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Chapter 4 Resource Masters
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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.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240 - 4027
ISBN: 978-0-07-890513-1MHID: 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 ISBN13Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4Homework Pratice Workbook 0-07-890849-3 978-0-07-890849-1
Spanish VersionHomework 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.
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ContentsTeacher’s Guide to Using the Chapter 4
Resource Masters .........................................iv
Chapter ResourcesStudent-Built Glossary ....................................... 1Anticipation Guide (English) .............................. 3Anticipation Guide (Spanish) ............................. 4
Lesson 4-1Classifying TrianglesStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10
Lesson 4-2Angles of TrianglesStudy Guide and Intervention .......................... 11Skills Practice .................................................. 13Practice ............................................................ 14Word Problem Practice ................................... 15Enrichment ...................................................... 16Cabri Junior. ................................................... 17Geometer’s Sketchpad Activity ....................... 18
Lesson 4-3Congruent TrianglesStudy Guide and Intervention .......................... 19Skills Practice .................................................. 21Practice ............................................................ 22Word Problem Practice ................................... 23Enrichment ...................................................... 24
Lesson 4-4Proving Triangles Congruent—SSS, SASStudy Guide and Intervention .......................... 25Skills Practice .................................................. 27Practice ............................................................ 28Word Problem Practice ................................... 29Enrichment ...................................................... 30
Lesson 4-5Proving Triangles Congruent—ASA, AASStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36
Lesson 4-6Isosceles and Equilateral TrianglesStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42
Lesson 4-7Congruence TransformationsStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice ............................................................ 46Word Problem Practice ................................... 47Enrichment ...................................................... 48
Lesson 4-8Triangles and Coordinate ProofStudy Guide and Intervention .......................... 49Skills Practice .................................................. 51Practice ............................................................ 52Word Problem Practice ................................... 53Enrichment ...................................................... 54
AssessmentStudent Recording Sheet ................................ 55Rubric for Extended Response ....................... 56Chapter 4 Quizzes 1 and 2 ............................. 57Chapter 4 Quizzes 3 and 4 ............................. 58Chapter 4 Mid-Chapter Test ............................ 59Chapter 4 Vocabulary Test ............................. 60Chapter 4 Test, Form 1 ................................... 61Chapter 4 Test, Form 2A ................................. 63Chapter 4 Test, Form 2B ................................. 65Chapter 4 Test, Form 2C ................................ 67Chapter 4 Test, Form 2D ................................ 69Chapter 4 Test, Form 3 ................................... 71Chapter 4 Extended-Response Test ............... 73Standardized Test Practice ............................. 74
Answers ........................................... A1–A40
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Teacher’s Guide to Using theChapter 4 Resource Masters
The 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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Chapter 4 2 Glencoe Geometry
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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Chapter 4 3 Glencoe Geometry
Anticipation GuideCongruent 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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Ejercicios preparatoriosTriá 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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Chapter 4 5 Glencoe Geometry
Study Guide and InterventionClassifying 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.
ExercisesClassify 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
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Chapter 4 6 Glencoe Geometry
Study Guide and Intervention (continued)
Classifying Triangles
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. 23
12
15X V
T
Two sides are congruent. All three sides are The triangle has no pairThe triangle is an congruent. The triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle.
ExercisesClassify 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
CA
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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Chapter 4 7 Glencoe Geometry
Skills PracticeClassifying 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
A B9
8
8√2 7. �ABE 8. �EDB
9. �EBC 10. �DBC
11. ALGEBRA Find x and the length of each side if �ABC is an isosceles triangle with
−− AB �
−− BC .
12. ALGEBRA Find x and the length of each side if �FGH is an equilateral triangle.
5x - 83x + 10
4x + 1
13. ALGEBRA Find x and the length of each side if �RST is an isosceles triangle with
−− RS �
−− TS .
3x - 1
9x + 2
6x + 8
14. ALGEBRA Find x and the length of each side if �DEF is an equilateral triangle.
5x - 212x + 6
7x - 39
4-1
4x - 3 2x + 5
x + 2
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Chapter 4 8 Glencoe Geometry
PracticeClassifying 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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Chapter 4 9 Glencoe Geometry
Word Problem PracticeClassifying Triangles
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
CAG H
IF
J
B
E D
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Chapter 4 10 Glencoe Geometry
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.
ExercisesDraw 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. Thex-coordinates of R and S are the y-coordinates of J and K are thesame. The y-coordinate of T is same. The x-coordinates of K and Lbetween 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 ID, 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-coordinateequilateral? of G. Is �GHI scalene, isosceles, or
equilateral?
BA
C
x
y
O
4-1
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Chapter 4 11 Glencoe Geometry
Study Guide and InterventionAngles 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°
12 3
E
DAC
B
ExercisesFind 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°
1 N
M
P
4-2
Example 1
35°
25°R T
S
Example 2
Q
O
NM
P58°
66°
50°
321
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Chapter 4 12 Glencoe Geometry
Study Guide and Intervention (continued)
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
D1m∠1 = m∠A + m∠B
Find m∠1.
m∠1 = m∠R + m∠S Exterior Angle Theorem
= 60 + 80 Substitution
= 140 Simplify.
R T
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∠FE
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
B C D
A
25°
35°
12Y Z W
X
65°
50°
1
4-2
Example 1 Example 2
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Chapter 4 13 Glencoe Geometry
Skills PracticeAngles 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
2D
A 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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Chapter 4 14 Glencoe Geometry
PracticeAngles 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
BC
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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Chapter 4 15 Glencoe Geometry
Word Problem PracticeAngles 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
3 4
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Chapter 4 16 Glencoe Geometry
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 = 43So, 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
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Chapter 4 17 Glencoe Geometry
Graphing Calculator ActivityCabri 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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Chapter 4 18 Glencoe Geometry
Geometer’s Sketchpad ActivityAngles 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.
ExercisesAnalyze 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
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Chapter 4 19 Glencoe Geometry
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
ExercisesShow 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 InterventionCongruent 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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Chapter 4 20 Glencoe Geometry
Study Guide and Intervention (continued)
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 � ∠BCD4. ∠ABD � ∠CBD5. ∠BDA � ∠BDC6. �ABD � �CBD
1. Given2. Reflexive Property of congruence3. Given4. Definition of angle bisector
5. Third Angles Theorem6. 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 Prove: �ABD � �CBD
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Chapter 4 21 Glencoe Geometry
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. 3. Find the value of x.
4. Find the value of y.
5. PROOF Write a two-column proof.
Given: −−−
AB � −−−
CB , −−−
AD � −−−
CD , ∠ABD � ∠CBD,∠ADB � ∠CDB
Prove: �ABD � �CBD
Skills PracticeCongruent Triangles
D
E
G
F
108° 48° y°
(2x - y)°
A F
BC ED
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Chapter 4 22 Glencoe Geometry
PracticeCongruent Triangles
Show that the polygons are congruent by indentifying all congruent corresponding parts. Then write a congruence statement.
1.
D
R
SCA
B 2. MN
L
P
Q
Polygon ABCD � polygon PQRS.
3. Find the value of x.
4. Find the value of y.
5. PROOF Write a two-column proof.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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Chapter 4 23 Glencoe Geometry
Word Problem PracticeCongruent 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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Chapter 4 24 Glencoe Geometry
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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Chapter 4 25 Glencoe Geometry
Study Guide and InterventionProving 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.
Write a two-column proof.
Given: −−
AB � −−−
DB and C is the midpoint of −−−
AD .Prove: �ABC � �DBC B
CDA
Statements Reasons
1. −−
AB � −−−
DB 1. Given2. C is the midpoint of
−−− AD . 2. Given
3. −−
AC � −−−
DC 3. Midpoint Theorem4.
−−− BC �
−−− BC 4. Reflexive Property of �
5. �ABC � �DBC 5. SSS Postulate
Exercises
Write a two-column proof.
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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Chapter 4 26 Glencoe Geometry
Study Guide and Intervention (continued)
Proving Triangles Congruent—SSS, SAS
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
2 R
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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Chapter 4 27 Glencoe Geometry
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 .
Prove: �ABD � �CBD
T
R
P
F
E
D
4-4 Skills PracticeProving Triangles Congruent—SSS, SAS
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Chapter 4 28 Glencoe Geometry
PracticeProving Triangles Congruent—SSS, SAS
Determine 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)
3. Write a flow proof.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
4-4
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Chapter 4 29 Glencoe Geometry
Word Problem PracticeProving Triangles Congruent—SSS, SAS
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.
4-4
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Chapter 4 30 Glencoe Geometry
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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Chapter 4 31 Glencoe Geometry
Study Guide and InterventionProving 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 � �CDBStatements Reasons1.
−− AB ‖
−−− CD
2. ∠CBD � ∠ADB3. ∠ABD � ∠BDC4. BD = BD5. �ABD � �CDB
1. Given2. Given3. Alternate Interior Angles Theorem4. Reflexive Property of congruence5. 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
ExercisesPROOF Write the specified type of proof. 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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Chapter 4 32 Glencoe Geometry
Study Guide and Intervention (continued)
Proving Triangles Congruent—ASA, AAS
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.
ExercisesPROOF Write the specified type of proof. 1. Write a two column proof. Given:
−−− BC ‖
−− EF
AB = ED ∠C � ∠F Prove: �ABD � �DEF
2. Write a flow proof. 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
C12A
B
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Chapter 4 33 Glencoe Geometry
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
3. Write a paragraph proof. Given:
−−− DE ‖
−−− FG
∠E � ∠G Prove: �DFG � �FDE
FG
D E
A C
B
D
N
J
M
K L
4-5 Skills PracticeProving Triangles Congruent—ASA, AAS
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Chapter 4 34 Glencoe Geometry
PROOF Write the specified type of proof. 1. Write a flow proof. Given: S is the midpoint of
−−−QT .
−−−
QR ‖ −−−
TU Prove: �QSR � �TSU
2. Write a paragraph proof. 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
UQ S
RT
4-5 PracticeProving Triangles Congruent—ASA, AAS
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Chapter 4 35 Glencoe Geometry
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 PraticeProving Triangles Congruent—ASA, AAS
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Chapter 4 36 Glencoe Geometry
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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Chapter 4 37 Glencoe Geometry
ExercisesALGEBRA 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
T3x°
x°
7. PROOF Write a two-column proof.
Given: ∠1 � ∠2 Prove:
−− AB �
−−− CB
Study Guide and InterventionIsosceles 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.
R T
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
4-6
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Chapter 4 38 Glencoe Geometry
Study Guide and Intervention (continued)
Isosceles and Equilateral Triangles
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. Given2. 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. Substitution5. �APQ is equilateral. 5. If a � is equiangular, then it is equilateral.
Exercises
ALGEBRA Find the value of each variable.
1. D
F E6x°
2. G
J H
6x - 5 5x
3.
3x°
4.
4x 4060°
5. X
Z Y4x - 4
3x + 8 60°
6.
4x°
60°
7. PROOF Write a two-column proof.
Given: �ABC is equilateral; ∠1 � ∠2.
Prove: ∠ADB � ∠CDB
A
D
C
B12
A
B
P Q
C
1 2
4-6
Example
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Chapter 4 39 Glencoe Geometry
Skills PracticeIsosceles and Equilateral Triangles
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°
ALGEBRA Find the value of each variable.
7.
2x + 4 3x - 10
8.
(2x + 3)°
9. PROOF Write a two-column proof.
Given: −−−
CD � −−−
CG
−−−
DE � −−−
GF
Prove: −−−
CE � −−
CF
D
E
FG
C
4-6
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Chapter 4 40 Glencoe Geometry
PracticeIsosceles and Equilateral Triangles
Lincoln Hawks
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.
10. PROOF Write a two-column proof.
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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Chapter 4 41 Glencoe Geometry
Word Problem PracticeIsosceles and Equilateral Triangles
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
4-6
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Chapter 4 42 Glencoe Geometry
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 DF 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
BE
4. Given: m∠N = 120, −−−
JN � −−−
MN , �JNM � �KLM.
Find m∠JKM.
J
K
L
MN
U VW
XYZ
4-6
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Chapter 4 43 Glencoe Geometry
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.
ExercisesIdentify 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
Study Guide and InterventionCongruence Transformations
Example
4-7
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Chapter 4 44 Glencoe Geometry
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)
Congruence Transformations
Example
x
y
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Chapter 4 45 Glencoe Geometry
4-7
Identify the type of congruence transformation shown as a reflection, translation, 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 PracticeCongruence Transformations
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Chapter 4 46 Glencoe Geometry
Identify the type of congruence transformation shown as a reflection, translation, 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. ΔABC has vertices A(−4, 2), B(−2, 0), C(−4, -2). ΔDEF 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 PracticeCongruence Transformations
x
y
x
y
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Chapter 4 47 Glencoe Geometry
4-7 Word Problem PracticeCongruence Transformations
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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Chapter 4 48 Glencoe Geometry
4-7 Enrichment
Transformations in The Coordinate PlaneThe 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
YO
(x, y) � (x + 6, y - 3)
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Chapter 4 49 Glencoe Geometry
Study Guide and InterventionTriangles 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) .
ExercisesName 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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Chapter 4 50 Glencoe Geometry
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)
Triangles and Coordinate Proof
Example
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Chapter 4 51 Glencoe Geometry
Position and label each triangle on the coordinate plane.
1. right �FGH with legs 2. isosceles �KLP with 3. isosceles �AND witha units and b units long base
−− KP 6b units long base
−−− AD 5a units long
x
y
x
y
x
y
Name the missing coordinates of each triangle. 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 PracticeTriangles and Coordinate Proof
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Chapter 4 52 Glencoe Geometry
Position and label each triangle on the coordinate plane.
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
Name the missing coordinates of each triangle.
4.
x
y S(?, ?)
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 PracticeTriangles and Coordinate Proof
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Chapter 4 53 Glencoe Geometry
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 locatesthe 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
xO
(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).
Word Problem PracticeTriangles and Coordinate Proof
4-8
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Chapter 4 54 Glencoe Geometry
Rectangle ParadoxUse 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
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Chapter 4 55 Glencoe Geometry
SCORE 4 Student Recording SheetUse 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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Chapter 4 56 Glencoe Geometry
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
4The 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.
3 A 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.
0 An 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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Chapter 4 57 Glencoe Geometry
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 34
56 107°
43°
Chapter 4 Quiz 2(Lessons 4-3 and 4-4)
Q
P TR
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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Chapter 4 58 Glencoe Geometry
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 � �AKCProof:
Statements Reasons1. ∠Z � ∠C −−−
AK bisects ∠ZKC. 1. Given2. ∠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
Chapter 4 Quiz 3(Lessons 4-5 and 4-6)
Chapter 4 Quiz 4(Lessons 4-7 and 4-8)
x
y
SCORE
21
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Chapter 4 59 Glencoe Geometry
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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Chapter 4 60 Glencoe Geometry
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
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Chapter 4 61 Glencoe Geometry
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 � �CBDD �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 13 460°
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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Chapter 4 62 Glencoe Geometry
N
K
L MJ
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 � �MNLProof:
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
Use the figure for Questions 11 and 12.
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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Chapter 4 63 Glencoe Geometry
SCORE
Write the letter for the correct answer in the blank at the right of each question.
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
Use the figure for Questions 3 and 4.
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 � �LMNJ �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
LK
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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Chapter 4 64 Glencoe Geometry
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
Use the proof for Questions 10 and 11.
Given: −−−
DA || −−−
YN
−−− DA �
−−− YN
Prove: ∠NDY � ∠DNAProof:
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
11. What is the reason for statement 6? 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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Chapter 4 65 Glencoe Geometry
SCORE
Write the letter for the correct answer in the blank at the right of each question.
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
Use the figure for Questions 3 and 4.
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°12
7x - 15
3x + 54x
1.
2.
3.
4.
5.
6.
7.
8.
4 Chapter 4 Test, Form 2B
Q
T S
R
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Chapter 4 66 Glencoe Geometry
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
Use the proof for Questions 10 and 11.
Given: −−−
EG � −−
IA ∠EGA � ∠IAG
Prove: ∠GEN � ∠AINProof:
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)
10. What is the reason for statement 4? F SSS G ASA H SAS J AAS
11. What is the reason for statement 5? 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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Chapter 4 67 Glencoe Geometry
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
GB
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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Chapter 4 68 Glencoe Geometry
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.
12. Find the value of x.
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
N MK
L
4 Chapter 4 Test, Form 2C (continued)
9.
10.
11.
12.
13.
14.
B:
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Chapter 4 69 Glencoe Geometry
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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Chapter 4 70 Glencoe Geometry
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.
12. Find the value of x.
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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Chapter 4 71 Glencoe Geometry
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.
