∠

EA

B,

nam

e t

wo c

on

gru

en

t se

gm

en

ts.

4

. If

∠C

ED

& ∠

CD

E,

nam

e t

wo c

on

gru

en

t se

gm

en

ts.

Fin

d e

ach

me

asu

re

.

5

. m

∠A

BC

6. m

∠E

DF

60

°

40°

ALG

EB

RA

F

ind

th

e v

alu

e o

f e

ach

va

ria

ble

.

7

.

2x

+4

3x

-10

8.

(2x

+3

)°

9

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

G

ive

n:

−−

−

CD

& −

−−

CG

−−

−

DE

& −−

− G

F

P

ro

ve

: −−

− C

E &

−−

CF

D E F G

C

4-6 ∠

AC

D $

∠C

DA

∠B

EC

$ ∠

BC

E

−−

EB

$ −

−

EA

−−

CE

$

−−

CD

P

roo

f:

S

tate

men

ts

Reaso

ns

1

. −

−

CD

$ −

−

CG

1.

Giv

en

3

. −

−

DE

$ −

−

GF

3.

Giv

en

4

. △

CD

E $

△C

GF

4.

SA

S

5

. −

−

CE

$ −

−

CF

5.

CP

CT

C

2

. ∠

D $

∠G

2

. If

2 s

ides o

f a △

are

$,

then

th

e &

op

po

sit

e

tho

se s

ides a

re $

.

60

70

14

21



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

40

G

len

co

e G

eo

me

try

Practi

ce

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Lin

co

ln H

aw

ks

E DBC

A12

3 4

UR

TV

S

Re

fer t

o t

he

fig

ure

at

the

rig

ht.

1

. If

−−−

RV

" −

−

RT

, n

am

e t

wo c

on

gru

en

t an

gle

s.

∠R

TV

! ∠

RV

T

2

. If

−−

RS

" −

−

SV

, n

am

e t

wo c

on

gru

en

t an

gle

s.

∠S

VR

! ∠

SR

V

3

. If

∠S

RT

" ∠

ST

R,

nam

e t

wo c

on

gru

en

t se

gm

en

ts. −

−

ST !

−−

SR

4

. If

∠S

TV

" ∠

SV

T,

nam

e t

wo c

on

gru

en

t se

gm

en

ts.

−−

ST

! −

−

SV

Fin

d e

ach

me

asu

re

.

5

. m

∠K

ML

6. m

∠H

MG

7. m

∠G

HM

8. If

m∠

HJ

M =

145,

fin

d m

∠M

HJ

.

9. If

m∠

G =

67,

fin

d m

∠G

HM

.

10

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

G

ive

n:

−−

−

DE

‖ −−

− B

C

∠1 "

∠2

P

ro

ve

: −

−

AB

" −

−

AC

11

. S

PO

RT

S A

pen

nan

t fo

r th

e s

port

s te

am

s at

Lin

coln

Hig

h

Sch

ool

is i

n t

he s

hap

e o

f an

iso

scele

s tr

ian

gle

. If

th

e m

easu

re

of

the v

ert

ex a

ngle

is

18,

fin

d t

he m

easu

re o

f each

base

an

gle

.

4-6

50°

81,

81

P

roo

f:

S

tate

men

ts

R

easo

ns

1. G

iven

2. C

orr

. $

are

!.

3. G

iven

4. C

on

gru

en

ce o

f $

is t

ran

sit

ive.

5. I

f 2 $

of

a △

are

!,

then

th

e s

ide

o

pp

osit

e t

ho

se s

ides a

re !

1. −

−

DE

‖

−−

BC

2. ∠

1 !

∠4

∠

2 !

∠3

3. ∠

1 !

∠2

4. ∠

3 !

∠4

5. −

−

AB

! −

−

AC

60

70

40

17.5

46

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Ch

ap

ter

4

41

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

1

1

. T

RIA

NG

LE

S A

t an

art

su

pp

ly s

tore

, tw

o

dif

fere

nt

tria

ngu

lar

rule

rs a

re a

vail

able

.

On

e h

as

an

gle

s 45°,

45°,

an

d 9

0°.

Th

e

oth

er

has

an

gle

s 30°,

60°,

an

d 9

0°.

Wh

ich

tri

an

gle

is

isosc

ele

s?

2. R

ULE

RS

A

fold

able

ru

ler

has

two h

inges

that

div

ide t

he r

ule

r in

to t

hir

ds.

If

the

en

ds

are

fold

ed

up

un

til

they t

ou

ch,

wh

at

kin

d o

f tr

ian

gle

resu

lts?

3

. H

EX

AG

ON

S Ju

an

ita p

lace

d o

ne e

nd

of

each

of

6 b

lack

sti

cks

at

a c

om

mon

poin

t

an

d t

hen

sp

ace

d t

he o

ther

en

ds

even

ly

aro

un

d t

hat

poin

t. S

he c

on

nect

ed

th

e

free e

nd

s of

the s

tick

s w

ith

lin

es.

Th

e

resu

lt w

as

a r

egu

lar

hexagon

. T

his

con

stru

ctio

n s

how

s th

at

a r

egu

lar

hexagon

can

be m

ad

e f

rom

six

con

gru

en

t

tria

ngle

s. C

lass

ify t

hese

tri

an

gle

s.

Exp

lain

.

4

. P

AT

HS

A

marb

le p

ath

is

con

stru

cted

ou

t of

severa

l co

ngru

en

t is

osc

ele

s

tria

ngle

s. T

he v

ert

ex a

ngle

s are

all

20.

Wh

at

is t

he m

easu

re o

f an

gle

1 i

n t

he

figu

re?

5. B

RID

GE

S E

very

day,

cars

dri

ve t

hro

ugh

isosc

ele

s tr

ian

gle

s w

hen

th

ey g

o o

ver

the

Leon

ard

Zak

im B

rid

ge i

n B

ost

on

. T

he

ten

-lan

e r

oad

way f

orm

s th

e b

ase

s of

the

tria

ngle

s.

a.

Th

e a

ngle

labele

d A

in

th

e p

ictu

re

has

a m

easu

re o

f 67.

Wh

at

is t

he

measu

re o

f ∠

B?

b.

Wh

at

is t

he m

easu

re o

f ∠

C?

c.

Nam

e t

he t

wo c

on

gru

en

t si

des.

the 4

5°-

45

°- 9

0°

tria

ng

le

A

B

C

−−

AB

! −

−

BC

4-6 e

qu

ilate

ral

tria

ng

le

Sam

ple

an

sw

er:

Sin

ce t

wo

sid

es

are

co

ng

ruen

t, t

hey a

re i

so

scele

s.