3. Find the value of x.
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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Chapter 4 72 Glencoe Geometry
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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Chapter 4 73 Glencoe Geometry
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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Chapter 4 74 Glencoe Geometry
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 SSSB 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
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Chapter 4 75 Glencoe Geometry
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 ResponsePart 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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Chapter 4 76 Glencoe Geometry
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
RT
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.
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NA
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RIO
D
Lesson 4-1
Cha
pte
r 4
5
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y Tr
ian
gle
s b
y A
ng
les
On
e w
ay t
o cl
assi
fy a
tri
angl
e is
by
the
mea
sure
s of
its
an
gles
.
• 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
one
of
the
an
gle
s 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
one
of
the
an
gle
s 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
lass
ify
each
tri
angl
e.
a.
60°
A
BC
All
th
ree
angl
es a
re c
ongr
uen
t, s
o al
l th
ree
angl
es h
ave
mea
sure
60°
.T
he
tria
ngl
e is
an
equ
ian
gula
r tr
ian
gle.
b.
25°
35°
120°
DF
E
Th
e tr
ian
gle
has
on
e an
gle
that
is
obtu
se. I
t is
an
obt
use
tri
angl
e.
c.
90°
60°
30°
G
HJ
Th
e tr
ian
gle
has
on
e ri
ght
angl
e. I
t is
a r
igh
t tr
ian
gle.
Exer
cise
sC
lass
ify
each
tri
angl
e as
acu
te, e
qu
ian
gula
r, o
btu
se, o
r ri
ght.
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
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018_
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:33:
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Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
3
Gle
ncoe
Geo
met
ry
Ant
icip
atio
n G
uide
Co
ng
ruen
t Tri
an
gle
s
B
efor
e yo
u b
egin
Ch
ap
ter
4
• R
ead
each
sta
tem
ent.
• D
ecid
e w
het
her
you
Agr
ee (
A)
or D
isag
ree
(D)
wit
h t
he
stat
emen
t.
• W
rite
A o
r D
in
th
e fi
rst
colu
mn
OR
if
you
are
not
su
re w
het
her
you
agr
ee o
r di
sagr
ee,
wri
te N
S (
Not
Su
re).
A
fter
you
com
ple
te C
ha
pte
r 4
• R
erea
d ea
ch s
tate
men
t an
d co
mpl
ete
the
last
col
um
n b
y en
teri
ng
an A
or
a D
.
• D
id a
ny
of y
our
opin
ion
s ab
out
the
stat
emen
ts c
han
ge f
rom
th
e fi
rst
colu
mn
?
• F
or t
hos
e st
atem
ents
th
at y
ou m
ark
wit
h a
D, u
se a
pie
ce o
f pa
per
to w
rite
an
ex
ampl
e of
wh
y yo
u d
isag
ree.
4 Step
1
ST
EP
1A
, D
, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1.
For
a t
rian
gle
to b
e ac
ute
, all
th
ree
angl
es m
ust
be
acu
te.
A
2.
All
sid
es o
f an
equ
ilat
eral
tri
angl
e ar
e co
ngr
uen
t.A
3.
An
ext
erio
r an
gle
is a
ny
angl
e ou
tsid
e of
th
e gi
ven
tr
ian
gle.
D
4.
Th
e su
m o
f th
e m
easu
res
of t
he
angl
es i
n a
tri
angl
e is
360
°.D
5.
An
obt
use
tri
angl
e is
a t
rian
gle
wit
h t
hre
e ob
tuse
an
gles
.D
6.
Tri
angl
es t
hat
are
th
e sa
me
shap
e an
d si
ze a
re
con
gru
ent.
A
7.
If t
he
angl
es o
f on
e tr
ian
gle
are
con
gru
ent
to t
he
angl
es o
f an
oth
er t
rian
gle,
th
en t
he
two
tria
ngl
es a
re
con
gru
ent.
D
8.
An
iso
scel
es t
rian
gle
is a
tri
angl
e w
ith
tw
o co
ngr
uen
t si
des.
A
9.
A t
rian
gle
can
be
equ
ian
gula
r an
d n
ot e
quil
ater
al.
D
10.
To
mak
e th
e ca
lcu
lati
ons
in a
coo
rdin
ate
proo
f ea
sier
, pl
ace
eith
er a
ver
tex
or t
he
cen
ter
of t
he
poly
gon
at
the
orig
in.
A
Step
2
001_
018_
GE
OC
RM
C04
_890
513.
indd
35/
22/0
88:
03:1
4P
M
Answers (Anticipation Guide and Lesson 4-1)
Chapter 4 A1 Glencoe Geometry
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Lesson 4-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
7
Gle
ncoe
Geo
met
ry
Skill
s Pr
acti
ceC
lassif
yin
g T
rian
gle
s
Cla
ssif
y ea
ch t
rian
gle
as a
cute
, eq
uia
ngu
lar,
obt
use
, or
righ
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 ea
ch t
rian
gle
as e
qu
ila
tera
l, i
sosc
eles
, or
sca
len
e.
EC
D
AB
9
88√2
7. �
AB
E
8. �
ED
B
s
cale
ne
iso
scele
s
9. �
EB
C
10. �
DB
C
s
cale
ne
eq
uala
tera
l
11. A
LGEB
RA
Fin
d x
and
the
len
gth
of
each
sid
e if
�A
BC
is
an i
sosc
eles
tr
ian
gle
wit
h −−
A
B �
−−
BC
.
12. A
LGEB
RA
Fin
d x
and
the
len
gth
of
each
si
de i
f �
FG
H i
s an
equ
ilat
eral
tri
angl
e.
5x-
83x
+10
4x+
1
13. A
LGEB
RA
Fin
d x
and
the
len
gth
of
each
sid
e if
�R
ST
is
an i
sosc
eles
tr
ian
gle
wit
h −
−
RS
� −
−
TS .
3x-
1
9x+
2
6x+
8
14. A
LGEB
RA
Fin
d x
and
the
len
gth
of
each
si
de i
f �
DE
F i
s an
equ
ilat
eral
tri
angl
e.
5x-
212x
+6
7x-
39
4-1 x =
4;
AB
=
B
C =
13;
AC
= 6
x =
9;
FG
=
G
H =
FH
= 3
7
x =
2;
RS
=
S
T =
20;
RT
= 5
x =
9;
DE
=
E
F =
DF =
24
4x-
32x
+5
x+
2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
6
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y Tr
ian
gle
s b
y Si
des
You
can
cla
ssif
y a
tria
ngl
e by
th
e n
um
ber
of c
ongr
uen
t si
des.
Equ
al n
um
bers
of
has
h m
arks
in
dica
te c
ongr
uen
t si
des.
• If
all
thre
e sid
es 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
leas
t tw
o sid
es 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
two
sid
es 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
lass
ify
each
tri
angl
e.
a. L
J
H
b.
N
RP
c.
23
12
15X
V
T
T
wo
side
s ar
e co
ngr
uen
t.
All
th
ree
side
s ar
e
Th
e tr
ian
gle
has
no
pair
Th
e tr
ian
gle
is a
n
con
gru
ent.
Th
e tr
ian
gle
o
f co
ngr
uen
t si
des.
It
is
isos
cele
s tr
ian
gle.
i
s an
equ
ilat
eral
tri
angl
e.
a s
cale
ne
tria
ngl
e.
Exer
cise
sC
lass
ify
each
tri
angl
e as
eq
uil
ate
ral,
iso
scel
es, o
r sc
ale
ne.
1.
21
3√
⎯G
CA
2.
G
KI
18
1818
3.
12
17
19Q
O
M
s
cale
ne
eq
uilate
ral
scale
ne
4.
UW
S
5. 8x
32x
32x
B CA
6.
DE
Fx
xx
i
so
scele
s
iso
scele
s
eq
uilate
ral
Exam
ple
4-1
7. A
LGEB
RA
Fin
d x
and
the
len
gth
of
each
si
de i
f �
RS
T i
s an
equ
ilat
eral
tri
angl
e. 8
. ALG
EBR
A F
ind
x an
d th
e le
ngt
h o
f ea
ch s
ide
if �
AB
C i
s is
osce
les
wit
h
AB
= B
C.
5x-
42x
+2
3x
3y+
24y
3y
2;
62;
AB
= 8;
BC
= 8, A
C =
6
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Answers (Lesson 4-1)
Chapter 4 A2 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 2A01_A26_GEOCRM04_890513.indd 2 5/23/08 11:38:17 PM5/23/08 11:38:17 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
8
Gle
ncoe
Geo
met
ry
Prac
tice
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y ea
ch t
rian
gle
as a
cute
, eq
uia
ngu
lar,
obt
use
, or
righ
t.
1.
2.
3.
Cla
ssif
y ea
ch t
rian
gle
in t
he
figu
re a
t th
e ri
ght
by
its
angl
es a
nd
sid
es.
4. �
AB
D
5. �
AB
C
e
qu
ian
gu
lar;
eq
uilate
ral
rig
ht;
scale
ne
6. �
ED
C
7. �
BD
C
r
igh
t; s
cale
ne
ob
tuse;
iso
scele
s
ALG
EBR
A F
or e
ach
tri
angl
e, f
ind
x a
nd
th
e m
easu
re o
f ea
ch s
ide.
8. �
FG
H i
s an
equ
ilat
eral
tri
angl
e w
ith
FG
= x
+ 5
, GH
= 3
x -
9, a
nd
FH
= 2
x -
2.
9. �
LM
N i
s an
iso
scel
es t
rian
gle,
wit
h L
M =
LN
, LM
= 3
x -
2, L
N =
2x
+ 1
, an
d M
N =
5x
- 2
.
Fin
d t
he
mea
sure
s of
th
e si
des
of
�K
PL
an
d c
lass
ify
each
tri
angl
e b
y it
s si
des
.
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
ESIG
N D
ian
a en
tere
d th
e de
sign
at
the
righ
t in
a l
ogo
con
test
sp
onso
red
by a
wil
dlif
e en
viro
nm
enta
l gr
oup.
Use
a p
rotr
acto
r.
How
man
y ri
ght
angl
es a
re t
her
e?
AC
DE
B
60°
30°
90°
65°
30°
85°
40°
100°
40°
4-1 x =
7;
FG
=
12,
GH
= 12,
FH
= 1
2
x =
3;
LM
=
7,
LN
= 7,
MN
= 1
3
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
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Lesson 4-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
9
Gle
ncoe
Geo
met
ry
Wor
d Pr
oble
m P
ract
ice
Cla
ssif
yin
g T
rian
gle
s
1. M
USE
UM
SP
aul
is s
tan
din
g in
fro
nt
of
a m
use
um
exh
ibit
ion
. Wh
en h
e tu
rns
his
h
ead
60°
to t
he
left
, he
can
see
a s
tatu
e by
Don
atel
lo. W
hen
he
turn
s h
is h
ead
60°
to t
he
righ
t, h
e ca
n s
ee a
sta
tue
by
Del
la R
obbi
a. T
he
two
stat
ues
an
d P
aul
form
th
e ve
rtic
es o
f a
tria
ngl
e. C
lass
ify
this
tri
angl
e as
acu
te, r
igh
t, o
r ob
tuse
.
2. P
APE
R M
arsh
a cu
ts a
rec
tan
gula
r pi
ece
of p
aper
in
hal
f al
ong
a di
agon
al. T
he
resu
lt i
s tw
o tr
ian
gles
. Cla
ssif
y th
ese
tria
ngl
es a
s ac
ute
, rig
ht,
or
obtu
se.
3. W
ATE
RSK
IING
Kim
an
d C
assa
ndr
a ar
e w
ater
skii
ng.
Th
ey a
re h
oldi
ng
on t
o ro
pes
that
are
th
e sa
me
len
gth
an
d ti
ed
to t
he
sam
e po
int
on t
he
back
of
a sp
eed
boat
. Th
e bo
at i
s go
ing
full
spe
ed a
hea
d an
d th
e ro
pes
are
full
y ta
ut.
K
im, C
assa
ndr
a, a
nd
the
poin
t w
her
e th
e ro
pes
are
tied
on
th
e bo
at f
orm
th
e ve
rtic
es o
f a
tria
ngl
e. T
he
dist
ance
be
twee
n K
im a
nd
Cas
san
dra
is n
ever
eq
ual
to
the
len
gth
of
the
rope
s. C
lass
ify
the
tria
ngl
e as
equ
ilat
eral
, iso
scel
es, o
r sc
alen
e.
4. B
OO
KEN
DS
Tw
o bo
oken
ds a
re s
hap
ed
like
rig
ht
tria
ngl
es.
Th
e bo
ttom
sid
e of
eac
h t
rian
gle
is
exac
tly
hal
f as
lon
g as
th
e sl
ante
d si
de
of t
he
tria
ngl
e. I
f al
l th
e bo
oks
betw
een
th
e bo
oken
ds a
re r
emov
ed a
nd
they
are
pu
shed
tog
eth
er, t
hey
wil
l fo
rm a
sin
gle
tria
ngl
e. C
lass
ify
the
tria
ngl
e th
at c
an
be f
orm
ed a
s eq
uil
ater
al, i
sosc
eles
, or
scal
ene.
5. D
ESIG
NS
Su
zan
ne
saw
th
is p
atte
rn o
n
a pe
nta
gon
al f
loor
til
e. S
he
not
iced
man
y di
ffer
ent
kin
ds o
f tr
ian
gles
wer
e cr
eate
d by
th
e li
nes
on
th
e ti
le.
a. I
den
tify
fiv
e tr
ian
gles
th
at a
ppea
r to
be
acu
te i
sosc
eles
tri
angl
es.
b.
Iden
tify
fiv
e tr
ian
gles
th
at a
ppea
r to
be
obt
use
iso
scel
es t
rian
gles
.
4-1
Kim
Cass
andr
a
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
001_
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Answers (Lesson 4-1)
Chapter 4 A3 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 3A01_A26_GEOCRM04_890513.indd 3 5/23/08 11:38:25 PM5/23/08 11:38:25 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
10
G
lenc
oe G
eom
etry
Enri
chm
ent
Read
ing
Math
em
ati
cs
Wh
en y
ou r
ead
geom
etry
, you
may
nee
d to
dra
w a
dia
gram
to
mak
e th
e te
xt
easi
er t
o u
nde
rsta
nd.
C
onsi
der
th
ree
poi
nts
, A, B
, an
d C
on
a c
oord
inat
e gr
id. T
he
y-co
ord
inat
es o
f A
an
d B
are
th
e sa
me.
Th
e x-
coor
din
ate
of B
is
grea
ter
than
th
e x-
coor
din
ate
of A
. Bot
h c
oord
inat
es o
f C
are
gre
ater
th
an t
he
corr
esp
ond
ing
coor
din
ates
of
B. I
s �
AB
C a
cute
, rig
ht,
or
obtu
se?
To
answ
er t
his
qu
esti
on, f
irst
dra
w a
sam
ple
tria
ngl
e
that
fit
s th
e de
scri
ptio
n.
Sid
e −
−
AB
mu
st b
e a
hor
izon
tal
segm
ent
beca
use
th
e y-
coor
din
ates
are
th
e sa
me.
Poi
nt
C m
ust
be
loca
ted
to t
he
righ
t an
d u
p fr
om p
oin
t B
.F
rom
th
e di
agra
m y
ou c
an s
ee t
hat
tri
angl
e A
BC
m
ust
be
obtu
se.
Exer
cise
sD
raw
a s
imp
le t
rian
gle
on g
rid
pap
er t
o h
elp
you
.
1. C
onsi
der
thre
e po
ints
, R, S
, an
d
2. C
onsi
der
thre
e n
onco
llin
ear
poin
ts,
T o
n a
coo
rdin
ate
grid
. Th
e
J, K
, an
d L
on
a c
oord
inat
e gr
id. T
he
x-co
ordi
nat
es o
f R
an
d S
are
th
e y
-coo
rdin
ates
of
J a
nd
K a
re t
he
sam
e. T
he
y-co
ordi
nat
e of
T i
s s
ame.
Th
e x-
coor
din
ates
of
K a
nd
Lbe
twee
n t
he
y-co
ordi
nat
es o
f R
a
re t
he
sam
e. I
s �
JK
L a
cute
,an
d S
. Th
e x-
coor
din
ate
of T
is
less
r
igh
t, o
r ob
tuse
? th
an t
he
x-co
ordi
nat
e of
R. I
s an
gle
R
of
�R
ST
acu
te, r
igh
t, o
r ob
tuse
?