Becau

se t

he s

ticks a

re s

paced

even

ly,

the v

ert

ex a

ng

les (

at

the

cen

ter)

are

all 3

60

−

6

or

60.

Th

us t

he

base a

ng

les m

ust

be

(18

0 -

60

) −

6

or

60.

Sin

ce a

ll a

ng

les o

f an

y o

ne

tria

ng

le i

n t

he f

igu

re a

re 6

0,

all

sid

es a

re e

qu

al

an

d t

he t

rian

gle

s

are

eq

uilate

ral.

80 4

6

67



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

42

G

len

co

e G

eo

me

try

En

rich

men

t

Tri

an

gle

Ch

allen

ges

Som

e p

roble

ms

incl

ud

e d

iagra

ms.

If

you

are

not

sure

how

to s

olv

e t

he p

roble

m,

begin

by

usi

ng t

he g

iven

in

form

ati

on

. F

ind

th

e m

easu

res

of

as

man

y a

ngle

s as

you

can

, w

riti

ng e

ach

m

easu

re o

n t

he d

iagra

m.

Th

is m

ay g

ive y

ou

m

ore

clu

es

to t

he s

olu

tion

.

1

. G

ive

n:

BE

= B

F,

∠B

FG

! ∠

BE

F !

∠

BE

D,

m∠

BF

E =

82 a

nd

A

BF

G a

nd

BC

DE

each

have

op

posi

te s

ides

para

llel

an

d

con

gru

en

t.

Fin

d m

∠A

BC

. A

GD

FE

CB

3

. G

ive

n:

m∠

UZ

Y =

90,

m∠

ZW

X =

45,

△Y

ZU

! △

VW

X,

UV

XY

is

a

squ

are

(all

sid

es

con

gru

en

t, a

ll

an

gle

s ri

gh

t an

gle

s).

Fin

d m

∠W

ZY

.

2

. G

ive

n:

AC

= A

D,

an

d −

−

AB

⊥

−−

−

BD

, m

∠D

AC

= 4

4 a

nd

−−

−

CE

bis

ect

s ∠

AC

D.

Fin

d m

∠D

EC

. A

DC

B

E

4

. G

ive

n:

m∠

N =

120,

−−

−

JN

! −

−−

MN

, △

JN

M !

△K

LM

.

Fin

d m

∠J

KM

.

J

K

L

MN

UV

W

XY

Z

4-6

15

78

148

45

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Ch

ap

ter

4

43

G

len

co

e G

eo

me

try

Ide

nti

fy C

on

gru

en

ce T

ran

sfo

rma

tio

ns

A c

on

gru

en

ce

tra

nsfo

rm

ati

on

is

a

tran

sform

ati

on

wh

ere

th

e o

rigin

al

figu

re,

or

pre

image,

an

d t

he t

ran

sform

ed

fig

ure

, or

image,

figu

re a

re s

till

con

gru

en

t. T

he t

hre

e t

yp

es

of

con

gru

en

ce t

ran

sform

ati

on

s are

re

fle

cti

on

(or

flip

), t

ra

nsla

tio

n (

or

slid

e),

an

d r

ota

tio

n (

or

turn

).

Id

en

tify

th

e t

yp

e o

f co

ng

ru

en

ce

tra

nsfo

rm

ati

on

sh

ow

n a

s a

reflection

, translation

, o

r rotation

.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re t

he

sa

me

dis

tan

ce

fro

m t

he

y-a

xis

.

Th

is is a

re

fle

ctio

n.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re in

the

sa

me

po

sitio

n,

just

two

un

its

do

wn

. T

his

is a

tra

nsla

tio

n.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re t

he

sa

me

dis

tan

ce

fro

m t

he

orig

in.

Th

e a

ng

les f

orm

ed

by e

ach

pa

ir o

f

co

rre

sp

on

din

g p

oin

ts a

nd

th

e

orig

in a

re c

on

gru

en

t. T

his

is a

rota

tio

n.

Exerc

ises

Ide

nti

fy t

he

ty

pe

of

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

sh

ow

n a

s a

reflection

, translation

,

or rotation

.

1

.

x

y

2.

x

y

3.

x

y

4

.

x

y

5.

x

y

6.

x

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

Exam

ple

4-7

refl

ecti

on

tran

sla

tio

nro

tati

on

ro

tati

on

refl

ecti

on

or t

ran

sla

tio

nrefl

ecti

on



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

44

G

len

co

e G

eo

me

try

Ve

rify

Co

ng

rue

nce

Y

ou

can

veri

fy t

hat

refl

ect

ion

s, t

ran

slati

on

s, a

nd

rota

tion

s of

tria

ngle

s p

rod

uce

con

gru

en

t tr

ian

gle

s u

sin

g S

SS

.

V

erif

y c

on

gru

en

ce

aft

er a

tra

nsfo

rm

ati

on

.

△W

XY

wit

h v

ert

ices

W(3

, -

7),

X(6

, -

7),

Y(6

, -

2)

is a

tra

nsf

orm

ati

on

of

△R

ST

wit

h v

ert

ices

R(2

, 0),

S(5

, 0),

an

d T

(5,

5).

Gra

ph

th

e

ori

gin

al

figu

re a

nd

its

im

age.

Iden

tify

th

e t

ran

sform

ati

on

an

d

veri

fy t

hat

it i

s a c

on

gru

en

ce t

ran

sform

ati

on

.

Gra

ph

each

fig

ure

. U

se t

he D

ista

nce

Form

ula

to s

how

th

e s

ides

are

con

gru

en

t an

d t

he t

rian

gle

s are

con

gru

en

t by S

SS

.

RS

= 3

, S

T =

5,

TR

= √

##

##

##

#

( 5

- 2

) 2 +

(5 -

0) 2

= √

##

34

WX

= 3

, X

Y =

5,

YW

= √

##

##

##

##

#

( 6

- 3

) 2 +

(-

2-

(-

7) )

2 =

√ #

#

34

−−

RS

% −−

− W

X ,

−−

ST

, %

−−

XY

, −

−

TR

% −−

− Y

W

By S

SS

, △

RS

T %

△W

XY

.

Exerc

ises

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

pa

ir o

f tr

ian

gle

s w

ith

th

e g

ive

n v

erti

ce

s.

Th

en

id

en

tify

th

e t

ra

nsfo

rm

ati

on

, a

nd

ve

rif

y t

ha

t it

is a

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

.

1

. A

(−3,

1),

B(−

1,

1),

C(−

1,

4)

2. Q

(−3,

0),

R(−

2,

2),

S(−

1,

0)

D

(3,

1),

E(1

, 1),

F(1

, 4)

T

(2,

−4),

U(3

, −

2),

V(4

, −

4)

x

y

x

y

4-7

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

Exam

ple

x

y

refl

ecti

on

; A

B =

2,

BC

= 3

,

CA

= √

""

13 ,

DE

= 2

, E

F =

3,

FD

= √

""

13 t

hu

s −

−

AB

$ −

−

DE

,

−−

BC

$ −

−

EF ,

−−

AC

$ −

−

DF B

y S

SS

,

△A

BC

$ △

DE

F

tran

sla

tio

n;

QR

= √

"

5 ,

RS

= √

"

5 ,

SQ

= 2

, TU

= √

"

5 ,

UV

= √

"

5 ,

VT

= 2

th

us,

−−

QR

$ −

−

TU

, −

−

RS

$ −

−

UV

,

−−

SQ

$ −

−

VT .

By S

SS

, △

QR

S $

△TU

V.

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Ch

ap

ter

4

45

G

len

co

e G

eo

me

try

4-7

Ide

nti

fy t

he

ty

pe

of

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

sh

ow

n a

s a

refl

ecti

on

, tr

an

sla

tio

n,

or r

ota

tio

n.

1

.

x

y

2.

x

y

r

ota

tio

n

re

flecti

on

CO

OR

DIN

AT

E G

EO

ME

TR

Y Id

en

tify

ea

ch

tra

nsfo

rm

ati

on

an

d v

erif

y t

ha

t it

is a

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

.

3

.

x

y

4.

x

y

T

ran

sla

tio

n;

AB

= √

"

5 ,

BC

= √

""

13 ,

CA

= 4

, D

E =

√ "

5 ,

EF =

√ "

"

13 ,

FD

= 4

. △

AB

C $

△D

EF b

y S

SS

.

R

efl

ecti

on

; Q

R =

5,

RS

= 4

,

SQ

= 3

, TU

= 5

, U

V =

4,

VT =

3.

△Q

RS

$ △

TU

V b

y S

SS

.

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

pa

ir o

f tr

ian

gle

s w

ith

th

e g

ive

n v

erti

ce

s.

Th

en

, id

en

tify

th

e t

ra

nsfo

rm

ati

on

, a

nd

ve

rif

y t

ha

t it

is a

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

.

5

. A

(1,

3),

B(1

, 1),

C(4

, 1);

6. J

(−3,

0),

K(−

2,

4),

L(−

1,

0);

D

(3,

-1),

E(1

, -

1),

F(1

, -

4)

Q

(2,

−4),

R(3

, 0),

S(4

, −

4)

x

y

x

y

4-7

Sk

ills

Pra

ctic

e

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

Tra

nsla

tio

n;

JK

= √

""

17 ,

KL =

√ "

"

17 ,

LJ =

2,

QR

= √

""

17 ,

RS

= √

""

17 ,

SQ

= 2

. △

JK

L $

△Q

RS

by S

SS

.

rota

tio

n;

AB

= 2

, B

C =

3,

CA

= √

##

13 ,

DE

= 2

, E

F =

3,

FD

= √

##

13

△A

BC

$ △

DE

F b

y S

SS

.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

46

G

len

co

e G

eo

me

try

Ide

nti

fy t

he

ty

pe

of

co

ng

ru

en

ce

tra

nsfo

rm

ati

on

sh

ow

n a

s a

reflection

, translation

,

or rotation

.

1

.

x

y

2.

x

y

3

. Id

en

tify

th

e t

yp

e o

f co

ngru

en

ce t

ran

sform

ati

on

sh

ow

n a

s a reflection

, translation

, or rotation

, an

d v

eri

fy t

hat

it

is a

con

gru

en

ce t

ran

sform

ati

on

.

R

ota

tio

n;

AB

= √

""

13 ,

BC

= 3

√ "

2 ,

CA

= 5

, AD

= √

""

13

D

E =

3 √

"

2 ,

EA

= 5

, △

AB

C $

△A

DE

by S

SS

.

4

. ∆ABC

has

vert

ices A

(−4,

2),

B(−

2,

0),

C(−

4,

-2).

∆DEF

has

vert

ices D

(4,

2),

E(2

, 0),

F(4

, −

2).

Gra

ph

th

e o

rigin

al

figu

re

an

d i

ts i

mage.

Th

en

id

en

tify

th

e t

ran

sform

ati

on

an

d v

eri

fy

that

it i

s a c

on

gru

en

ce t

ran

sform

ati

on

.

R

efl

ecti

on

; A

B =

2 √

"

2 ,

BC

= 2

√ "

2 ,

CA

= 4

, D

E =

2 √

"

2 ,

EF =

2 √

"

2 ,

FD

= 4

. △

AB

C $

△D

EF b

y S

SS

.

5

. S

TE

NC

ILS

C

arl

y i

s p

lan

nin

g o

n s

ten

cili

ng a

patt

ern

of

flow

ers

alo

ng t

he c

eil

ing i

n h

er

bed

room

. S

he w

an

ts a

ll o

f th

e f

low

ers

to l

ook

exact

ly t

he s

am

e.

Wh

at

typ

e o

f co

ngru

en

ce

tran

sform

ati

on

sh

ou

ld s

he u

se?

Wh

y?

T

ran

sla

tio

n;

A t

ran

sla

tio

n c

on

tain

s c

on

gru

en

t sh

ap

es w

ith

th

e s

am

e

ori

en

tati

on

.

4-7

Practi

ce

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

x

y

tran

sla

tio

nro

tati

on

x

y

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Ch

ap

ter

4

47

G

len

co

e G

eo

me

try

4-7

Wo

rd

Pro

ble

m P

racti

ce

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

1

. Q

UIL

TIN

G Y

ou

an

d y

ou

r tw

o f

rien

ds

vis

it a

cra

ft f

air

an

d n

oti

ce a

qu

ilt,

part

of

wh

ose

patt

ern

is

show

n b

elo

w.

You

r fi

rst

frie

nd

says

the p

att

ern

is

rep

eate

d

by t

ran

slati

on

. Y

ou

r se

con

d f

rien

d s

ays

the p

att

ern

is

rep

eate

d b

y r

ota

tion

. W

ho

is c

orr

ect

?

B

oth

fri

en

ds a

re c

orr

ect.

Th

e

tria

ng

les a

re b

ein

g r

ota

ted

an

d

the h

exag

on

s a

re b

ein

g

tran

sla

ted

.

2

. R

EC

YC

LIN

G T

he i

nte

rnati

on

al

sym

bol

for

recy

clin

g i

s sh

ow

n b

elo

w.

Wh

at

typ

e

of

con

gru

en

ce t

ran

sform

ati

on

does

this

il

lust

rate

?