3. C
onsi
der
thre
e n
onco
llin
ear
poin
ts,
4. C
onsi
der
thre
e po
ints
, G, H
, an
d I
D, E
, an
d F
on
a c
oord
inat
e gr
id.
on
a c
oord
inat
e gr
id. P
oin
ts G
an
d T
he
x-co
ordi
nat
es o
f D
an
d E
are
H
are
on
th
e po
siti
ve y
-axi
s, a
nd
oppo
site
s. T
he
y-co
ordi
nat
es o
f D
an
d t
he
y-co
ordi
nat
e of
G i
s tw
ice
the
E a
re t
he
sam
e. T
he
x-co
ordi
nat
e of
y
-coo
rdin
ate
of H
. Poi
nt
I is
on
th
e F
is
0. W
hat
kin
d of
tri
angl
e m
ust
p
osit
ive
x-ax
is, a
nd
the
x-co
ordi
nat
e�
DE
F b
e: s
cale
ne,
iso
scel
es, o
r o
f I
is g
reat
er t
han
th
e y-
coor
din
ate
equ
ilat
eral
?
of
G. I
s �
GH
I sc
alen
e, i
sosc
eles
, or
equ
ilat
eral
?
BA
C x
y
O
4-1
Exam
ple
iso
scele
s
rig
ht
scale
ne
acu
te
001_
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Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
11
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
An
gle
s o
f Tri
an
gle
s
Tria
ng
le A
ng
le-S
um
Th
eore
m I
f th
e m
easu
res
of t
wo
angl
es o
f a
tria
ngl
e ar
e kn
own
, th
e m
easu
re o
f th
e th
ird
angl
e ca
n a
lway
s be
fou
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
T
riangle
Angle
-
Sum
Theore
m
25
+ 3
5 +
m∠
T =
180
S
ubstitu
tion
60
+ m
∠T
= 1
80
Sim
plif
y.
m
∠T
= 1
20
Subtr
act
60 f
rom
each s
ide.
F
ind
th
e m
issi
ng
angl
e m
easu
res.
m∠
1 +
m∠
A +
m∠
B =
180
T
riangle
Angle
- S
um
T
heore
m
m∠
1 +
58
+ 9
0 =
180
S
ubstitu
tion
m∠
1 +
148
= 1
80
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
T
riangle
Angle
- S
um
T
heore
m
m∠
3 +
32
+ 1
08 =
180
S
ubstitu
tion
m∠
3 +
140
= 1
80
Sim
plif
y.
m∠
3 =
40
Subtr
act
140 f
rom
each s
ide.
58°90
°
108°
12
3
E
DA
C
B
Exer
cise
sF
ind
th
e m
easu
res
of e
ach
nu
mb
ered
an
gle.
1.
2.
3.
4.
5.
6.
20°
152°
DG
A
130
°60
°
12
S
R
TW
V WT
U
30°
60°
2
1
S
QR
30°
1
90°
62°
1NM
P4-2
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°
32
1
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Answers (Lesson 4-1 and Lesson 4-2)
Chapter 4 A4 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 4A01_A26_GEOCRM04_890513.indd 4 5/23/08 11:38:31 PM5/23/08 11:38:31 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
12
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
An
gle
s o
f Tri
an
gle
s
Exte
rio
r A
ng
le T
heo
rem
At
each
ver
tex
of a
tri
angl
e, t
he
angl
e fo
rmed
by
one
side
an
d an
ext
ensi
on o
f th
e ot
her
sid
e is
cal
led
an e
xter
ior
angl
e of
th
e tr
ian
gle.
For
eac
h
exte
rior
an
gle
of a
tri
angl
e, t
he
rem
ote
inte
rior
an
gles
are
th
e in
teri
or a
ngl
es t
hat
are
not
ad
jace
nt
to t
hat
ext
erio
r an
gle.
In
th
e di
agra
m b
elow
, ∠B
an
d ∠
A a
re t
he
rem
ote
inte
rior
an
gles
for
ext
erio
r ∠
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
D1
m∠
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
∠P
QS
= m
∠R
+ m
∠S
E
xte
rior
Angle
Theore
m
78
= 5
5 +
x
Substitu
tion
23
= x
S
ubtr
act
55 f
rom
each
sid
e.
SR
Q
P
55°
78°
x°
Exer
cise
s
Fin
d t
he
mea
sure
s of
eac
h n
um
ber
ed a
ngl
e.
1.
2.
3.
4.
Fin
d e
ach
mea
sure
.
5. m
∠A
BC
6.
m∠
FE
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° 1
2Y
ZW
X 65°50
°
1
4-2
Exam
ple
1Ex
amp
le 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
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Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
13
G
lenc
oe G
eom
etry
Skill
s Pr
acti
ceA
ng
les o
f Tri
an
gle
s
Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle.
1.
2.
Fin
d e
ach
mea
sure
.
3. m
∠1
4. m
∠2
5. m
∠3
Fin
d e
ach
mea
sure
.
6. m
∠1
7. m
∠2
8. m
∠3
Fin
d e
ach
mea
sure
.
9. m
∠1
10. m
∠2
11. m
∠3
12. m
∠4
13. m
∠5
Fin
d e
ach
mea
sure
.
14. m
∠1
15. m
∠2
63°
1
2D
AC
B
80°
60°
40°
105°
14
52
3
150°
55°
70°
12
3
85°
55°
40°
12
3
146°
12
TIG
ERS
80° 73°
1
4-2 m
∠1 =
27
m∠
1 =
m∠
2 =
17
55
55
70
125
55
95
140
40
65
75
115
27
27
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2P
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Answers (Lesson 4-2)
Chapter 4 A5 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 5A01_A26_GEOCRM04_890513.indd 5 5/23/08 11:38:46 PM5/23/08 11:38:46 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
14
G
lenc
oe G
eom
etry
Prac
tice
An
gle
s o
f Tri
an
gle
s
Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle.
1.
2.
Fin
d e
ach
mea
sure
.
3. m
∠1
97
4. m
∠2
83
5. m
∠3
62
Fin
d e
ach
mea
sure
.
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
mea
sure
.
12. m
∠1
26
13. m
∠2
32
14. C
ON
STR
UC
TIO
N T
he
diag
ram
sh
ows
an
exam
ple
of t
he
Pra
tt T
russ
use
d in
bri
dge
con
stru
ctio
n. U
se t
he
diag
ram
to
fin
d 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 55
m
∠1 =
18,
m∠
1 =
90
m∠
1 =
85
001_
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Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
15
G
lenc
oe G
eom
etry
Wor
d Pr
oble
m P
ract
ice
An
gle
s o
f Tri
an
gle
s
62˚
136
˚
1. P
ATH
S E
ric
wal
ks a
rou
nd
a tr
ian
gula
r pa
th. A
t ea
ch c
orn
er, h
e re
cord
s th
e m
easu
re o
f th
e an
gle
he
crea
tes.
He
mak
es o
ne
com
plet
e ci
rcu
it a
rou
nd
the
path
. Wh
at i
s th
e su
m o
f th
e th
ree
angl
e m
easu
res
that
he
wro
te
dow
n?
2. S
TAN
DIN
G S
am, K
endr
a, a
nd
Ton
y ar
e st
andi
ng
in s
uch
a w
ay t
hat
if
lin
es w
ere
draw
n t
o co
nn
ect
the
frie
nds
th
ey w
ould
fo
rm a
tri
angl
e.
If S
am i
s lo
okin
g at
Ken
dra
he
nee
ds t
o tu
rn h
is h
ead
40°
to l
ook
at T
ony.
If
Ton
y is
loo
kin
g at
Sam
he
nee
ds t
o tu
rn
his
hea
d 50
° to
loo
k at
Ken
dra.
How
m
any
degr
ees
wou
ld K
endr
a h
ave
to
turn
her
hea
d to
loo
k at
Ton
y if
sh
e is
lo
okin
g at
Sam
?
3. T
OW
ERS
A l
ooko
ut
tow
er s
its
on a
n
etw
ork
of s
tru
ts a
nd
post
s. L
esli
e m
easu
red
two
angl
es o
n t
he
tow
er.
W
hat
is
the
mea
sure
of
angl
e 1?
4. Z
OO
S T
he
zoo
ligh
ts u
p th
e ch
impa
nze
e pe
n w
ith
an
ove
rhea
d li
ght
at n
igh
t. T
he
cros
s se
ctio
n o
f th
e li
ght
beam
mak
es a
n
isos
cele
s tr
ian
gle.
Th
e to
p an
gle
of t
he
tria
ngl
e is
52
and
the
exte
rior
an
gle
is 1
16. W
hat
is
the
mea
sure
of
angl
e 1?
5. D
RA
FTIN
G C
hlo
e bo
ugh
t a
draf
tin
g ta
ble
and
set
it u
p so
th
at s
he
can
dra
w
com
fort
ably
fro
m h
er s
tool
. Ch
loe
mea
sure
d th
e tw
o an
gles
cre
ated
by
the
legs
an
d th
e ta
blet
op i
n c
ase
she
had
to
dism
antl
e th
e ta
ble.
a. W
hic
h o
f th
e fo
ur
nu
mbe
red
angl
es
can
Ch
loe
dete
rmin
e by
kn
owin
g th
e tw
o an
gles
for
med
wit
h t
he
tabl
etop
? W
hat
are
th
eir
mea
sure
s?
b.
Wh
at c
oncl
usi
on c
an C
hlo
e m
ake
abou
t th
e u
nkn
own
an
gles
bef
ore
she
mea
sure
s th
em t
o fi
nd
thei
r ex
act
mea
sure
men
ts?
4-2
52˚
116 ˚
1
Kend
ra
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
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Answers (Lesson 4-2)
Chapter 4 A6 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 6A01_A26_GEOCRM04_890513.indd 6 5/23/08 11:38:52 PM5/23/08 11:38:52 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
16
G
lenc
oe G
eom
etry
Enri
chm
ent
Fin
din
g A
ng
le M
easu
res i
n T
rian
gle
sY
ou c
an u
se a
lgeb
ra t
o so
lve
prob
lem
s in
volv
ing
tria
ngl
es.
In
tri
angl
e A
BC
, m∠
A i
s tw
ice
m∠
B, a
nd
m∠
C
is 8
mor
e th
an m
∠B
. Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
Wri
te a
nd
solv
e an
equ
atio
n. L
et x
= m
∠B
.m
∠A
+ m
∠B
+ m
∠C
= 1
80
2x +
x +
(x +
8) =
180
4x
+ 8
= 1
80
4x =
172
x =
43
So,
m∠
A =
2(4
3) o
r 86
, m∠
B =
43,
an
d m
∠C
= 4
3 +
8 o
r 51
.
Sol
ve e
ach
pro
ble
m.
1. I
n t
rian
gle
DE
F, m
∠E
is
thre
e ti
mes
2.
In
tri
angl
e R
ST
, m∠
T i
s 5
mor
e th
an
m∠
D, a
nd
m∠
F i
s 9
less
th
an m
∠E
. m
∠R
, an
d m
∠S
is
10 l
ess
than
m∠
T.
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
3. I
n t
rian
gle
JK
L, m
∠K
is
fou
r ti
mes
4.
In
tri
angl
e X
YZ
, m∠
Z i
s 2
mor
e th
an t
wic
e m
∠J
, an
d m
∠L
is
five
tim
es m
∠J
. m
∠X
, an
d m
∠Y
is
7 le
ss t
han
tw
ice
m∠
X.
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
5. I
n t
rian
gle
GH
I, m
∠H
is
20 m
ore
than
6.
In
tri
angl
e M
NO
, m∠
M i
s eq
ual
to
m∠
N,
m∠
G, a
nd
m∠
G i
s 8
mor
e th
an m
∠I.
an
d m
∠O
is
5 m
ore
than
th
ree
tim
es
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
m
∠N
. Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
7. I
n t
rian
gle
ST
U, m
∠U
is
hal
f m
∠T
, 8.
In
tri
angl
e P
QR
, m∠
P i
s eq
ual
to
and
m∠
S i
s 30
mor
e th
an m
∠T
. Wh
at
m∠
Q, a
nd
m∠
R i
s 24
les
s th
an m
∠P
.is
th
e m
easu
re o
f ea
ch a
ngl
e?
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
9. W
rite
you
r ow
n p
robl
ems
abou
t m
easu
res
of t
rian
gles
.
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.
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M
Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
17
G
lenc
oe G
eom
etry
Gra
phin
g Ca
lcul
ator
Act
ivit
yC
ab
ri J
un
ior:
An
gle
s o
f Tri
an
gle
s
Cab
ri J
un
ior
can
be
use
d to
in
vest
igat
e th
e re
lati
onsh
ips
betw
een
th
e an
gles
in
a t
rian
gle.
U
se C
abri
Ju
nio
r to
in
vest
igat
e th
e su
m o
f th
e m
easu
res
of t
he
angl
es o
f a
tria
ngl
e an
d t
o in
vest
igat
e th
e m
easu
re o
f th
e ex
teri
or a
ngl
e in
a
tria
ngl
e in
rel
atio
n t
o it
s re
mot
e in
teri
or a
ngl
es.
Ste
p 1
U
se C
abri
Ju
nio
r to
dra
w a
tri
angl
e.
• S
elec
t F
2 T
rian
gle
to d
raw
a t
rian
gle.
•
Mov
e th
e cu
rsor
to
wh
ere
you
wan
t th
e fi
rst
vert
ex. T
hen
pre
ss E
NT
ER
.
• R
epea
t th
is p
roce
dure
to
dete
rmin
e th
e n
ext
two
vert
ices
of
the
tria
ngl
e.
Ste
p 2
L
abel
eac
h v
erte
x of
th
e tr
ian
gle.
•
Sel
ect
F5
Alp
h-n
um
.
• M
ove
the
curs
or t
o a
vert
ex a
nd
pres
s E
NT
ER
. En
ter
A a
nd
pres
s E
NT
ER
aga
in.
•
Rep
eat
this
pro
cedu
re t
o la
bel
vert
ex B
an
d ve
rtex
C.
Ste
p 3
D
raw
an
ext
erio
r an
gle
of �
AB
C.
•
Sel
ect
F2
Lin
e to
dra
w a
lin
e th
rou
gh −−
− B
C .
•
Sel
ect
F2
Poi
nt,
Poi
nt
on t
o dr
aw a
poi
nt
on �
��
BC
so
that
C i
s be
twee
n B
an
d th
e n
ew p
oin
t.
• S
elec
t F
5 A
lph
-nu
m t
o la
bel
the
poin
t D
.
Ste
p 4
F
ind
the
mea
sure
of
the
thre
e in
teri
or a
ngl
es o
f �
AB
C a
nd
the
exte
rior
an
gle,
∠A
CD
.
• S
elec
t F
5 M
easu
re, A
ngl
e.
• T
o fi
nd
the
mea
sure
of
∠A
BC
, sel
ect
poin
ts A
, B, a
nd
C
(wit
h t
he
vert
ex B
as
the
seco
nd
poin
t se
lect
ed).
•
Use
th
is m
eth
od t
o fi
nd
the
rem
ain
ing
angl
e m
easu
res.
Exer
cise
s
An
alyz
e yo
ur
dra
win
g.
1. U
se F
5 C
alcu
late
to
fin
d th
e su
m o
f th
e m
easu
res
of t
he
thre
e in
teri
or a
ngl
es.
2. M
ake
a co
nje
ctu
re a
bou
t th
e su
m o
f th
e in
teri
or a
ngl
es o
f a
tria
ngl
e.
3. C
han
ge t
he
dim
ensi
ons
of t
he
tria
ngl
e by
mov
ing
poin
t A
. (P
ress
CLE
AR
so
the
poin
ter
beco
mes
a b
lack
arr
ow. M
ove
the
poin
ter
clos
e to
poi
nt
A u
nti
l th
e ar
row
bec
omes
tr
ansp
aren
t an
d po
int
A i
s bl
inki
ng.
Pre
ss A
LP
HA
to
chan
ge t
he
arro
w t
o a
han
d. T
hen
m
ove
the
poin
t.)
Is y
our
con
ject
ure
sti
ll t
rue?
4. W
hat
is
the
mea
sure
of
∠A
CD
? W
hat
is
the
sum
of
the
mea
sure
s of
th
e tw
o re
mot
e in
teri
or a
ngl
es (
∠A
BC
an
d ∠
BA
C)?
5. M
ake
a co
nje
ctu
re a
bou
t th
e re
lati
onsh
ip b
etw
een
th
e m
easu
re o
f an
ext
erio
r an
gle
of a
tr
ian
gle
and
its
two
rem
ote
inte
rior
an
gles
.
6. C
han
ge t
he
dim
ensi
ons
of t
he
tria
ngl
e by
mov
ing
poin
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
°
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Answers (Lesson 4-2)
Chapter 4 A7 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 7A01_A26_GEOCRM04_890513.indd 7 5/23/08 11:39:00 PM5/23/08 11:39:00 PM
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
18
G
lenc
oe G
eom
etry
Geo
met
er’s
Ske
tchp
ad A
ctiv
ity
An
gle
s o
f Tri
an
gle
s
Th
e G
eom
eter
’s S
ketc
hpa
d ca
n b
e u
sed
to i
nve
stig
ate
the
rela
tion
ship
s be
twee
n t
he
angl
es
in a
tri
angl
e.