R

ota

tio

n

3

. A

NA

TO

MY

C

arl

noti

ces

that

wh

en

he

hold

s h

is h

an

ds

palm

up

in

fro

nt

of

him

, th

ey l

ook

th

e s

am

e.

Wh

at

typ

e o

f co

ngru

en

ce t

ran

sform

ati

on

does

this

il

lust

rate

?

R

efl

ecti

on

4

. S

TA

MP

S A

sh

eet

of

post

age s

tam

ps

con

tain

s 20 s

tam

ps

in 4

row

s of

5

iden

tica

l st

am

ps

each

. W

hat

typ

e o

f co

ngru

en

ce t

ran

sform

ati

on

does

this

il

lust

rate

?

T

ran

sla

tio

n

5

. M

OS

AIC

S José

cu

t re

ctan

gle

s ou

t of

tiss

ue p

ap

er

to c

reate

a p

att

ern

. H

e c

ut

ou

t fo

ur

con

gru

en

t blu

e r

ect

an

gle

s an

d

then

cu

t ou

t fo

ur

con

gru

en

t re

d

rect

an

gle

s th

at

were

sli

gh

tly s

mall

er

to

create

th

e p

att

ern

sh

ow

n b

elo

w.

a.

Wh

ich

con

gru

en

ce t

ran

sform

ati

on

d

oes

this

patt

ern

ill

ust

rate

?

Ro

tati

on

b.

If t

he w

hole

squ

are

were

rota

ted

90,

wou

ld t

he t

ran

sform

ati

on

sti

ll b

e t

he

sam

e?

Yes



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

48

G

len

co

e G

eo

me

try

4-7

En

rich

men

t

Tra

nsfo

rmati

on

s i

n T

he C

oo

rdin

ate

Pla

ne

Th

e f

oll

ow

ing s

tate

men

t te

lls

on

e w

ay t

o r

ela

te p

oin

ts t

o

corr

esp

on

din

g t

ran

slate

d p

oin

ts i

n t

he c

oord

inate

pla

ne.

(x,

y)

→ (

x +

6,

y -

3)

Th

is c

an

be r

ead

, “T

he p

oin

t w

ith

coord

inate

s (x

, y)

is

map

ped

to t

he p

oin

t w

ith

coord

inate

s (x

+ 6

, y -

3).

” W

ith

th

is t

ran

sform

ati

on

, fo

r exam

ple

, (3

, 5)

is m

ap

ped

or

moved

to (

3 +

6,

5 -

3)

or

(9,

2).

Th

e f

igu

re s

how

s h

ow

th

e

tria

ngle

AB

C i

s m

ap

ped

to t

rian

gle

XY

Z.

1

. D

oes

the t

ran

sform

ati

on

above a

pp

ear

to b

e a

con

gru

en

ce t

ran

sform

ati

on

? E

xp

lain

you

r an

swer.

Dra

w t

he

tra

nsfo

rm

ati

on

im

ag

e f

or e

ach

fig

ure

. T

he

n t

ell

wh

eth

er t

he

tra

nsfo

rm

ati

on

is o

r i

s n

ot

a c

on

gru

en

ce

tra

nsfo

rm

ati

on

.

2

. (x

, y)

→ (

x -

4,

y)

3. (x

, y)

→ (

x +

5,

y +

4)

4

. (x

, y)

→ (

-x ,

-y)

5. (x

, y)

→ (

- 1

−

2 x

, y)

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C

X

ZY

O

(x,

y)#

(x

+ 6

, y

- 3

)

yes

yes

yes

no

Yes;

the t

ran

sfo

rmati

on

slid

es t

he f

igu

re t

o t

he l

ow

er

rig

ht

wit

ho

ut

ch

an

gin

g i

ts s

ize o

r sh

ap

e.

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Ch

ap

ter

4

49

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Po

siti

on

an

d L

ab

el

Tri

an

gle

s A

coord

inate

pro

of

use

s p

oin

ts,

dis

tan

ces,

an

d s

lop

es

to

pro

ve g

eom

etr

ic p

rop

ert

ies.

Th

e f

irst

ste

p i

n w

riti

ng a

coord

inate

pro

of

is t

o p

lace

a f

igu

re

on

th

e c

oord

inate

pla

ne a

nd

label

the v

ert

ices.

Use

th

e f

oll

ow

ing g

uid

eli

nes.

1.

Use

th

e o

rig

in a

s a

ve

rte

x o

r ce

nte

r o

f th

e f

igu

re.

2.

Pla

ce

at

lea

st

on

e s

ide

of

the

po

lyg

on

on

an

axis

.

3.

Ke

ep

th

e f

igu

re in

th

e f

irst

qu

ad

ran

t if p

ossib

le.

4.

Use

co

ord

ina

tes t

ha

t m

ake

co

mp

uta

tio

ns a

s s

imp

le a

s p

ossib

le.

Po

sit

ion

an

iso

sce

les t

ria

ng

le o

n t

he

co

ord

ina

te p

lan

e s

o t

ha

t it

s s

ide

s a

re

a u

nit

s l

on

g a

nd

on

e s

ide

is o

n t

he

po

sit

ive

x-a

xis

.

Sta

rt w

ith

R(0

, 0).

If

RT

is

a,

then

an

oth

er

vert

ex i

s T

(a,

0).

For

vert

ex S

, th

e x

-coord

inate

is

a −

2 .

Use

b f

or

the y

-coord

inate

,

so t

he v

ert

ex i

s S

( a

−

2 ,

b) .

Exerc

ises

Na

me

th

e m

issin

g c

oo

rd

ina

tes o

f e

ach

tria

ng

le.

1

.

x

y

B( 2

p, 0

)

C( ?

,q)

A( 0

, 0)

2

.

x

y

S( 2

a, 0

)

T( ?

, ?)

R( 0

, 0)

3

.

x

y

G( 2

g, 0

)

F( ?

,b)

E( ?

, ?)

Po

sit

ion

an

d l

ab

el

ea

ch

tria

ng

le o

n t

he

co

ord

ina

te p

lan

e.

4

. is

osc

ele

s tr

ian

gle

5

. is

osc

ele

s ri

gh

t △

DE

F

6. equ

ilate

ral

tria

ngle

△

RS

T w

ith

base

−−

RS

w

ith

legs

e u

nit

s lo

ng

△

EQ

I w

ith

vert

ex

4a

un

its

lon

g

Q

(0,

√ &

3

b)

an

d s

ides

2b u

nit

s lo

ng

x

y

I(b,

0)

E( –

b, 0

)

Q( 0

,√

3b)

x

y

E( e

, 0)

F( e

,e)

D( 0

, 0)

x

yT

( 2a,

b)

R( 0

, 0)

S( 4

a, 0

)

x

y

T( a

, 0)

R( 0

, 0

)S(a – 2

,b )

Sam

ple

an

sw

ers

are

giv

en

.