U
se T
he
Geo
met
er’s
Sk
etch
pad
to
inve
stig
ate
the
sum
of
the
mea
sure
s of
th
e an
gles
of
a tr
ian
gle
and
to
inve
stig
ate
the
mea
sure
of
the
exte
rior
an
gle
in a
tri
angl
e in
rel
atio
n t
o it
s re
mot
e in
teri
or a
ngl
es.
Ste
p 1
U
se T
he
Geo
met
er’s
Ske
tch
pad
to d
raw
a t
rian
gle.
• C
onst
ruct
a r
ay b
y se
lect
ing
the
Ray
too
l fr
om t
he
tool
bar.
Fir
st, c
lick
wh
ere
you
wan
t th
e fi
rst
poin
t. T
hen
cli
ck o
n a
sec
ond
poin
t to
dra
w t
he
ray.
•
Nex
t, s
elec
t th
e S
egm
ent
tool
fro
m t
he
tool
bar.
Use
th
e en
dpoi
nt
of t
he
ray
as
the
firs
t po
int
for
the
segm
ent
and
clic
k on
a s
econ
d po
int
to c
onst
ruct
th
e se
gmen
t.
• C
onst
ruct
an
oth
er s
egm
ent
join
ing
the
seco
nd
poin
t of
th
e n
ew s
egm
ent
to a
po
int
on t
he
ray.
Ste
p 2
D
ispl
ay t
he
labe
ls f
or e
ach
poi
nt.
U
se t
he
poin
ter
tool
to
sele
ct a
ll
fou
r po
ints
. Dis
play
th
e la
bels
by
sel
ecti
ng
Sh
ow L
abel
s fr
om
the
Dis
pla
y m
enu
.
Ste
p 3
F
ind
the
mea
sure
s of
th
e th
ree
inte
rior
an
gles
an
d th
e on
e ex
teri
or
angl
e. T
o fi
nd
the
mea
sure
of
angl
e A
BC
, use
th
e S
elec
tion
Arr
ow t
ool
to s
elec
t po
ints
A, B
, an
d C
(w
ith
th
e ve
rtex
B a
s th
e se
con
d po
int
sele
cted
). T
hen
, un
der
the
Mea
sure
m
enu
, sel
ect
An
gle.
Use
th
is m
eth
od
to f
ind
the
rem
ain
ing
angl
e m
easu
res.
Exer
cise
sA
nal
yze
you
r d
raw
ing.
1. W
hat
is
the
sum
of
the
mea
sure
s of
th
e th
ree
inte
rior
an
gles
?
2. M
ake
a co
nje
ctu
re a
bou
t th
e su
m o
f th
e in
teri
or a
ngl
es o
f a
tria
ngl
e.
3. C
han
ge t
he
dim
ensi
ons
of t
he
tria
ngl
e by
sel
ecti
ng
poin
t A
wit
h t
he
poin
ter
tool
an
d m
ovin
g it
. Is
you
r co
nje
ctu
re s
till
tru
e?
4. W
hat
is
the
mea
sure
of ∠
AC
D?
Wh
at i
s th
e su
m o
f th
e m
easu
res
of t
he
two
rem
ote
inte
rior
an
gles
(∠
AB
C a
nd
∠B
AC
)? 5
. Mak
e a
con
ject
ure
abo
ut
the
rela
tion
ship
bet
wee
n t
he
mea
sure
of
an
exte
rior
an
gle
of a
tri
angl
e an
d it
s tw
o re
mot
e in
teri
or a
ngl
es.
6. C
han
ge t
he
dim
ensi
ons
of t
he
tria
ngl
e by
sel
ecti
ng
poin
t A
wit
h t
he
poin
ter
tool
an
d m
ovin
g it
. Is
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 =
11
8.19
°m
/A
BC
+ m
/B
AC
+ m
/A
CB
= 18
0.00
°m
/A
BC
+ m
/B
AC
= 11
8.19
°
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
001_
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-3
Cha
pte
r 4
19
G
lenc
oe G
eom
etry
Co
ng
ruen
ce a
nd
Co
rres
po
nd
ing
Par
ts
Tri
angl
es t
hat
hav
e th
e sa
me
size
an
d sa
me
shap
e ar
e co
ngr
uen
t tr
ian
gles
. Tw
o tr
ian
gles
are
con
gru
ent
if a
nd
only
if
all
thre
e pa
irs
of c
orre
spon
din
g an
gles
are
con
gru
ent
and
all
thre
e pa
irs
of c
orre
spon
din
g si
des
are
con
gru
ent.
In
th
e fi
gure
, �A
BC
� �
RS
T.
If
�X
YZ
� �
RS
T, n
ame
the
pai
rs o
f co
ngr
uen
t an
gles
an
d c
ongr
uen
t si
des
.
∠X
� ∠
R, ∠
Y �
∠S
, ∠Z
� ∠
T −
−
XY
� −
−
RS
, −−
XZ
� −
−
RT
, −−
YZ
� −
−
ST
Exer
cise
sS
how
th
at t
he
pol
ygon
s ar
e co
ngr
uen
t b
y id
enti
fyin
g al
l co
ngr
uen
t co
rres
pon
din
g p
arts
. Th
en w
rite
a c
ongr
uen
ce s
tate
men
t.
1.
CA
B
LJK
2.
C
D
A
B
3. K J
L M
4. F
GL
K JE
5.
BD
CA
6.
R TUS
�A
BC
� �
DE
F.
7. F
ind
the
valu
e of
x.
8. F
ind
the
valu
e of
y.
Y
XZ
T
SR
AC
B
R
TS
4-3
Stud
y G
uide
and
Inte
rven
tion
Co
ng
ruen
t Tri
an
gle
s
Th
ird
An
gle
s
Th
eo
rem
If t
wo
an
gle
s o
f o
ne
tria
ng
le a
re c
on
gru
en
t to
tw
o a
ng
les
of
a s
eco
nd
tria
ng
le,
the
n t
he
th
ird
an
gle
s o
f th
e t
ria
ng
les a
re c
on
gru
en
t.
Exam
ple
2x+
y(2
y-
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; −
−
A
B �
−−
JK
;
∠
AC
B �
∠C
BD
; −
−
A
C �
−
−
B
D
∠
KM
J �
∠M
KL; −
−
K
J �
−
−
LM
−
−
B
C �
−−
K
L ; −
−
A
C �
−−
JL
−
−
A
B �
−
−
C
D
−
−
K
L �
−
−
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; −
−
E
F �
−−
JK
;
∠
AB
C �
∠D
CB
;
∠
RS
U �
∠T
SU
;
−
−
E
G �
−−
JL ; −
−
FG
� −
−
K
L
∠
AC
B �
∠D
BC
;
∠
RU
S �
∠T
US
;
�FG
E �
�K
LJ
−
−
A
B �
−−
D
C ; −
−
A
C �
−−
D
B ;
−
−
R
U �
−−
TU
; −
−
R
S �
−−
TS ;
−
−
B
C �
−−
C
B ;
�A
BC
� �
DC
B
−
−
S
U �
−−
S
U ;
�R
SU
� �
TS
U
019_
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:31:
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M
Answers (Lesson 4-2 and Lesson 4-3)
Chapter 4 A8 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 8A01_A26_GEOCRM04_890513.indd 8 5/23/08 11:39:05 PM5/23/08 11:39:05 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
20
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Co
ng
ruen
t Tri
an
gle
s
4-3
Exam
ple
Pro
ve T
rian
gle
s C
on
gru
ent
Tw
o tr
ian
gles
are
con
gru
ent
if a
nd
only
if
thei
r co
rres
pon
din
g pa
rts
are
con
gru
ent.
Cor
resp
ondi
ng
part
s in
clu
de c
orre
spon
din
g an
gles
an
d co
rres
pon
din
g si
des.
Th
e ph
rase
“if
an
d on
ly i
f” m
ean
s th
at b
oth
th
e co
ndi
tion
al a
nd
its
con
vers
e ar
e tr
ue.
For
tri
angl
es, w
e sa
y, “
Cor
resp
ondi
ng
part
s of
con
gru
ent
tria
ngl
es a
re
con
gru
ent,
” or
CP
CT
C.
W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:
−−
AB
� −−
− C
B ,
−−−
AD
� −−
− C
D , ∠
BA
D �
∠B
CD
−−
− B
D b
isec
ts ∠
AB
C P
rove
: �A
BD
� �
CB
D
Pro
of:
Sta
tem
ent
Rea
son
1. −
−
AB
� −−
− C
B ,
−−−
AD
� −−
− C
D
2. −−
− B
D �
−−−
BD
3.
∠B
AD
� ∠
BC
D4.
∠A
BD
� ∠
CB
D5.
∠B
DA
� ∠
BD
C6.
�A
BD
� �
CB
D
1. G
iven
2. R
efle
xive
Pro
pert
y of
con
gru
ence
3. G
iven
4. D
efin
itio
n o
f an
gle
bise
ctor
5. T
hir
d A
ngl
es T
heo
rem
6. C
PC
TC
Exer
cise
sW
rite
a t
wo-
colu
mn
pro
of.
1. G
iven
: ∠A
� ∠
C, ∠
D �
∠B
, −−
− A
D �
−−−
CB
, −
−
AE
� −−
− C
E ,
−−
AC
bis
ects
−−−
BD
P
rove
: �A
ED
� �
CE
BP
roo
f:
Sta
tem
en
tsR
easo
ns
1.
∠A
� ∠
C,
∠D
� ∠
B
2.
∠A
ED
� ∠
CE
B
3.
−−
A
D �
−−
C
B ,
−−
A
E �
−−
C
E
4.
−−
D
E �
−
−
B
E
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
Wri
te a
par
agra
ph
pro
of.
2. G
iven
: −−
− B
D b
isec
ts ∠
AB
C a
nd
∠A
DC
, −
−
AB
� −−
− C
B ,
−−
AB
� −−
− A
D ,
−−−
CB
� −−
− D
C
Pro
ve: �
AB
D �
�C
BD
We a
re g
iven
−−
B
D b
isects
∠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
−−
A
B �
−−
C
B ,
−−
A
B �
−
−
A
D ,
an
d −
−
C
B �
−−
D
C .
Usin
g t
he s
ub
sti
tuti
on
pro
pert
y,
we c
an
dete
rmin
e t
hat
−−
A
D �
−−
C
D .
Fin
ally,
−−
B
D �
−
−
B
D u
sin
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
.
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NA
ME
DA
TE
PE
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D
Lesson 4-3
Cha
pte
r 4
21
G
lenc
oe G
eom
etry
4-3
Sh
ow t
hat
pol
ygon
s ar
e co
ngr
uen
t b
y id
enti
fyin
g al
l co
ngr
uen
t co
rres
pon
din
g p
arts
. Th
en w
rite
a c
ongr
uen
ce s
tate
men
t.
1.
L
P
J
S
V
T
2.
In
th
e fig
ur
e, �
AB
C �
�F
DE
. 3
. Fin
d th
e va
lue
of x
. 36
4. F
ind
the
valu
e of
y.
48
5. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: −
−−
AB
� −
−−
CB
, −−
−
AD
� −
−−
CD
, ∠A
BD
� ∠
CB
D,
∠A
DB
� ∠
CD
B
Pro
ve: �
AB
D �
�C
BD
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
−−
A
B �
−−
C
B ,
−−
A
D �
−
−
C
D
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
Skill
s Pr
acti
ceC
on
gru
en
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
; −
−
D
E �
−
−
D
G ;
−−
P
L �
−−
V
S ;
−−
JL �
−−
TS
−
−
G
F �
−−
E
F ;
�D
EF �
�D
GF
�JP
L �
�TV
S
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Answers (Lesson 4-3)
Chapter 4 A9 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 9A01_A26_GEOCRM04_890513.indd 9 5/29/08 7:14:12 PM5/29/08 7:14:12 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
22
G
lenc
oe G
eom
etry
Prac
tice
Co
ng
ruen
t Tri
an
gle
s
Sh
ow t
hat
th
e p
olyg
ons
are
con
gru
ent
by
ind
enti
fyin
g al
l co
ngr
uen
t co
rres
pon
din
g p
arts
. Th
en w
rite
a c
ongr
uen
ce s
tate
men
t.
1.
D
R
SC
A
B
2.
MN
L
P
Q
Pol
ygon
AB
CD
� p
olyg
on P
QR
S.
3. F
ind
the
valu
e of
x.
4. F
ind
the
valu
e of
y.
5. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.G
iven
: ∠P
� ∠
R, ∠
PS
Q �
∠R
SQ
, −−
− P
Q �
−−−
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.
−−
P
Q �
−−
R
Q ,
−−
P
S �
−−
R
S
3.
Giv
en
4.
−−
Q
S �
−−
Q
S
4.
Refl
exiv
e P
rop
ert
y
5.
�P
QS
� �
RQ
S
5.
CP
CT
C
6. Q
UIL
TIN
G
a. I
ndi
cate
th
e tr
ian
gles
th
at a
ppea
r to
be
con
gru
ent.
b.
Nam
e th
e co
ngr
uen
t an
gles
an
d co
ngr
uen
t si
des
of a
pai
r of
co
ngr
uen
t tr
ian
gles
.
B
A I
E FH
G
CD
4-3
(2x
+4)
° (3y
-3)
80°
100°
1012
∠A
� ∠
D;
∠B
� ∠
R
∠L �
∠Q
; ∠
M �
∠P
∠C
� ∠
S;
−−
A
B �
−−
R
D
∠M
NL �
∠P
NQ
;
−−
B
C �
−−
R
S ;
−−
A
C �
−−
S
D
−−
LM
� −
−
P
Q ;
−−
M
N �
−−
N
P
�A
BC
� �
DR
S
−−
LM
� −
−
N
Q
�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;
−−
A
B �
−−
E
B ,
−−
B
I �
−−
B
F ,
−−
A
I �
−−
E
F
x =
48
y =
5
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NA
ME
DA
TE
PE
RIO
D
Lesson 4-3
Cha
pte
r 4
23
G
lenc
oe G
eom
etry
Wor
d Pr
oble
m P
ract
ice
Co
ng
ruen
t Tri
an
gle
s
1. P
ICTU
RE
HA
NG
ING
Can
dice
hu
ng
a pi
ctu
re t
hat
was
in
a t
rian
gula
r fr
ame
on h
er b
edro
om w
all.
Befo
reAB
C
A
B
CAfte
r
O
ne
day,
it
fell
to
the
floo
r, b
ut
did
not
br
eak
or b
end.
Th
e fi
gure
sh
ows
the
obje
ct b
efor
e an
d af
ter
the
fall
. Lab
el
the
vert
ices
on
th
e fr
ame
afte
r th
e fa
ll
acco
rdin
g to
th
e “b
efor
e” f
ram
e.
2. S
IER
PIN
SKI’S
TR
IAN
GLE
Th
e fi
gure
be
low
is
a po
rtio
n o
f S
ierp
insk
i’s
Tri
angl
e. T
he
tria
ngl
e h
as t
he
prop
erty
th
at a
ny
tria
ngl
e m
ade
from
an
y co
mbi
nat
ion
of
edge
s is
equ
ilat
eral
. H
ow m
any
tria
ngl
es i
n t
his
por
tion
are
co
ngr
uen
t to
th
e bl
ack
tria
ngl
e at
th
e bo
ttom
cor
ner
?
3. Q
UIL
TIN
G S
tefa
n d
rew
th
is p
atte
rn f
or
a pi
ece
of h
is q
uil
t. I
t is
mad
e u
p of
co
ngr
uen
t is
osce
les
righ
t tr
ian
gles
. He
drew
on
e tr
ian
gle
and
then
rep
eate
dly
drew
it
all
the
way
aro
un
d.
45˚
W
hat
are
th
e m
issi
ng
mea
sure
s of
th
e an
gles
of
the
tria
ngl
e?
4. M
OD
ELS
Dan
a bo
ugh
t a
mod
el a
irpl
ane
kit.
Wh
en h
e op
ened
th
e bo
x, t
hes
e tw
o co
ngr
uen
t tr
ian
gula
r pi
eces
of
woo
d fe
ll
out
of i
t.
A
B
CQ
R
P
I
den
tify
th
e tr
ian
gle
that
is
con
gru
ent
to
�A
BC
.
5. G
EOG
RA
PHY
Igo
r n
otic
ed o
n a
map
th
at t
he
tria
ngl
e w
hos
e ve
rtic
es a
re t
he
supe
rmar
ket,
th
e li
brar
y, a
nd
the
post
of
fice
(�
SL
P)
is c
ongr
uen
t to
th
e tr
ian
gle
wh
ose
vert
ices
are
Igo
r’s
hom
e,
Jaco
b’s
hom
e, a
nd
Ben
’s h
ome
(�IJ
B).
T
hat
is,
�S
LP
� �
IJB
.
a. T
he
dist
ance
bet
wee
n t
he
supe
rmar
ket
and
the
post
off
ice
is
1 m
ile.
Wh
ich
pat
h a
lon
g th
e tr
ian
gle
�IJ
B i
s co
ngr
uen
t to
th
is?
b
. T
he
mea
sure
of
∠L
PS
is
40. I
den
tify
th
e an
gle
that
is
con
gru
ent
to t
his
an
gle
in �
IJB
.
4-3 11
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
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Answers (Lesson 4-3)
Chapter 4 A10 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 10A01_A26_GEOCRM04_890513.indd 10 5/23/08 11:39:22 PM5/23/08 11:39:22 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
24
G
lenc
oe G
eom
etry
Enri
chm
ent
4-3
Overl
ap
pin
g T
rian
gle
sS
omet
imes
wh
en t
rian
gles
ove
rlap
eac
h o
ther
, th
ey s
har
e co
rres
pon
din
g pa
rts.
Wh
en t
ryin
g to
pic
ture
th
e co
rres
pon
din
g pa
rts,
try
red
raw
ing
the
tria
ngl
es s
epar
atel
y.
In t
he
figu
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, r
edra
w �
AB
D a
nd
�D
CA
lik
e th
is.
30°
Now
you
can
see
th
at ∠
BD
A �
∠C
AD
bec
ause
th
ey a
re c
orre
spon
din
g pa
rts
of c
ongr
uen
t tr
ian
gles
, so
m∠
CA
D =
30.
Usi
ng
the
Tri
angl
e S
um
Th
eore
m, 1
80 -
(90
+ 3
0) =
60,
so
m∠
CD
A =
60.
Exer
cise
s 1
. In
th
e fi
gure
�A
BC
� �
CD
B, m
∠A
BC
= 1
10, a
nd
m∠
D =
20.
Fin
d m
∠A
CB
an
d m
∠D
CB
.
2. W
rite
a t
wo
colu
mn
pro
of.
Giv
en: ∠
A a
nd
∠B
are
rig
ht
angl
es, ∠
AC
D �
∠B
DC
, −−
− A
D �
−−
− B
C ,
−−
AC
� −−
− B
D
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.
−−
A
D �
−−
B
C ,
−−
A
C �
−−
B
D
5.
Giv
en
6.
−−
D
C �
−−
C
D
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
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NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Cha
pte
r 4
25
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
SSS
Post
ula
te Y
ou k
now
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t if
cor
resp
ondi
ng
side
s ar
e co
ngr
uen
t an
d co
rres
pon
din
g an
gles
are
con
gru
ent.
Th
e S
ide-
Sid
e-S
ide
(SS
S)
Pos
tula
te l
ets
you
sh
ow t
hat
tw
o tr
ian
gles
are
con
gru
ent
if y
ou k
now
on
ly t
hat
th
e si
des
of o
ne
tria
ngl
e ar
e co
ngr
uen
t to
th
e si
des
of t
he
seco
nd
tria
ngl
e.
W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:
−−
AB
� −−
− D
B a
nd
C i
s th
e m
idpo
int
of −−
− A
D .
Pro
ve: �
AB
C �
�D
BC
B C
DA
Sta
tem
ents
R
easo
ns
1. −
−
AB
� −−
− D
B
1. G
iven
2. C
is
the
mid
poin
t of
−−−
AD
. 2.
Giv
en3.
−−
AC
� −−
− D
C
3. M
idpo
int
Th
eore
m4.
−−−
BC
� −−
− B
C
4. R
efle
xive
Pro
pert
y of
�5.
�A
BC
� �
DB
C
5. S
SS
Pos
tula
te
Exer
cise
s
Wri
te a
tw
o-co
lum
n p
roof
.
1.
BY
CA
XZ
Giv
en:
AB
�−
−X
Y ,
−−
AC
�−
−X
Z ,
−−−
BC
�−
−Y
Z
Pro
ve:�
AB
C�
�X
YZ
2.
TU
RS
Giv
en:
−−
RS
� −−
− U
T ,
−−
RT
� −−
− U
S
Pro
ve:
�R
ST
� �
UT
S
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.
−−
A
B �
−−
X
Y
1.
Giv
en
−
−
A
C �
−−
X
Z
−
−
B
C �
−−
Y
Z
2.
�A
BC
� �
XY
Z
2.
SS
S P
ost.
1.
−−
R
S �
−−
U
T
1.
Giv
en
−
−
R
T �
−−
U
S
2.
−−
S
T �
−−
TS
2
. R
efl
. P
rop
.
3.
�R
ST �
�U
TS
3.
SS
S P
ost.
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Answers (Lesson 4-3 and Lesson 4-4)
Chapter 4 A11 Glencoe Geometry
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
26
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
SAS
Post
ula
te A
not
her
way
to
show
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t is
to
use
th
e S
ide-
An
gle-
Sid
e (S
AS
) P
ostu
late
.
F
or e
ach
dia
gram
, det
erm
ine
wh
ich
pai
rs o
f tr
ian
gles
can
be
pro
ved
con
gru
ent
by
the
SA
S P
ostu
late
.
a.
A BC
X YZ
b
. D
H
FE
G
J
c.
P
Q
S
1
2R
I
n �
AB
C, t
he
angl
e is
not
T
he
righ
t an
gles
are
T
he
incl
ude
d an
gles
, “i
ncl
ude
d” b
y th
e si
des
−−
AB
c
ongr
uen
t an
d th
ey a
re t
he
∠1
and
∠2,
are
con
gru
ent
and
−−
AC
. So
the
tria
ngl
es
in
clu
ded
angl
es f
or t
he
b
ecau
se t
hey
are
ca
nn
ot b
e pr
oved
con
gru
ent
c
ongr
uen
t si
des.
a
lter
nat
e in
teri
or a
ngl
es
by t
he
SA
S P
ostu
late
. �
DE
F �
�J
GH
by
the
f
or t
wo
para
llel
lin
es.
S
AS
Pos
tula
te.
�P
SR
� �
RQ
P b
y th
e
S
AS
Pos
tula
te.
Exer
cise
s
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
Wri
te t
he
spec
ifie
d t
ype
of p
roof
. 1
. Wri
te a
tw
o co
lum
n p
roof
. 2
. Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: NP
= P
M,
−−−
NP
⊥ −
−
PL
G
iven
: AB
= C
D,
−−
AB
‖ −−
− C
D
Pro
ve: �
NP
L �
�M
PL
P
rove
: �A
CD
� �
CA
B
Pro
of:
Pro
of:
3. W
rite
a p
arag
raph
pro
of.
Giv
en: V
is
the
mid
poin
t of
−−
YZ
.
V i
s th
e m
idpo
int
of −−
− W
X .
Pro
ve: �
XV
Z �
�W
VY
S
ince V
is t
he m
idp
oin
t o
f −
−
Y
Z a
nd
th
e m
idp
oin
t o
f −
−
W
X ,
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, −
−
N
P ⊥
−−
P
L
2.
∠M
PL i
s a
rig
ht
an
gle
∠N
PL i
s a
rig
ht
an
gle
3.
∠M
PL
� ∠
NP
L
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, −
−
A
B ‖
−−
C
D
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
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NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Cha
pte
r 4
27
G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er �
AB
C �
�K
LM
. Exp
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)
PRO
OF
Wri
te t
he
spec
ifie
d t
ype
of p
roof
.
3. W
rite
a f
low
pro
of.
G
iven
: −
−
PR
� −−
− D
E ,
−−
PT
� −−
− D
F
∠R
� ∠
E, ∠
T �
∠F
P
rove
: �
PR
T �
�D
EF
PR�
DE
Give
n
PT�
DF
Give
n
∠R
�∠
EGi
ven
∠P
�∠
DTh
ird A
ngle
Theo
rem
�PR
T�
�D
EFSA
S
∠T
�∠
FGi
ven
4. W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:
−−
AB
� −−
− C
B , D
is
the
mid
poin
t of
−−
AC
.P
rove
: �A
BD
� �
CB
D
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
−−
A
B �
−−
C
B ,
D i
s t
he m
idp
oin
t o
f −
−
A
C 1
. G
iven
2.
−−
A
D �
−−
D
C
2.
Defi
nit
ion
of
Mid
po
int
3.
−−
B
D �
−−
B
D
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
Skill
s Pr
acti
ceP
rovin
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 ,
AC
= 2
, K
M =
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:
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Answers (Lesson 4-4)
Chapter 4 A12 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 12A01_A26_GEOCRM04_890513.indd 12 5/29/08 7:14:31 PM5/29/08 7:14:31 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
28
G
lenc
oe G
eom
etry
Prac
tice
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
Det
erm
ine
wh
eth
er �
DE
F �
�P
QR
giv
en t
he
coor
din
ates
of
the
vert
ices
. Exp
lain
.
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
is
the
mid
poin
t of
−−
RT
.P
rove
: �
RS
V �
�T
SV
Det
erm
ine
wh
ich
pos
tula
te c
an b
e u
sed
to
pro
ve t
hat
th
e tr
ian
gles
are
con
gru
ent.
If
it
is n
ot p
ossi
ble
to
pro
ve c
ongr
uen
ce, w
rite
not
pos
sibl
e.
4.
5.
6.
7. I
ND
IREC
T M
EASU
REM
ENT
To
mea
sure
th
e w
idth
of
a si
nkh
ole
on
h
is p
rope
rty,
Har
mon
mar
ked
off
con
gru
ent
tria
ngl
es a
s sh
own
in
th
e di
agra
m. H
ow d
oes
he
know
th
at t
he
len
gth
s A
'B' a
nd
AB
are
equ
al?
A'
B'
AB
C
S
R V T
4-4 DE
= 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
, −
−
A
C �
−−
−
A
′C
an
d −
−
B
C �
−−
−
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
Give
n
SV�
SVRe
flexi
vePr
oper
ty
RV
�V
TDe
finiti
onof
mid
poin
t
V is
th
em
idp
oin
t o
f R
T.Gi
ven
ΔR
SV�
ΔTS
VSS
S
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Cha
pte
r 4
29
G
lenc
oe G
eom
etry
Wor
d Pr
oble
m P
ract
ice
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
1. S
TIC
KS
Tys
on h
ad t
hre
e st
icks
of
len
gth
s 24
in
ches
, 28
inch
es, a
nd
30 i
nch
es. I
s it
pos
sibl
e to
mak
e tw
o n
onco
ngr
uen
t tr
ian
gles
usi
ng
the
sam
e th
ree
stic
ks?
Exp
lain
.
2. B
AK
ERY
Son
ia m
ade
a sh
eet
of
bakl
ava.
Sh
e h
as m
arki
ngs
on
her
pan
so
th
at s
he
can
cu
t th
em i
nto
lar
ge
squ
ares
. Aft
er s
he
cuts
th
e pa
stry
in
sq
uar
es, s
he
cuts
th
em d
iago
nal
ly t
o fo
rm t
wo
con
gru
ent
tria
ngl
es. W
hic
h
post
ula
te c
ould
you
use
to
prov
e th
e tw
o tr
ian
gles
con
gru
ent?
3. C
AK
E C
arl
had
a p
iece
of
cake
in
th
e sh
ape
of a
n i
sosc
eles
tri
angl
e w
ith
an
gles
26,
77,
an
d 77
. He
wan
ted
to
divi
de i
t in
to t
wo
equ
al p
arts
, so
he
cut
it t
hro
ugh
th
e m
iddl
e of
th
e 26
an
gle
to
the
mid
poin
t of
th
e op
posi
te s
ide.
He
says
th
at b
ecau
se h
e is
div
idin
g it
at
the
mid
poin
t of
a s
ide,
th
e tw
o pi
eces
are
co
ngr
uen
t. I
s th
is e
nou
gh i
nfo
rmat
ion
? E
xpla
in.
4. T
ILES
Tam
my
inst
alls
bat
hro
om t
iles
. H
er c
urr
ent
job
requ
ires
til
es t
hat
are
eq
uil
ater
al t
rian
gles
an
d al
l th
e ti
les
hav
e to
be
con
gru
ent
to e
ach
oth
er. S
he
has
a b
ig s
ack
of t
iles
all
in
th
e sh
ape
of e
quil
ater
al t
rian
gles
. Alt
hou
gh s
he
know
s th
at a
ll t
he
tile
s ar
e eq
uil
ater
al,
she
is n
ot s
ure
th
ey a
re a
ll t
he
sam
e si
ze. W
hat
mu
st s
he
mea
sure
on
eac
h
tile
to
be s
ure
th
ey a
re c
ongr
uen
t?
Exp
lain
.
5. I
NV
ESTI
GA
TIO
NA
n i
nve
stig
ator
at
a cr
ime
scen
e fo
un
d a
tria
ngu
lar
piec
e of
to
rn f
abri
c. T
he
inve
stig
ator
re
mem
bere
d th
at o
ne
of t
he
susp
ects
h
ad a
tri
angu
lar
hol
e in
th
eir
coat
. P
erh
aps
it w
as a
mat
ch. U
nfo
rtu
nat
ely,
to
avo
id t
ampe
rin
g w
ith
evi
den
ce, t
he
inve
stig
ator
did
not
wan
t to
tou
ch t
he
fabr
ic a
nd
cou
ld n
ot f
it i
t to
th
e co
at
dire
ctly
.
a. I
f th
e in
vest
igat
or m
easu
res
all
thre
e si
de l
engt
hs
of t
he
fabr
ic a
nd
the
hol
e, c
an t
he
inve
stig
ator
mak
e a
con
clu
sion
abo
ut
wh
eth
er o
r n
ot t
he
hol
e co
uld
hav
e be
en f
ille
d by
th
e fa
bric
?
b.
If t
he
inve
stig
ator
mea
sure
s tw
o si
des
of t
he
fabr
ic a
nd
the
incl
ude
d an
gle
and
then
mea
sure
s tw
o si
des
of t
he
hol
e an
d th
e in
clu
ded
angl
e ca
n h
e de
term
ine
if i
t is
a m
atch
? E
xpla
in.
4-4
No
. T
he 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
p
rove 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.
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Answers (Lesson 4-4)
Chapter 4 A13 Glencoe Geometry
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ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
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cGraw
-Hill C
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panies, Inc.
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NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
30
G
lenc
oe G
eom
etry
Enri
chm
ent
4-4
Co
ng
ruen
t Tri
an
gle
s i
n t
he C
oo
rdin
ate
Pla
ne
If y
ou k
now
th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f tw
o tr
ian
gles
in
th
e co
ordi
nat
e pl
ane,
you
can
oft
en d
ecid
e w
het
her
th
e tw
o tr
ian
gles
are
con
gru
ent.
Th
ere
may
be
mor
e th
an o
ne
way
to
do t
his
.
1. C
onsi
der
�A
BD
an
d �
CD
B w
hos
e ve
rtic
es h
ave
coor
din
ates
A(0
, 0),
B
(2, 5
), C
(9, 5
), a
nd
D(7
, 0).
Bri
efly
des
crib
e h
ow y
ou c
an u
se w
hat
you
kn
ow a
bou
t co
ngr
uen
t tr
ian
gles
an
d th
e co
ordi
nat
e pl
ane
to s
how
th
at
�A
BD
� �
CD
B. Y
ou m
ay w
ish
to
mak
e a
sket
ch t
o h
elp
get
you
sta
rted
.
2. C
onsi
der
�P
QR
an
d �
KL
M w
hos
e ve
rtic
es a
re t
he
foll
owin
g po
ints
.
P
(1, 2
) Q
(3, 6
) R
(6, 5
)
K
(-2,
1)
L(-
6, 3
) M
(-5,
6)
B
rief
ly d
escr
ibe
how
you
can
sh
ow t
hat
�P
QR
� �
KL
M.
3. I
f yo
u k
now
th
e co
ordi
nat
es o
f al
l th
e ve
rtic
es o
f tw
o tr
ian
gles
, is
it
alw
ays
poss
ible
to
tell
wh
eth
er t
he
tria
ngl
es a
re c
ongr
uen
t? E
xpla
in.
Sam
ple
an
sw
er:
Sh
ow
th
at
the s
lop
es o
f −
−
A
D a
nd
−−
B
C a
re e
qu
al. C
on
clu
de
that
−−
B
C |
| −
−
A
D .
Use 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
−−
B
D i
s 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
.