C

(p,

q)

T(2

a,

2a)

E(-

2g

, 0);

F(0

, b

)

Exam

ple

4-8



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

50

G

len

co

e G

eo

me

try

Writ

e C

oo

rd

ina

te

Pro

ofs

Coord

inate

pro

ofs

can

be u

sed

to p

rove t

heore

ms

an

d t

o

veri

fy p

rop

ert

ies.

Man

y c

oord

inate

pro

ofs

use

th

e D

ista

nce

Form

ula

, S

lop

e F

orm

ula

, or

Mid

poin

t T

heore

m.

P

ro

ve

th

at

a s

eg

me

nt

fro

m t

he

ve

rte

x

an

gle

of

an

iso

sce

les t

ria

ng

le t

o t

he

mid

po

int

of

the

ba

se

is p

erp

en

dic

ula

r t

o t

he

ba

se

.

Fir

st,

posi

tion

an

d l

abel

an

iso

scele

s tr

ian

gle

on

th

e c

oord

inate

p

lan

e.

On

e w

ay i

s to

use

T(a

, 0),

R(-

a,

0),

an

d S

(0,

c).

Th

en

U(0

, 0)

is t

he m

idp

oin

t of

−−

RT

.

Giv

en

: Is

osc

ele

s △

RS

T;

U i

s th

e m

idp

oin

t of

base

−−

RT

.

Pro

ve

: −−

− S

U ⊥

−−

RT

Pro

of:

U i

s th

e m

idp

oin

t of

−−

RT

so t

he c

oord

inate

s of

U a

re ( -

a +

a

−

2

, 0

+ 0

−

2

) = (

0,

0).

Th

us

−−−

SU

lie

s on

the y

-axis

, an

d △

RS

T w

as

pla

ced

so −

−

RT

lie

s on

th

e x

-axis

. T

he a

xes

are

perp

en

dic

ula

r, s

o

−−−

SU

⊥ −

−

RT

.

Exerc

ise

PR

OO

F

Writ

e a

co

ord

ina

te p

ro

of

for t

he

sta

tem

en

t.

Pro

ve t

hat

the s

egm

en

ts j

oin

ing t

he m

idp

oin

ts o

f th

e s

ides

of

a r

igh

t tr

ian

gle

form

a r

igh

t tr

ian

gle

.

x

y

C( 2

a,

0)

B( 0

, 2

b)

P

Q

R

A( 0

, 0)

x

y

T( a

, 0)

U( 0

, 0)

R( –

a,

0)

S( 0

,c)

Sam

ple

an

sw

er:

Po

sit

ion

an

d l

ab

el

rig

ht

△A

BC

wit

h t

he c

oo

rdin

ate

s

A(0

, 0),

B(0

, 2b

), a

nd

C(2

a,

0).

Th

e m

idp

oin

t P

of

BC

is

( 0 +

2a

−

2

, 2

b +

0

−

2

) = (

a,

b).

Th

e m

idp

oin

t Q

of

AC

is ( 0

+ 2

a

−

2

, 2

+ 0

−

2

) = (

a,

0).

Th

e m

idp

oin

t R

of

AB

is

( 0 +

0

−

2

, 0

+ 2

b

−

2

) = (

0,

b).

Th

e s

lop

e o

f −

−

RP

is b

- b

−

a -

0 =

0

−

a =

0,

so

th

e s

eg

men

t is

ho

rizo

nta

l.

Th

e s

lop

e o

f −

−

PQ

is

b -

0

−

a -

a =

b

−

0 ,

wh

ich

is u

nd

efi

ned

, so

th

e s

eg

men

t is

vert

ical.

∠R

PQ

is a

rig

ht

an

gle

becau

se a

ny h

ori

zo

nta

l lin

e i

s p

erp

en

dic

ula

r to

an

y

vert

ical

lin

e.

△P

RQ

has a

rig

ht

an

gle

, so

△P

RQ

is a

rig

ht

tria

ng

le.

4-8

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Exam

ple

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Ch

ap

ter

4

51

G

len

co

e G

eo

me

try

Po

sit

ion

an

d l

ab

el

ea

ch

tria

ng

le o

n t

he

co

ord

ina

te p

lan

e.

1

. ri

gh

t △

FG

H w

ith

legs

2. is

osc

ele

s △

KL

P w

ith

3

. is

osc

ele

s △

AN

D w

ith

a u

nit

s an

d b

un

its

lon

g

base

−−

KP

6b

un

its

lon

g

base

−−−

AD

5a

un

its

lon

g

x

y F( 0

,a)

G( 0

, 0)

H( b

, 0)

x

yL

( 3b,

c)

K( 0

, 0)

P( 6

b,

0)

x

y

N(5 – 2

a,

b)

A( 0

, 0)

D( 5

a,

0)

Na

me

th

e m

issin

g c

oo

rd

ina

tes o

f e

ach

tria

ng

le.

4

.

x

y

B( 2

a,

0)

A( 0

, ?)

C( 0

, 0)

5

.

x

y

Y( 2

b,

0)

Z( ?

, ?)

X( 0

, 0)

6

.

x

y

N( 3

b,

0)

M( ?

, ?)

O( 0

, 0)

7

.

x

y

Q( ?

, ?)

R( 2

a,

b)

P( 0

, 0)

8

.

x

y

P( 7

b,

0)

R( ?

, ?)

N( 0

, 0)

9

.

x

y

U( a

, 0)

T( ?

, ?)

S( –

a,

0)

10

. P

RO

OF W

rite

a c

oord

inate

pro

of

to p

rove t

hat

in a

n i

sosc

ele

s ri

gh

t tr

ian

gle

, th

e s

egm

en

t fr

om

th

e v

ert

ex o

f th

e r

igh

t an

gle

to t

he m

idp

oin

t of

the h

yp

ote

nu

se i

s p

erp

en

dic

ula

r to

th

e h

yp

ote

nu

se.

G

ive

n:

isosc

ele

s ri

gh

t △

AB

C w

ith

∠A

BC

th

e r

igh

t an

gle

an

d M

th

e m

idp

oin

t of

−−

AC

P

ro

ve

: −−

− B

M ⊥

−−

AC

x

y

C( 2

a,

0)

A( 0

, 2

a)

M

B( 0

, 0

)

Sam

ple

an

sw

ers

are

giv

en

.