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Cha
pte
r 4
31
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
ASA
Po
stu
late
Th
e A
ngl
e-S
ide-
An
gle
(AS
A)
Pos
tula
te l
ets
you
sh
ow t
hat
tw
o tr
ian
gles
ar
e co
ngr
uen
t.
W
rite
a t
wo
colu
mn
pro
of.
Giv
en:
−−
AB
‖ −−
− C
D
∠
CB
D �
∠A
DB
Pro
ve: �
AB
D �
�C
DB
Sta
tem
ents
Rea
son
s1.
−−
AB
‖ −−
− C
D
2. ∠
CB
D �
∠A
DB
3. ∠
AB
D �
∠B
DC
4. B
D =
BD
5. �
AB
D �
�C
DB
1. G
iven
2. G
iven
3. A
lter
nat
e In
teri
or A
ngl
es T
heo
rem
4. R
efle
xive
Pro
pert
y of
con
gru
ence
5. A
SA
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
Exer
cise
sPR
OO
F W
rite
th
e sp
ecif
ied
typ
e of
pro
of.
1. W
rite
a t
wo
colu
mn
pro
of.
2. W
rite
a p
arag
raph
pro
of.
S
V
U
RT
AC
B
E
D
Giv
en: ∠
S �
∠V
,
G
iven
: −−
− C
D b
isec
ts −
−
AE
, −
−
AB
‖ −−
− C
D
T
is
the
mid
poin
t of
−−
SV
∠
E �
∠B
CA
Pro
ve: �
RT
S �
�U
TV
P
rove
: �A
BC
� �
CD
E
Pro
of:
W
e 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
�
∠C
DE
by 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
−−
S
V
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
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Answers (Lesson 4-4 and Lesson 4-5)
Chapter 4 A14 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 14A01_A26_GEOCRM04_890513.indd 14 5/23/08 11:39:53 PM5/23/08 11:39:53 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
32
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
AA
S Th
eore
m A
not
her
way
to
show
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t is
th
e A
ngl
e-A
ngl
e-S
ide
(AA
S)
Th
eore
m.
You
now
hav
e fi
ve w
ays
to s
how
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t.•
defi
nit
ion
of
tria
ngl
e co
ngr
uen
ce
• A
SA
Pos
tula
te•
SS
S P
ostu
late
•
AA
S T
heo
rem
• S
AS
Pos
tula
te
In
th
e d
iagr
am, ∠
BC
A �
∠D
CA
. Wh
ich
sid
es
ar
e co
ngr
uen
t? W
hic
h a
dd
itio
nal
pai
r of
cor
resp
ond
ing
par
ts
nee
ds
to b
e co
ngr
uen
t fo
r th
e tr
ian
gles
to
be
con
gru
ent
by
the
AA
S T
heo
rem
?
−−
AC
� −
−
AC
by
the
Ref
lexi
ve P
rope
rty
of c
ongr
uen
ce. T
he
con
gru
ent
angl
es c
ann
ot b
e ∠
1 an
d ∠
2, b
ecau
se −
−
AC
wou
ld b
e th
e in
clu
ded
side
. If
∠B
� ∠
D, t
hen
�A
BC
� �
AD
C b
y th
e A
AS
Th
eore
m.
Exer
cise
sPR
OO
F W
rite
th
e sp
ecif
ied
typ
e of
pro
of.
1. W
rite
a t
wo
colu
mn
pro
of.
G
iven
: −−
− B
C ‖
−−
EF
AB
= E
D
∠C
� ∠
F
Pro
ve: �
AB
D �
�D
EF
Sta
tem
en
tsR
easo
ns
1.
−−
B
C ‖
−−
E
F
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
iven
: ∠S
� ∠
U;
−−
TR
bis
ects
∠S
TU
.
Pro
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
Give
n
Give
n
RT
�R
T
Refl.
Pro
p. o
f �
Def.o
f∠ b
isec
tor
TR b
isec
ts ∠
STU
.
SRT
� U
RT
∠ST
R�
∠U
TR
AAS
∠SR
T�
∠U
RT
CPCT
C∠
S�
∠U
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Cha
pte
r 4
33
G
lenc
oe G
eom
etry
PRO
OF
Wri
te a
flo
w p
roof
.
1. G
iven
: ∠
N �
∠L
−
−
JK
� −
−−
MK
Pro
ve:
�J
KN
� �
MK
L
2. G
iven
: −
−
AB
� −−
− C
B
∠
A �
∠C
−−
− D
B b
isec
ts ∠
AB
C.
P
rove
: −−
− A
D �
−−−
CD
3. W
rite
a p
arag
raph
pro
of.
G
iven
: −−
− D
E ‖
−−−
FG
∠
E �
∠G
P
rove
: �
DF
G �
�F
DE
FG
DE
AC
B D
N
J
M
KL
4-5
Skill
s Pr
acti
ceP
rovin
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
−−
D
E ‖
−
−
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,
−−
D
F �
−−
FD
. S
o �
DFG
� �
FD
E b
y A
AS
.
∠N
�∠
LGi
ven
JK�
MK
Give
n
∠JK
N�
∠M
KL
Verti
cal �
are
�.
�JK
N�
�M
KL
AAS
Pro
of:
∠A
�∠
C
Give
nA
B�
CB
Give
nCP
CTC
AD
�C
D
DB
bis
ects
∠A
BC
.Gi
ven
�A
BD
��
CB
DAS
A
∠A
BD
�∠
CB
DDe
f. of
∠ b
isec
tor
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Answers (Lesson 4-5)
Chapter 4 A15 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 15A01_A26_GEOCRM04_890513.indd 15 5/23/08 11:39:59 PM5/23/08 11:39:59 PM
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
34
G
lenc
oe G
eom
etry
PRO
OF
Wri
te t
he
spec
ifie
d t
ype
of p
roof
. 1
. Wri
te a
flo
w p
roof
.
Giv
en:
S i
s th
e m
idpo
int
of −−
−Q
T .
−−−
QR
‖ −−
− T
U
P
rove
: �
QS
R �
�T
SU
2. W
rite
a p
arag
raph
pro
of.
G
iven
: ∠
D �
∠F
−−−
GE
bis
ects
∠D
EF
.
Pro
ve:
−−−
DG
� −−
− F
G
AR
CH
ITEC
TUR
E F
or E
xerc
ises
3 a
nd
4, u
se t
he
foll
owin
g
info
rmat
ion
.A
n a
rch
itec
t u
sed
the
win
dow
des
ign
in
th
e di
agra
m w
hen
rem
odel
ing
an a
rt s
tudi
o. −
−
AB
an
d −−
− C
B e
ach
mea
sure
3 f
eet.
3. S
upp
ose
D i
s th
e m
idpo
int
of −
−
AC
. Det
erm
ine
wh
eth
er �
AB
D �
�C
BD
. Ju
stif
y yo
ur
answ
er.
4. S
upp
ose
∠A
� ∠
C. D
eter
min
e w
het
her
�A
BD
� �
CB
D. J
ust
ify
you
r an
swer
.DB
AC
D
G
F
E
UQ
S
RT
4-5
Prac
tice
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
−−
G
E b
isects
∠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,
−−
G
E �
−−
G
E .
So
�D
EG
� �
FE
G b
y A
AS
. T
here
fore
−−
D
G �
−−
FG
by C
PC
TC
.
Sin
ce D
is t
he m
idp
oin
t o
f −
−
A
C ,
−−
A
D �
−−
C
D b
y t
he M
idp
oin
t T
heo
rem
.
−−
A
B �
−−
C
B b
y t
he d
efi
nit
ion
of
co
ng
ruen
t seg
men
ts.
By t
he R
efl
exiv
e
Pro
pert
y −
−
B
D �
−−
B
D .
So
�A
BD
� �
CB
D b
y S
SS
.
We a
re g
iven
−−
A
B �
−−
C
B a
nd
∠A
� ∠
C.
−−
B
D �
−−
B
D b
y 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
Give
n
QR
||
TUGi
ven
Mid
poin
t The
orem
Alt.
Int.
� a
re �
.
QS
�TS
S is
th
e m
idp
oin
t o
f Q
T.
QSR
�
TSU
ASA
∠Q
SR�
∠TS
UVe
rtica
l � a
re �
.
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Cha
pte
r 4
35
G
lenc
oe G
eom
etry
1. D
OO
R S
TOPS
Tw
o do
or s
tops
hav
e cr
oss-
sect
ion
s th
at a
re r
igh
t tr
ian
gles
. T
hey
bot
h h
ave
a 20
an
gle
and
the
len
gth
of
the
side
bet
wee
n t
he
90 a
nd
20 a
ngl
es a
re e
qual
. Are
th
e cr
oss-
sect
ion
s co
ngr
uen
t? E
xpla
in.
2. M
APP
ING
Tw
o pe
ople
dec
ide
to t
ake
a w
alk.
On
e pe
rson
is
in B
omba
y an
d th
e ot
her
is
in M
ilw
auke
e. T
hey
sta
rt b
y w
alki
ng
stra
igh
t fo
r 1
kilo
met
er. T
hen
th
ey b
oth
tu
rn r
igh
t at
an
an
gle
of 1
10,
and
con
tin
ue
to w
alk
stra
igh
t ag
ain
. A
fter
a w
hil
e, t
hey
bot
h t
urn
rig
ht
agai
n, b
ut
this
tim
e at
an
an
gle
of 1
20.
Th
ey e
ach
wal
k st
raig
ht
for
a w
hil
e in
th
is n
ew d
irec
tion
un
til
they
en
d u
p w
her
e th
ey s
tart
ed. E
ach
per
son
wal
ked
in a
tri
angu
lar
path
at
thei
r lo
cati
on.
Are
th
ese
two
tria
ngl
es c
ongr
uen
t?
Exp
lain
.
3. C
ON
STR
UC
TIO
N T
he
roof
top
of
An
gelo
’s h
ouse
cre
ates
an
equ
ilat
eral
tr
ian
gle
wit
h t
he
atti
c fl
oor.
An
gelo
w
ants
to
divi
de h
is a
ttic
in
to 2
equ
al
part
s. H
e th
inks
he
shou
ld d
ivid
e it
by
plac
ing
a w
all
from
th
e ce
nte
r of
th
e ro
of
to t
he
floo
r at
a 9
0 an
gle.
If
An
gelo
doe
s th
is, t
hen
eac
h s
ecti
on w
ill
shar
e a
side
an
d h
ave
corr
espo
ndi
ng
90 a
ngl
es. W
hat
el
se m
ust
be
expl
ain
ed t
o pr
ove
that
th
e tw
o tr
ian
gula
r se
ctio
ns
are
con
gru
ent?
90˚
90˚
4. L
OG
IC W
hen
Car
olyn
fin
ish
ed h
er
mu
sica
l tr
ian
gle
clas
s, h
er t
each
er g
ave
each
stu
den
t in
th
e cl
ass
a ce
rtif
icat
e in
th
e sh
ape
of a
gol
den
tri
angl
e. E
ach
st
ude
nt
rece
ived
a d
iffe
ren
t sh
aped
tr
ian
gle.
Car
olyn
los
t h
er t
rian
gle
on h
er
way
hom
e. L
ater
sh
e sa
w p
art
of a
go
lden
tri
angl
e u
nde
r a
grat
e.
Is e
nou
gh o
f th
e tr
ian
gle
visi
ble
to a
llow
C
arol
yn t
o de
term
ine
that
th
e tr
ian
gle
is
inde
ed h
ers?
Exp
lain
.
5. P
AR
K M
AIN
TEN
AN
CE
Par
k of
fici
als
nee
d a
tria
ngu
lar
tarp
to
cove
r a
fiel
d sh
aped
lik
e an
equ
ilat
eral
tri
angl
e 20
0 fe
et o
n a
sid
e.
a. S
upp
ose
you
kn
ow t
hat
a t
rian
gula
r ta
rp h
as t
wo
60 a
ngl
es a
nd
one
side
of
len
gth
200
fee
t. W
ill
this
tar
p co
ver
the
fiel
d? E
xpla
in.
b.
Su
ppos
e yo
u k
now
th
at a
tri
angu
lar
tarp
has
th
ree
60 a
ngl
es. W
ill
this
ta
rp n
eces
sari
ly c
over
th
e fi
eld?
E
xpla
in.
4-5
Wor
ld P
robl
em P
rati
ceP
rovin
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 b
y 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.
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Answers (Lesson 4-5)
Chapter 4 A16 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 16A01_A26_GEOCRM04_890513.indd 16 5/23/08 11:40:05 PM5/23/08 11:40:05 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 4
36
G
lenc
oe G
eom
etry
Enri
chm
ent
4-5
Wh
at
Ab
ou
t A
AA
to
Pro
ve T
rian
gle
Co
ng
ruen
ce?
You
hav
e le
arn
ed a
bou
t th
e S
SS
an
d A
SA
pos
tula
tes
to p
rove
th
at t
wo
tria
ngl
es a
re
con
gru
ent.
Do
you
th
ink
that
tri
angl
es c
an b
e pr
oven
con
gru
ent
by A
AA
?
Inve
stig
ate
the
AA
S t
heo
rem
. AA
S r
epre
sen
ts t
he
con
ject
ure
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t if
th
ey h
ave
two
pair
s of
co
rres
pon
din
g an
gles
con
gru
ent
and
corr
espo
ndi
ng
non
-in
clu
ded
side
s co
ngr
uen
t. F
or e
xam
ple,
in
th
e tr
ian
gles
at
the
righ
t, ∠
A �
∠Y
, ∠B
� ∠
X, a
nd
−−
AC
� −
−
YZ
.
1. R
efer
to
the
draw
ing
abov
e. W
hat
is
the
mea
sure
of
the
mis
sin
g an
gle
in t
he
firs
t tr
ian
gle?
Wh
at i
s th
e m
easu
re o
f th
e m
issi
ng
angl
e in
th
e se
con
d tr
ian
gle?
2. W
ith
th
e in
form
atio
n f
rom
Exe
rcis
e 1,
wh
at p
ostu
late
can
you
now
use
to
prov
e th
at t
he
two
tria
ngl
es a
re c
ongr
uen
t?
Now
in
vest
igat
e th
e A
AA
con
ject
ure
. AA
A r
epre
sen
ts t
he
con
ject
ure
th
at t
wo
tria
ngl
es a
re
con
gru
ent
if t
hey
hav
e th
ree
pair
s of
cor
resp
ondi
ng
angl
es c
ongr
uen
t. T
he
tria
ngl
es b
elow
de
mon
stra
te A
AA
.
25˚
25˚
70˚
70˚
85˚
85˚
3. A
re t
he
tria
ngl
es a
bove
con
gru
ent?
Exp
lain
.
4. C
an y
ou d
raw
tw
o tr
ian
gles
th
at h
ave
all
thre
e an
gles
con
gru
ent
that
are
not
con
gru
ent?
C
an A
AA
be
use
d to
pro
ve t
hat
tw
o tr
ian
gles
are
con
gru
ent?
5. W
hat
ch
arac
teri
stic
s do
th
ese
two
tria
ngl
es d
raw
n b
y A
AA
see
m t
o po
sses
s?
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.
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NA
ME
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D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Cha
pte
r 4
37
G
lenc
oe G
eom
etry
Exer
cise
sA
LGEB
RA
Fin
d t
he
valu
e of
eac
h v
aria
ble
.
1.
R
P
Q2x
°
40°
2.
S
V
T3x
- 6
2x+
6
3.
W
YZ
3x°
4.
2x°
(6x
+6)
°
5.
5x°
6.
RS
T 3x°
x°
7. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: ∠1
� ∠
2
Pro
ve:
−−
AB
� −−
− C
B
Pro
of:
Sta
tem
en
ts
Reaso
ns
Stud
y G
uide
and
Inte
rven
tion
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Pro
per
ties
of
Iso
scel
es T
rian
gle
s A
n i
sosc
eles
tri
angl
e h
as t
wo
con
gru
ent
side
s ca
lled
th
e le
gs. T
he
angl
e fo
rmed
by
the
legs
is
call
ed t
he
vert
ex a
ngl
e. T
he
oth
er t
wo
angl
es a
re c
alle
d b
ase
angl
es. Y
ou c
an p
rove
a t
heo
rem
an
d it
s co
nve
rse
abou
t is
osce
les
tria
ngl
es.
•
If t
wo
side
s of
a t
rian
gle
are
con
gru
ent,
th
en t
he
angl
es o
ppos
ite
th
ose
side
s ar
e co
ngr
uen
t. (
Isos
cele
s T
rian
gle
Th
eore
m)
•
If t
wo
angl
es o
f a
tria
ngl
e ar
e co
ngr
uen
t, t
hen
th
e si
des
oppo
site
th
ose
angl
es a
re c
ongr
uen
t. (
Con
vers
e of
Iso
scel
es T
rian
gle
Th
eore
m)
If −
−
AB
� −−
− C
B ,
then ∠
A �
∠C
.