2

a

Z(b

, b

√ %

3 )

M(0

, c)

Q

(4a,

0)

R ( 7

−

2 b

, c

) T

(∅,

b )

P

roo

f:

T

he M

idp

oin

t F

orm

ula

sh

ow

s t

hat

the c

oo

rdin

ate

s o

f

M

are

( 0 +

2a

−

2

, 2

a +

0

−

2

) or

(a,

a).

Th

e s

lop

e o

f −

−

AC

is

2a

- 0

−

0 -

2a

= -

1.

Th

e s

lop

e o

f −

−

BM

is

a -

0

−

a -

0 =

1.

Th

e p

rod

uct

o

f th

e s

lop

es i

s -

1,

so

−−

BM

⊥ −

−

AC

.

4-8

Sk

ills

Pra

ctic

e

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

52

G

len

co

e G

eo

me

try

Po

sit

ion

an

d l

ab

el

ea

ch

tria

ng

le o

n t

he

co

ord

ina

te p

lan

e.

1

. equ

ilate

ral

△S

WY

wit

h

2. is

osc

ele

s △

BL

P w

ith

3. is

osc

ele

s ri

gh

t △

DG

J

si

des

1 −

4 a

un

its

lon

g

base

−−

BL

3b u

nit

s lo

ng

w

ith

hyp

ote

nu

se −

−

DJ

an

d

legs

2a

un

its

lon

g

x

y D( 0

, 2

a)

G( 0

, 0)

J( 2

a,

0)

x

yP

(3 – 2b,

c)

B( 0

, 0)

L( 3

b, 0

)x

yY

(1 – 8a,

b)

W(1 – 4

a, 0)

S( 0

, 0

)

Na

me

th

e m

issin

g c

oo

rd

ina

tes o

f e

ach

tria

ng

le.

4

.

x

yS

( ?, ?)

J( 0

, 0

)R

( b

, 0)

1 – 3

5

.

x

y

C( ?

, 0

)

E( 0

, ?

)

B( –

3a, 0

)

6

.

x

y

P( 2

b,

0)

M( 0

, ?)

N( ?

, 0)

NE

IGH

BO

RH

OO

DS

F

or E

xe

rcis

es 7

an

d 8

, u

se

th

e f

oll

ow

ing

in

form

ati

on

.

Kari

na l

ives

6 m

iles

east

an

d 4

mil

es

nort

h o

f h

er

hig

h s

chool.

Aft

er

sch

ool

she w

ork

s p

art

ti

me a

t th

e m

all

in

a m

usi

c st

ore

. T

he m

all

is

2 m

iles

west

an

d 3

mil

es

nort

h o

f th

e s

chool.

7

. P

ro

of W

rite

a c

oord

inate

pro

of

to p

rove t

hat

Kari

na’s

hig

h s

chool,

her

hom

e,

an

d t

he

mall

are

at

the v

ert

ices

of

a r

igh

t tr

ian

gle

.

G

ive

n:

△S

KM

P

ro

ve

: △

SK

M i

s a r

igh

t tr

ian

gle

.

8

. F

ind

th

e d

ista

nce

betw

een

th

e m

all

an

d K

ari

na’s

hom

e.

x

y S( 0

, 0)

K( 6

, 4)

M( –

2, 3

)

S

( 1

−

6 b

, c

) C

(3a,

0),

E(0

, c)

M(0

, c

), N

(-2b

, 0)

P

roo

f:

S

lop

e o

f S

K =

4 -

0

−

6 -

0 o

r 2

−

3

S

lop

e o

f S

M =

3

- 0

−

-2

- 0

or

- 3

−

2

S

ince t

he s

lop

e o

f −

−

SM

is t

he n

eg

ati

ve r

ecip

rocal

of

the

s

lop

e o

f −

−

SK

, −

−

SM

⊥ −

−

SK

. T

here

fore

, △

SK

M i

s r

igh

t tr

ian

gle

.

K

M =

√ %

%%

%%

%%

%%

(-

2 -

6) 2

+ (

3 -

4) 2

= √

%%

%

64 +

1 =

√ %

%

65 o

r ≈

8.1

miles

4-8

Practi

ce

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Sam

ple

an

sw

ers

are

g

iven

.

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Ch

ap

ter

4

53

G

len

co

e G

eo

me

try

1

. S

HE

LV

ES

M

art

ha h

as

a s

helf

bra

cket

shap

ed

lik

e a

rig

ht

isosc

ele

s tr

ian

gle

. S

he w

an

ts t

o k

now

th

e l

en

gth

of

the

hyp

ote

nu

se r

ela

tive t

o t

he s

ides.

Sh

e

does

not

have a

ru

ler,

bu

t re

mem

bers

th

e D

ista

nce

Form

ula

. S

he p

lace

s th

e

bra

cket

on

a c

oord

inate

gri

d w

ith

th

e

righ

t an

gle

at

the o

rigin

. T

he l

en

gth

of

each

leg i

s a

. W

hat

are

th

e c

oord

inate

s of

the v

ert

ices

mak

ing t

he t

wo a

cute

an

gle

s?(a

, 0)

an

d (

0,

a)

2

. FLA

GS

A

fla

g i

s sh

ap

ed

lik

e a

n i

sosc

ele

s tr

ian

gle

. A

desi

gn

er

wou

ld

lik

e t

o m

ak

e a

dra

win

g o

f th

e

flag o

n a

coord

inate

pla

ne.

Sh

e p

osi

tion

s it

so t

hat

the

base

of

the t

rian

gle

is

on

th

e

y-a

xis

wit

h o

ne e

nd

poin

t lo

cate

d a

t (0

, 0).

Sh

e l

oca

tes

the t

ip o

f th

e f

lag a

t (a

, b − 2

).

Wh

at

are

th

e c

oord

inate

s of

the t

hir

d v

ert

ex?

(0,

b)

3

. B

ILLIA

RD

S T

he f

igu

re s

how

s a s

itu

ati

on

on

a b

illi

ard

table

.

(–b,

?)

y

x

O

( 0,

0)

Wh

at

are

th

e c

oord

inate

s of

the c

ue b

all

befo

re i

t is

hit

an

d t

he p

oin

t w

here

th

e

cue b

all

hit

s th

e e

dge o

f th

e t

able

?

Sam

ple

an

sw

er:

(-

2b

, 0),

(-

b,

a)

4

. T

EN

TS

T

he e

ntr

an

ce t

o M

att

’s t

en

t m

ak

es

an

iso

scele

s tr

ian

gle

. If

pla

ced

on

a c

oord

inate

gri

d w

ith

th

e b

ase

on

th

e

x–axis

an

d t

he l

eft

corn

er

at

the o

rigin

, th

e r

igh

t co

rner

wou

ld b

e a

t (6

, 0)

an

d

the v

ert

ex a

ngle

wou

ld b

e a

t (3

, 4).