If ∠
A �
∠C
, th
en −
−
AB
� −−
− C
B .
A
B
C
F
ind
x, g
iven
−−
B
C �
−−
B
A .
B
AC( 4
x+
5) °
( 5x
- 1
0)°
BC
= B
A, s
o
m∠
A =
m∠
C
Isos.
Triangle
Theore
m
5x
- 1
0 =
4x
+ 5
S
ubstitu
tion
x
- 1
0 =
5
subtr
act
4x
from
each s
ide.
x
= 1
5 A
dd 1
0 t
o e
ach s
ide.
Exam
ple
1Ex
amp
le 2
F
ind
x.
RT
S 3x-
13
2x
m∠
S =
m∠
T, s
o
SR
= T
R
Convers
e o
f Is
os. �
Thm
.
3x
- 1
3 =
2x
Substitu
tion
3x
= 2
x +
13
Add 1
3 t
o e
ach s
ide.
x
= 1
3 S
ubtr
act
2x
from
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.
−−
A
B �
−−
C
B
4. C
on
vers
e o
f Is
os.
Tri
an
gle
Th
m.
12
936
35
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15
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Answers (Lesson 4-5 and Lesson 4-6)
Chapter 4 A17 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 17A01_A26_GEOCRM04_890513.indd 17 5/23/08 11:40:15 PM5/23/08 11:40:15 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
38
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Pro
per
ties
of
Equ
ilate
ral T
rian
gle
s A
n e
qu
ilat
eral
tri
angl
e h
as t
hre
e co
ngr
uen
t si
des.
Th
e Is
osce
les
Tri
angl
e T
heo
rem
lea
ds t
o tw
o co
roll
arie
s ab
out
equ
ilat
eral
tri
angl
es.
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
rove
th
at i
f a
lin
e is
par
alle
l to
on
e si
de
of a
n
equ
ilat
eral
tri
angl
e, t
hen
it
form
s an
oth
er e
qu
ilat
eral
tri
angl
e.
Pro
of:
Sta
tem
ents
R
easo
ns
1. �
AB
C i
s eq
uil
ater
al; −−
− P
Q ‖
−−−
BC
. 1.
Giv
en2.
m∠
A =
m∠
B =
m∠
C =
60
2. E
ach
∠ o
f an
equ
ilat
eral
� m
easu
res
60°.
3. ∠
1 �
∠B
, ∠2
� ∠
C
3. I
f ‖
lin
es, t
hen
cor
res.
� a
re �
.4.
m∠
1 =
60,
m∠
2 =
60
4. S
ubs
titu
tion
5. �
AP
Q i
s eq
uil
ater
al.
5.
If a
� i
s eq
uia
ngu
lar,
th
en i
t is
equ
ilat
eral
.
Exer
cise
s
ALG
EBR
A F
ind
th
e va
lue
of e
ach
var
iab
le.
1.
D
FE
6x°
2.
G
JH
6x-
55x
3.
3x°
4.
4x40
60°
5.
X
ZY
4x-
4
3x+
860
°
6.
4x°
60°
7. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: �A
BC
is
equ
ilat
eral
; ∠1
� ∠
2.
P
rove
: ∠A
DB
� ∠
CD
B
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
. −
−
A
B �
−−
C
B ;
∠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 �
�
CB
D
4.
AS
A P
ostu
late
5
. ∠
AD
B �
∠
CD
B
5.
CP
CT
C
10
12
15
10
520
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Lesson 4-6
Cha
pte
r 4
39
G
lenc
oe G
eom
etry
Skill
s Pr
acti
ceIs
oscele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
D
C
B
AE
Ref
er t
o th
e fi
gure
at
the
righ
t.
1. I
f −
−
AC
� −−
− A
D , n
ame
two
con
gru
ent
angl
es.
2. I
f −−
− B
E �
−−−
BC
, nam
e tw
o co
ngr
uen
t an
gles
.
3. I
f ∠
EB
A �
∠E
AB
, nam
e tw
o co
ngr
uen
t se
gmen
ts.
4. I
f ∠
CE
D �
∠C
DE
, nam
e tw
o co
ngr
uen
t se
gmen
ts.
Fin
d e
ach
mea
sure
. 5
. m∠
AB
C
6. m
∠E
DF
60°
40°
ALG
EBR
A F
ind
th
e va
lue
of e
ach
var
iab
le.
7.
2x+
43x
-10
8.
(2x
+3)
°
9. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: −−
− C
D �
−−−
CG
−−−
DE
� −−
− G
F
P
rove
: −−
− C
E �
−−
CF
D E F G
C
4-6
∠A
CD
� ∠
CD
A
∠B
EC
� ∠
BC
E
−−
E
B �
−−
E
A
−−
C
E �
−
−
C
D
P
roo
f:
Sta
tem
en
ts
Reaso
ns
1
. −
−
C
D �
−−
C
G
1.
Giv
en
3
. −
−
D
E �
−−
G
F
3.
Giv
en
4
. �
CD
E �
�
CG
F
4.
SA
S
5
. −
−
C
E �
−
−
C
F
5.
CP
CT
C
2
. ∠
D �
∠
G
2.
If 2
sid
es o
f a �
are
�,
then
th
e �
op
po
sit
e
tho
se s
ides a
re �
.
60
70
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Answers (Lesson 4-6)
Chapter 4 A18 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 18A01_A26_GEOCRM04_890513.indd 18 5/23/08 11:40:22 PM5/23/08 11:40:22 PM
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swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
NA
ME
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TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
40
G
lenc
oe G
eom
etry
Prac
tice
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Linc
oln
Haw
ks
E DBC
A12
3 4
UR
TV
S
Ref
er t
o th
e fi
gure
at
the
righ
t.
1. I
f −−
− R
V �
−−
RT
, nam
e tw
o co
ngr
uen
t an
gles
. ∠
RTV
� ∠
RV
T
2. I
f −
−
RS
� −
−
SV
, nam
e tw
o co
ngr
uen
t an
gles
. ∠
SV
R �
∠S
RV
3. I
f ∠
SR
T �
∠S
TR
, nam
e tw
o co
ngr
uen
t se
gmen
ts. −
−
S
T �
−−
S
R
4. I
f ∠
ST
V �
∠S
VT
, nam
e tw
o co
ngr
uen
t se
gmen
ts.
−−
S
T �
−−
S
V
Fin
d e
ach
mea
sure
.
5. m
∠K
ML
6.
m∠
HM
G
7. m
∠G
HM
8. I
f m
∠H
JM
= 1
45, f
ind
m∠
MH
J.
9. I
f m
∠G
= 6
7, f
ind
m∠
GH
M.
10. P
RO
OF
Wri
te a
tw
o-co
lum
n p
roof
.
G
iven
: −−
− D
E ‖
−−−
BC
∠
1 �
∠2
P
rove
: −
−
AB
� −
−
AC
11. S
POR
TS A
pen
nan
t fo
r th
e sp
orts
tea
ms
at L
inco
ln H
igh
S
choo
l is
in
th
e sh
ape
of a
n i
sosc
eles
tri
angl
e. I
f th
e m
easu
re
of t
he
vert
ex a
ngl
e is
18,
fin
d th
e m
easu
re o
f ea
ch b
ase
angl
e.
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
�
, th
en
th
e s
id
e
op
po
sit
e t
ho
se s
ides a
re �
1. −
−
D
E ‖
−
−
B
C
2. ∠
1 �
∠
4
∠
2 �
∠
3
3. ∠
1 �
∠
2
4. ∠
3 �
∠
4
5. −
−
A
B �
−−
A
C
60
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Cha
pte
r 4
41
G
lenc
oe G
eom
etry
Wor
d Pr
oble
m P
ract
ice
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
1
1. T
RIA
NG
LES
At
an a
rt s
upp
ly s
tore
, tw
o di
ffer
ent
tria
ngu
lar
rule
rs a
re a
vail
able
. O
ne
has
an
gles
45°
, 45°
, an
d 90
°. T
he
oth
er h
as a
ngl
es 3
0°, 6
0°, a
nd
90°.
W
hic
h t
rian
gle
is i
sosc
eles
?
2. R
ULE
RS
A f
olda
ble
rule
r h
as t
wo
hin
ges
that
div
ide
the
rule
r in
to t
hir
ds. I
f th
e en
ds a
re f
olde
d u
p u
nti
l th
ey t
ouch
, w
hat
kin
d of
tri
angl
e re
sult
s?
3. H
EXA
GO
NS
Juan
ita
plac
ed o
ne
end
of
each
of
6 bl
ack
stic
ks a
t a
com
mon
poi
nt
and
then
spa
ced
the
oth
er e
nds
eve
nly
ar
oun
d th
at p
oin
t. S
he
con
nec
ted
the
free
en
ds o
f th
e st
icks
wit
h l
ines
. Th
e re
sult
was
a r
egu
lar
hex
agon
. Th
is
con
stru
ctio
n s
how
s th
at a
reg
ula
r h
exag
on c
an b
e m
ade
from
six
con
gru
ent
tria
ngl
es. C
lass
ify
thes
e tr
ian
gles
. E
xpla
in.
4. P
ATH
S A
mar
ble
path
is
con
stru
cted
ou
t of
sev
eral
con
gru
ent
isos
cele
s tr
ian
gles
. Th
e ve
rtex
an
gles
are
all
20.
W
hat
is
the
mea
sure
of
angl
e 1
in t
he
figu
re?
5. B
RID
GES
Eve
ry d
ay, c
ars
driv
e th
rou
gh
isos
cele
s tr
ian
gles
wh
en t
hey
go
over
th
e L
eon
ard
Zak
im B
ridg
e in
Bos
ton
. Th
e te
n-l
ane
road
way
for
ms
the
base
s of
th
e tr
ian
gles
.
a. T
he
angl
e la
bele
d A
in
th
e pi
ctu
re
has
a m
easu
re o
f 67
. Wh
at i
s th
e m
easu
re o
f ∠
B?
b.
Wh
at i
s th
e m
easu
re o
f ∠
C?
c. N
ame
the
two
con
gru
ent
side
s.
the 4
5°-4
5°- 9
0° t
rian
gle
A
B
C
−−
A
B �
−
−
B
C
4-6 eq
uilate
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
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Answers (Lesson 4-6)
Chapter 4 A19 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 19A01_A26_GEOCRM04_890513.indd 19 5/23/08 11:40:30 PM5/23/08 11:40:30 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
42
G
lenc
oe G
eom
etry
Enri
chm
ent
Tri
an
gle
Ch
allen
ges
Som
e pr
oble
ms
incl
ude
dia
gram
s. I
f yo
u a
re n
ot s
ure
how
to
solv
e th
e pr
oble
m, b
egin
by
usi
ng
the
give
n i
nfo
rmat
ion
. Fin
d th
e m
easu
res
of a
s m
any
angl
es a
s yo
u c
an, w
riti
ng
each
m
easu
re o
n t
he
diag
ram
. Th
is m
ay g
ive
you
m
ore
clu
es t
o th
e so
luti
on.
1. G
iven
: B
E =
BF
, ∠B
FG
� ∠
BE
F �
∠
BE
D, m
∠B
FE
= 8
2 an
d A
BF
G a
nd
BC
DE
eac
h h
ave
oppo
site
sid
es p
aral
lel
and
con
gru
ent.
F
ind
m∠
AB
C. A
GD
FE
CB
3. G
iven
: m
∠U
ZY
= 9
0, m
∠Z
WX
= 4
5,
�Y
ZU
� �
VW
X, U
VX
Y i
s a
squ
are
(all
sid
es c
ongr
uen
t, a
ll
angl
es r
igh
t an
gles
).
F
ind
m∠
WZ
Y.
2. G
iven
: A
C =
AD
, an
d −
−
AB
⊥ −−
− B
D ,
m∠
DA
C =
44
and
−−−
CE
bis
ects
∠A
CD
.
Fin
d m
∠D
EC
. A
DC
BE
4. G
iven
: m
∠N
= 1
20,
−−
−
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
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Cha
pte
r 4
43
G
lenc
oe G
eom
etry
Iden
tify
Co
ng
ruen
ce T
ran
sfo
rmat
ion
s A
con
gru
ence
tra
nsf
orm
atio
n i
s a
tran
sfor
mat
ion
wh
ere
the
orig
inal
fig
ure
, or
prei
mag
e, a
nd
the
tran
sfor
med
fig
ure
, or
imag
e, f
igu
re a
re s
till
con
gru
ent.
Th
e th
ree
type
s of
con
gru
ence
tra
nsf
orm
atio
ns
are
refl
ecti
on (
or f
lip)
, tra
nsl
atio
n (
or s
lide
), a
nd
rota
tion
(or
tu
rn).
Id
enti
fy t
he
typ
e of
con
gru
ence
tra
nsf
orm
atio
n s
how
n a
s a
refl
ecti
on, t
ran
sla
tion
, or
rota
tion
.
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.
Exer
cise
sId
enti
fy t
he
typ
e of
con
gru
ence
tra
nsf
orm
atio
n s
how
n a
s a
refl
ecti
on, t
ran
sla
tion
, or
rot
ati
on.
1.
x
y
2.
x
y
3.
x
y
4.
x
y
5.
x
y
6.
x
y
Stud
y G
uide
and
Inte
rven
tion
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
Exam
ple
4-7
refl
ecti
on
tran
sla
tio
nro
tati
on
rota
tio
nre
flecti
on
or
tran
sla
tio
nre
flecti
on
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Answers (Lesson 4-6 and Lesson 4-7)
Chapter 4 A20 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 20A01_A26_GEOCRM04_890513.indd 20 5/23/08 11:40:39 PM5/23/08 11:40:39 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
NA
ME
DA
TE
PE
RIO
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
44
G
lenc
oe G
eom
etry
Ver
ify
Co
ng
ruen
ce Y
ou c
an v
erif
y th
at r
efle
ctio
ns,
tra
nsl
atio
ns,
an
d ro
tati
ons
of
tria
ngl
es p
rodu
ce c
ongr
uen
t tr
ian
gles
usi
ng
SS
S.
V
erif
y co
ngr
uen
ce a
fter
a t
ran
sfor
mat
ion
.
�W
XY
wit
h v
erti
ces
W(3
, -7)
, X(6
, -7)
, Y(6
, -2)
is
a tr
ansf
orm
atio
n
of �
RS
T w
ith
ver
tice
s R
(2, 0
), S
(5, 0
), a
nd
T(5
, 5).
Gra
ph t
he
orig
inal
fig
ure
an
d it
s im
age.
Ide
nti
fy t
he
tran
sfor
mat
ion
an
d ve
rify
th
at i
t is
a c
ongr
uen
ce t
ran
sfor
mat
ion
.
Gra
ph e
ach
fig
ure
. Use
th
e D
ista
nce
For
mu
la t
o sh
ow t
he
side
s ar
e co
ngr
uen
t an
d th
e tr
ian
gles
are
con
gru
ent
by S
SS
.
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
� −−
− W
X ,
−−
ST
, � −
−
XY
, −
−
TR
� −−
− Y
W
By
SS
S, �
RS
T �
�W
XY
.
Exer
cise
s
CO
OR
DIN
ATE
GEO
MET
RY
Gra
ph
eac
h p
air
of t
rian
gles
wit
h t
he
give
n v
erti
ces.
T
hen
id
enti
fy t
he
tran
sfor
mat
ion
, an
d v
erif
y th
at i
t is
a c
ongr
uen
ce
tran
sfor
mat
ion
.
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-
7St
udy
Gui
de a
nd In
terv
enti
on (
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 −
−
A
B �
−−
D
E ,
−−
B
C �
−−
E
F ,
−−
A
C �
−−
D
F B
y S
SS
,
�A
BC
�
�D
EF
tran
sla
tio
n;
QR
= √
�
5 ,
RS
= √
�
5 ,
SQ
= 2
, TU
= √
�
5 ,
UV
= √
�
5 ,
VT
= 2
th
us,
−−
Q
R �
−
−
TU
, −
−
R
S �
−
−
U
V ,
−−
S
Q �
−
−
V
T .
By S
SS
, �
QR
S �
�TU
V.
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Cha
pte
r 4
45
G
lenc
oe G
eom
etry
4-7
Iden
tify
th
e ty
pe
of c
ongr
uen
ce t
ran
sfor
mat
ion
sh
own
as
a re
flec
tion
, tra
nsl
ati
on,
or r
ota
tion
.