P

rove t

hat

it i

s an

iso

scele

s tr

ian

gle

.

Sam

ple

an

sw

er:

Bo

th s

eg

men

ts

of

the t

rian

gle

fro

m t

he b

ase t

o

the v

ert

ex a

re 5

un

its l

on

g.

Sin

ce

the t

wo

sid

es a

re c

on

gru

en

t, i

t is

an

iso

scele

s t

rian

gle

.

5

. D

RA

FT

ING

An

en

gin

eer

is d

esi

gn

ing

a r

oad

way.

Th

ree r

oad

s in

ters

ect

to

form

a t

rian

gle

. T

he e

ngin

eer

mark

s tw

o p

oin

ts o

f th

e t

rian

gle

at

(-5,

0)

an

d

(5,

0)

on

a c

oord

inate

pla

ne.

a.

Desc

ribe t

he s

et

of

poin

ts i

n t

he

coord

inate

pla

ne t

hat

cou

ld n

ot

be

use

d a

s th

e t

hir

d v

ert

ex o

f th

e

tria

ngle

.

the x

-axis

b.

Desc

ribe t

he s

et

of

poin

ts i

n t

he

coord

inate

pla

ne t

hat

wou

ld m

ak

e

the v

ert

ex o

f an

iso

scele

s tr

ian

gle

to

geth

er

wit

h t

he t

wo c

on

gru

en

t si

des.

the y

-axis

excep

t fo

r th

e o

rig

in

c.

Desc

ribe t

he s

et

of

poin

ts i

n t

he

coord

inate

pla

ne t

hat

wou

ld m

ak

e a

ri

gh

t tr

ian

gle

wit

h t

he o

ther

two

poin

ts i

f th

e r

igh

t an

gle

is

loca

ted

at

(-5,

0).

the p

oin

ts w

ith

x-c

oo

rdin

ate

-

5,

excep

t fo

r (-

5,

0)

Wo

rd

Pro

ble

m P

racti

ce

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

4-8



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

4

54

G

len

co

e G

eo

me

try

Recta

ng

le P

ara

do

x

Use

gri

d p

ap

er

to d

raw

th

e t

wo r

ect

an

gle

s belo

w.

Cu

t th

em

in

to p

iece

s alo

ng t

he

dash

ed

lin

es.

Th

en

rearr

an

ge t

he p

iece

s to

form

a n

ew

rect

an

gle

.

1

. M

ak

e a

sk

etc

h o

f th

e n

ew

rect

an

gle

. L

abel

the w

hole

nu

mber

len

gth

s

2

. W

hat

is t

he c

om

bin

ed

are

a o

f th

e t

wo o

rigin

al

rect

an

gle

s?

3

9 u

nit

s2

3

. W

hat

is t

he a

rea o

f th

e n

ew

rect

an

gle

?

4

0 u

nit

s2

4

. M

ak

e a

con

ject

ure

as

to w

hy t

here

is

a d

iscr

ep

an

cy b

etw

een

th

e a

nsw

ers

to

Exerc

ises

2 a

nd

3.

S

ee s

tud

en

ts’

an

sw

ers

.

5

. U

se a

str

aig

ht

ed

ge t

o p

reci

sely

dra

w t

he f

ou

r sh

ap

es

that

mak

e u

p t

he l

arg

er

rect

an

gle

. W

hat

does

you

dra

win

g s

how

?

I

t sh

ow

s a

gap

. S

o,

the r

ecta

ng

les d

on

’t q

uit

e

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Chapter 4 Assessment Answer KeyQuiz 1 (Lessons 4-1 and 4-2) Quiz 3 (Lessons 4-5 and 4-6) Mid-Chapter Test

Page 57 Page 58 Page 59

Quiz 2 (Lessons 4-3 and 4-4) Quiz 4 (Lessons 4-7 and 4-8)

Page 57 Page 58

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

obtuse scalene

B

x = 2, AB = 15,

BC = 15, AC = 10

AB = √ "" 53 , BC = 2 √ "" 10 ,

AC = √ "" 41 ; scalene

37

42

132

73

73

30

1.

2.

3.

4.

5.

△KMN $ △KML

∠O

x = 8; y = 14

SSS

∠Q $ ∠S

1.

2.

3.

4.

5.

Def. of angle bisector

AAS

60

45

120

1.

2.

3.

4.

5.

translation

x

y

J(a–2, 0)

D(0, a)

L(0, 0)

(a √ " 3 , 0)

AC = √ "" 34 , AB = 6,

and CB = √ "" 34

−−

AC $ −−

CB

1.

2.

3.

4.

D

H

A

G

5.

6.

7.

8.

AB = √ "" 41 ,

BC = √ "" 29 ,

AC = 5 √ " 2 ; scalene

SAS

3

congruent

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Chapter 4 Assessment Answer KeyVocabulary Test Form 1

Page 60 Page 61 Page 62

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

false, equiangular triangle

true

congruence transformation

exterior

included angle

flow proof

right triangle

included side

coordinate proof

vertex angle

a statement that can easily be

proved using a theorem

two triangles in which all

corresponding parts are

congruent

1.

2.

3.

4.

5.

6.

7.

8.

C

J

A

H

B

J

C

G

9.

10.

11.

12.

13.

C

F

C

F

B

isoscelesB:

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Chapter 4 Assessment Answer KeyForm 2A Form 2B

Page 63 Page 64 Page 65 Page 66

9.

10.

11.

12.

13.

B

H

B

F

A

B: A(0, 0), C(-a, 0)

1.

2.

3.

4.

5.

6.

7.

8.

D

G

D

F

C

G

D

J

9.

10.

11.

12.

13.

C

H

D

F

D

B: -2

1.

2.

3.

4.

5.

6.

7.

8.

C

H

A

J

B

J

D

F

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Chapter 4 Assessment Answer KeyForm 2C

Page 67 Page 68

1.

2.

3.

4.

5.

6.

7.

8.

acute scalene

x = 3, AB =

BC = AC = 24

EF = FG = 4 √ " 2 ,

EG = 8, isosceles

70

140

50

△DFG $ △BAC,∠D $ ∠B, ∠F $

∠A, ∠G $ ∠C

reflection;

AB = √ " 29 , BC = √ " 10 ,

CA = √ "" 17 , QT = √ " 29 ,

ST = √ " 10 , QS = √ " 17 ,

9.

10.

11.

12.

13.

14.

B:

45, 45, 90

Isosceles △

Theorem, AAS

50

x = 4

P ( b − 2 , b −

2 ) , AB = 2CP

△XYZ $ △MNO

x

y

C(b–2, c)

B(b, 0)A(0, 0)

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Chapter 4 Assessment Answer KeyForm 2D

Page 69 Page 70

1.

2.

3.

4.

5.

6.

7.

8.

9.

obtuse isosceles

x = 5, AB =

BC = 30, AC = 40

EF = FG = 5,

EG = 5 √ " 2 ,isosceles

110

70

30

△ABC $ △FDE,

∠A $ ∠F, ∠B $

∠D, ∠C $ ∠E

45, 45, 90

rotation; JK = √ "" 10 ,

KL = √ "" 37 , LJ = √ "" 29 ,

WX = √ "" 10 , XY =

√ "" 37 ,

YW = √ " 29

10.

11.

12.

13.

14.

B:

Isosceles △ Theorem, AAS

50

x = 6

M ( a −

2 , 0) , N (0,

b −

2 )

△ABC $ △DEF

x

y

M(3a, 0)

L(1.5a, b)

K(0, 0)

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Chapter 4 Assessment Answer KeyForm 3

Page 71 Page 72

1.

2.

3.

4.

5.

6.

7.

8.

x = 4,

AB = BC = 78, AC = 100

AB = 5, BC = 10, AC = 3 √ " 5 ; scalene

obtuse

20

90

40

reflection

GH = JK = 2 √ " 5 , IG = LJ

= 2 √ " 5 , IH = LK = 2 √ " 2 ;

△GHI $ △JKL by SSS.

AAS

9.

10.

11.

12.

13.

B:

Isosceles

Triangle Theorem

SAS

x = 3

Hypotenuse-leg

congruence for

right △s

m∠1 = m∠3 = m∠4 = m∠6 =

m∠9 = 20, m∠2 = m∠5 = 40,

m∠8 = 60, m∠7 = 140

x

y

B(a + b, 0)

C(a + b, c)2

A(0, 0)

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Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73

Scoring Rubric

Score General Description Specific Criteria

4 Superior

A correct solution that

is supported by well-

developed, accurate

explanations

• Shows thorough understanding of the concepts of using

the Distance Formula to classify triangles and verify

congruence, finding missing angles, solving algebraic

equations in isosceles and equilateral triangles, proving

triangles congruent, congruence and writing coordinate

proofs.

• Uses appropriate strategies to solve problems

• Written explanations are exemplary.

• Figures are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution,

but may contain minor flaws

in reasoning or computation

• Shows understanding of the concepts of using the Distance

Formula to classify triangles and verify congruence, finding

missing angles, solving algebraic equations in isosceles and

equilateral triangles, proving triangles congruent, congruence

and writing proofs.

• Uses appropriate strategies to solve problems

• Computations are mostly correct.

• Written explanations are effective.

• Figures are mostly accurate and appropriate.

• Satisfies all requirements of all problems.

2 Nearly Satisfactory

A partially correct

interpretation and/or

solution to the problem

• Shows understanding of most of the concepts of using the

Distance Formula to classify triangles and verify congruence,

finding missing angles, solving algebraic equations in

isosceles and equilateral triangles, proving triangles

congruent, congrueence and writing proofs.

• May not use appropriate strategies to solve problems

• Computations are mostly correct.

• Written explanations are satisfactory.

• Figures are mostly accurate.

• Satisfies the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no

supporting evidence or

explanation

• Final computation is correct.

• No written explanations or work is shown to substantiate the

final computation.

• Figures may be accurate but lack detail or explanation.

• Satisfies minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution

indicating no mathematical

understanding of the concept

or task, or no solution is given

• Shows little or no understanding of most of the concepts

of using the Distance Formula to classify triangles and

verify congruence, finding missing angles, solving algebraic

equations in isosceles and equilateral triangles, proving

triangles congruent, congruence and writing proofs.

• Does not use appropriate strategies to solve problems

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Figures are inaccurate or inappropriate.

• Does not satisfy the requirements of the problems.

• No answer may be given.

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Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73

Sample Answers

1. a. The figure is an acute isosceles triangle.

b. 9x + 4 + 2(20x - 10) = 180

9x + 4 + 40x - 20 = 180

49x - 16 = 180

49x = 196

x = 4

2. a. To determine whether a triangle is equilateral, the coordinates should be graphed first to see if they form a triangle. Then list the segments which form the triangle and use the Distance Formula to find their lengths. A triangle is equilateral only if all three sides have equal measures.

b. This triangle is not equilateral since AB = AC = 5 and BC = 5 √ " 2 , which makes this an isosceles triangle.

3. Since ∠DBA, ∠ABC, and ∠EBC form a straight line, the sum of the angle measures is 180. Therefore m∠ABC = 180 - 40 - 62 or 78. Then since the sum of the measures of the angles of a triangle is 180, m∠1 = 180 - 78 - 58 or 44. Lastly, since ∠2 is an exterior angle, its measure is equivalent to the sum of the measures of the two remote interior angles, 58 + 78, which equals 136.

4. a. SSS postulate

b. ∠J $ ∠G, ∠D $ ∠E, ∠L $ ∠S

−−−

DJ $ −−−

EG , −−−

DL $ −−−

ES , −−

JL $ −−−

GS

5. Statements Reasons

1. −−−

AB || −−−

DE 1. Given

2. ∠ABC $ ∠DEC 2. Alt. int. % are $.

3. −−−

AD bisects −−−

BE . 3. Given

4. −−−

BC $ −−−

EC 4. Definition of segment bisector

5. ∠ACB $ ∠DCE 5. Vert. % are $.

6. △ABC $ △DEC 6. ASA

In addition to the scoring rubric found on page A33, the following sample answers

may be used as guidance in evaluating open-ended assessment items.

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Chapter 4 Assessment Answer KeyStandardized Test Practice

Page 74 Page 75

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

A B C D

A B C D

8.

9.

10.

11.

12.

13.

14. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

15.

16. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

6

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 8

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Chapter 4 Assessment Answer KeyStandardized Test Practice (continued)

Page 76

17.

18.

19.

20.

21.

22a.

22b.

22c.

29 ft

35

62

40

∠P ! ∠H, ∠Q ! ∠G,

∠R ! ∠B, −−

PQ ! −−

HG ,

−−

QR ! −−

GB , −−

PR ! −−

HB

−−

FD

right triangle

15

Documents

Chapter 4 Resource Masters · PDF file · 2015-06-19Chapter 4 Mid-Chapter Test .....59 Chapter 4 Vocabulary ... assessment and summative (final) assessment. ... • Form 3 is a free-response