1.
x
y
2.
x
y
r
ota
tio
n
re
flecti
on
CO
OR
DIN
ATE
GEO
MET
RY
Id
enti
fy e
ach
tra
nsf
orm
atio
n a
nd
ver
ify
that
it
is a
co
ngr
uen
ce t
ran
sfor
mat
ion
.
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
ATE
GEO
MET
RY
Gra
ph
eac
h p
air
of t
rian
gles
wit
h t
he
give
n v
erti
ces.
T
hen
, id
enti
fy t
he
tran
sfor
mat
ion
, an
d v
erif
y th
at i
t is
a c
ongr
uen
ce
tran
sfor
mat
ion
.
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
Skill
s Pr
acti
ceC
on
gru
en
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 �
�
QR
S b
y 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
.
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Answers (Lesson 4-7)
Chapter 4 A21 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 21A01_A26_GEOCRM04_890513.indd 21 5/23/08 11:40:49 PM5/23/08 11:40:49 PM
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ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
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cGraw
-Hill C
om
panies, Inc.
Pdf Pass
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ME
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TE
PE
RIO
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
46
G
lenc
oe G
eom
etry
Iden
tify
th
e ty
pe
of c
ongr
uen
ce t
ran
sfor
mat
ion
sh
own
as
a re
flec
tion
, tra
nsl
ati
on,
or r
ota
tion
. 1
.
x
y
2.
x
y
3. I
den
tify
th
e ty
pe o
f co
ngr
uen
ce t
ran
sfor
mat
ion
sh
own
as
a re
flec
tion
, tra
nsl
atio
n, o
r ro
tati
on, a
nd
veri
fy t
hat
it
is a
con
gru
ence
tra
nsf
orm
atio
n.
R
ota
tio
n;
AB
= √
��
13 ,
BC
= 3
√ �
2 ,
CA
=
5
, A
D =
√ �
�
13
D
E =
3 √
�
2 ,
EA
= 5
, �
AB
C �
�A
DE
by S
SS
.
4. Δ
AB
C h
as v
erti
ces
A(−
4, 2
), B
(−2,
0),
C(−
4, -
2). Δ
DE
F h
as
vert
ices
D(4
, 2),
E(2
, 0),
F(4
, −2)
. Gra
ph t
he
orig
inal
fig
ure
an
d it
s im
age.
Th
en i
den
tify
th
e tr
ansf
orm
atio
n a
nd
veri
fy
that
it
is a
con
gru
ence
tra
nsf
orm
atio
n.
R
efl
ecti
on
; A
B =
2 √
�
2 ,
BC
= 2
√ �
2 ,
CA
= 4
, D
E =
2 √
�
2 ,
EF =
2 √
�
2 ,
FD
= 4
. �
AB
C �
�
DE
F b
y S
SS
.
5. S
TEN
CIL
S C
arly
is
plan
nin
g on
ste
nci
lin
g a
patt
ern
of
flow
ers
alon
g th
e ce
ilin
g in
her
be
droo
m. S
he
wan
ts a
ll o
f th
e fl
ower
s to
loo
k ex
actl
y th
e sa
me.
Wh
at t
ype
of c
ongr
uen
ce
tran
sfor
mat
ion
sh
ould
sh
e 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
Prac
tice
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
x
y
tran
sla
tio
nro
tati
on
x
y
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Cha
pte
r 4
47
G
lenc
oe G
eom
etry
4-7
Wor
d Pr
oble
m P
ract
ice
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
1. Q
UIL
TIN
G Y
ou a
nd
you
r tw
o fr
ien
ds
visi
t a
craf
t fa
ir a
nd
not
ice
a qu
ilt,
par
t of
wh
ose
patt
ern
is
show
n b
elow
. You
r fi
rst
frie
nd
says
th
e pa
tter
n i
s re
peat
ed
by t
ran
slat
ion
. You
r se
con
d fr
ien
d sa
ys
the
patt
ern
is
repe
ated
by
rota
tion
. Wh
o is
cor
rect
?
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
ECY
CLI
NG
Th
e in
tern
atio
nal
sym
bol
for
recy
clin
g is
sh
own
bel
ow. W
hat
typ
e of
con
gru
ence
tra
nsf
orm
atio
n d
oes
this
il
lust
rate
?
R
ota
tio
n
3. A
NA
TOM
Y C
arl
not
ices
th
at w
hen
he
hol
ds h
is h
ands
pal
m u
p in
fro
nt
of h
im,
they
loo
k th
e sa
me.
Wh
at t
ype
of
con
gru
ence
tra
nsf
orm
atio
n d
oes
this
il
lust
rate
?
R
efl
ecti
on
4. S
TAM
PS A
sh
eet
of p
osta
ge s
tam
ps
con
tain
s 20
sta
mps
in
4 r
ows
of 5
id
enti
cal
stam
ps e
ach
. Wh
at t
ype
of
con
gru
ence
tra
nsf
orm
atio
n d
oes
this
il
lust
rate
?
T
ran
sla
tio
n
5. M
OSA
ICS
José
cu
t re
ctan
gles
ou
t of
ti
ssu
e pa
per
to c
reat
e a
patt
ern
. He
cut
out
fou
r co
ngr
uen
t bl
ue
rect
angl
es a
nd
then
cu
t ou
t fo
ur
con
gru
ent
red
rect
angl
es t
hat
wer
e sl
igh
tly
smal
ler
to
crea
te t
he
patt
ern
sh
own
bel
ow.
a. W
hic
h c
ongr
uen
ce t
ran
sfor
mat
ion
do
es t
his
pat
tern
ill
ust
rate
?
Ro
tati
on
b.
If t
he
wh
ole
squ
are
wer
e ro
tate
d 90
, w
ould
th
e tr
ansf
orm
atio
n s
till
be
the
sam
e?
Yes
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Answers (Lesson 4-7)
Chapter 4 A22 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 22A01_A26_GEOCRM04_890513.indd 22 5/23/08 11:40:58 PM5/23/08 11:40:58 PM
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swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
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ME
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PE
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
48
G
lenc
oe G
eom
etry
4-7
Enri
chm
ent
Tra
nsfo
rmati
on
s i
n T
he C
oo
rdin
ate
Pla
ne
Th
e fo
llow
ing
stat
emen
t te
lls
one
way
to
rela
te p
oin
ts t
o
corr
espo
ndi
ng
tran
slat
ed p
oin
ts i
n t
he
coor
din
ate
plan
e.
(x, y
) →
(x
+ 6
, y -
3)
Th
is c
an b
e re
ad, “
Th
e po
int
wit
h c
oord
inat
es (
x, y
) is
m
appe
d to
th
e po
int
wit
h c
oord
inat
es (
x +
6, y
- 3
).”
Wit
h t
his
tra
nsf
orm
atio
n, f
or e
xam
ple,
(3,
5)
is m
appe
d or
m
oved
to
(3 +
6, 5
- 3
) or
(9,
2).
Th
e fi
gure
sh
ows
how
th
e tr
ian
gle
AB
C i
s m
appe
d to
tri
angl
e X
YZ
.
1. D
oes
the
tran
sfor
mat
ion
abo
ve a
ppea
r to
be
a co
ngr
uen
ce t
ran
sfor
mat
ion
? E
xpla
in y
our
answ
er.
Dra
w t
he
tran
sfor
mat
ion
im
age
for
each
fig
ure
. Th
en t
ell
wh
eth
er t
he
tran
sfor
mat
ion
is
or i
s n
ot a
con
gru
ence
tra
nsf
orm
atio
n.
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
ZYO
(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.
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ME
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Cha
pte
r 4
49
G
lenc
oe G
eom
etry
Stud
y G
uide
and
Inte
rven
tion
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Posi
tio
n a
nd
Lab
el T
rian
gle
s A
coo
rdin
ate
proo
f u
ses
poin
ts, d
ista
nce
s, a
nd
slop
es t
o pr
ove
geom
etri
c pr
oper
ties
. Th
e fi
rst
step
in
wri
tin
g a
coor
din
ate
proo
f is
to
plac
e a
figu
re
on t
he
coor
din
ate
plan
e an
d la
bel
the
vert
ices
. Use
th
e fo
llow
ing
guid
elin
es.
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.
Pos
itio
n a
n i
sosc
eles
tri
angl
e on
th
e co
ord
inat
e p
lan
e so
th
at i
ts s
ides
are
a u
nit
s lo
ng
and
on
e si
de
is o
n t
he
pos
itiv
e x-
axis
.
Sta
rt w
ith
R(0
, 0).
If
RT
is
a, t
hen
an
oth
er v
erte
x is
T(a
, 0).
For
ver
tex
S, t
he
x-co
ordi
nat
e is
a −
2 .
Use
b f
or t
he
y-co
ordi
nat
e,
so t
he
vert
ex i
s S
( a −
2 ,
b ) .
Exer
cise
sN
ame
the
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
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( ?
, ?)
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
4. i
sosc
eles
tri
angl
e
5. is
osce
les
righ
t �
DE
F
6. e
quil
ater
al t
rian
gle
�R
ST
wit
h b
ase
−−
RS
w
ith
leg
s e
un
its
lon
g �
EQ
I w
ith
ver
tex
4a u
nit
s lo
ng
Q
(0, √
�
3 b)
an
d si
des
2b
un
its
lon
g
x
y
I( b, 0
)E
( –b,
0)
Q( 0
,√3⎯b
)
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
037_
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:59:
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Answers (Lesson 4-7 and Lesson 4-8)
Chapter 4 A23 Glencoe Geometry
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ht © G
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cGraw
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ivision o
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cGraw
-Hill C
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ME
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PE
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
50
G
lenc
oe G
eom
etry
Wri
te C
oo
rdin
ate
Pro
ofs
Coo
rdin
ate
proo
fs c
an b
e u
sed
to p
rove
th
eore
ms
and
to
veri
fy p
rope
rtie
s. M
any
coor
din
ate
proo
fs u
se t
he
Dis
tan
ce F
orm
ula
, Slo
pe F
orm
ula
, or
Mid
poin
t T
heo
rem
.
P
rove
th
at a
seg
men
t fr
om t
he
vert
ex
angl
e of
an
iso
scel
es t
rian
gle
to t
he
mid
poi
nt
of t
he
bas
e is
per
pen
dic
ula
r to
th
e b
ase.
Fir
st, p
osit
ion
an
d la
bel
an i
sosc
eles
tri
angl
e on
th
e co
ordi
nat
e pl
ane.
On
e w
ay i
s to
use
T(a
, 0),
R(-
a, 0
), a
nd
S(0
, c).
Th
en U
(0, 0
) is
th
e m
idpo
int
of −
−
RT
.
Giv
en: I
sosc
eles
�R
ST
; U i
s th
e m
idpo
int
of b
ase
−−
RT
.P
rove
: −−
− S
U ⊥
−−
RT
Pro
of:
U i
s th
e m
idpo
int
of −
−
RT
so
the
coor
din
ates
of
U a
re ( -
a +
a
−
2 ,
0 +
0
−
2 ) =
(0,
0).
Th
us
−−−
SU
lie
s on
the
y-ax
is, a
nd
�R
ST
was
pla
ced
so −
−
RT
lie
s on
th
e x-
axis
. Th
e ax
es a
re p
erpe
ndi
cula
r, s
o −−
− S
U ⊥
−−
RT
.
Exer
cise
PRO
OF
Wri
te a
coo
rdin
ate
pro
of f
or t
he
stat
emen
t.
Pro
ve t
hat
th
e se
gmen
ts jo
inin
g th
e m
idpo
ints
of
the
side
s of
a r
igh
t tr
ian
gle
form
a r
igh
t tr
ian
gle.
x
y
C( 2
a, 0
)
B( 0
, 2b)
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
+ 2
a
−
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 −
−
R
P i
s 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 −
−
P
Q i
s
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
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Exam
ple
037_
054_
GE
OC
RM
C04
_890
513.
indd
504/
12/0
812
:37:
18A
M
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Cha
pte
r 4
51
G
lenc
oe G
eom
etry
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1. r
igh
t �
FG
H w
ith
leg
s
2. is
osce
les
�K
LP
wit
h
3. is
osce
les
�A
ND
wit
ha
un
its
and
b u
nit
s lo
ng
bas
e −
−
KP
6b
un
its
lon
g b
ase
−−−
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
yN
(5 – 2a,
b )
A( 0
, 0)
D( 5
a, 0
)
Nam
e th
e m
issi
ng
coor
din
ates
of
each
tri
angl
e. 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
Wri
te a
coo
rdin
ate
proo
f to
pro
ve t
hat
in
an
iso
scel
es r
igh
t tr
ian
gle,
th
e se
gmen
t fr
om t
he
vert
ex o
f th
e ri
ght
angl
e to
th
e m
idpo
int
of t
he
hyp
oten
use
is
perp
endi
cula
r to
th
e h
ypot
enu
se.
G
iven
: is
osce
les
righ
t �
AB
C w
ith
∠A
BC
th
e ri
ght
angl
e an
d M
th
e m
idpo
int
of −
−
AC
P
rove
: −
−−
BM
⊥ −
−
AC
x
y
C( 2
a, 0
)
A( 0
, 2a)
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
) o
r (a
, a).
Th
e s
lop
e o
f −
−
A
C i
s
2a -
0
−
0 -
2
a
= -
1.
Th
e s
lop
e o
f −
−
B
M i
s
a -
0
−
a -
0
= 1
. T
he p
rod
uct
o
f th
e s
lop
es i
s -
1,
so
−−
B
M ⊥
−−
A
C .
4-8
Skill
s Pr
acti
ceTri
an
gle
s a
nd
Co
ord
inate
Pro
of
037_
054_
GE
OC
RM
C04
_890
513.
indd
515/
22/0
88:
07:5
2P
M
Answers (Lesson 4-8)
Chapter 4 A24 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 24A01_A26_GEOCRM04_890513.indd 24 5/23/08 11:41:17 PM5/23/08 11:41:17 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
Pdf Pass
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
52
G
lenc
oe G
eom
etry
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1. e
quil
ater
al �
SW
Y w
ith
2.
isos
cele
s �
BL
P w
ith
3.
isos
cele
s ri
ght
�D
GJ
s
ides
1 −
4 a
un
its
lon
g b
ase
−−
BL
3b
un
its
lon
g w
ith
hyp
oten
use
−−
DJ
an
d le
gs 2
a u
nit
s lo
ng
x
y D( 0
, 2a)
G( 0
, 0)
J(2a
, 0)
x
yP
(3 – 2b,
c )
B( 0
, 0)
L(3b
, 0)
x
yY
(1 – 8a,b )
W(1 – 4a,
0)
S( 0
, 0)
Nam
e th
e m
issi
ng
coor
din
ates
of
each
tri
angl
e.
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)
NEI
GH
BO
RH
OO
DS
For
Exe
rcis
es 7
an
d 8
, use
th
e fo
llow
ing
info
rmat
ion
.K
arin
a li
ves
6 m
iles
eas
t an
d 4
mil
es n
orth
of
her
hig
h s
choo
l. A
fter
sch
ool
she
wor
ks p
art
tim
e at
the
mal
l in
a m
usic
sto
re. T
he m
all
is 2
mil
es w
est
and
3 m
iles
nor
th o
f th
e sc
hool
.
7. P
roo
f W
rite
a c
oord
inat
e pr
oof
to p
rove
th
at K
arin
a’s
hig
h s
choo
l, h
er h
ome,
an
d th
e m
all
are
at t
he
vert
ices
of
a ri
ght
tria
ngl
e.
G
iven
: �
SK
M
P
rove
: �
SK
M i
s a
righ
t tr
ian
gle.
8. F
ind
the
dist
ance
bet
wee
n t
he
mal
l an
d K
arin
a’s
hom
e.
x
y S( 0
, 0)
K( 6
, 4)
M( –
2, 3
)
S
( 1
−
6 b
, c
)
C(3
a,
0),
E(0
, c)
M(0
, c
), N
(-2
b,
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 o
r -
3
−
2
S
ince t
he s
lop
e o
f −
−
S
M i
s t
he n
eg
ati
ve r
ecip
rocal
of
the
s
lop
e o
f −
−
S
K ,
−−
S
M ⊥
−−
S
K .
Th
ere
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
Prac
tice
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Sam
ple
an
sw
ers
are
g
iven
.
037_
054_
GE
OC
RM
C04
_890
513.
indd
524/
12/0
812
:37:
29A
M
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Cha
pte
r 4
53
G
lenc
oe G
eom
etry
1. S
HEL
VES
Mar
tha
has
a s
hel
f br
acke
t sh
aped
lik
e a
righ
t is
osce
les
tria
ngl
e.
Sh
e w
ants
to
know
th
e le
ngt
h o
f th
e h
ypot
enu
se r
elat
ive
to t
he
side
s. S
he
does
not
hav
e a
rule
r, b
ut
rem
embe
rs
the
Dis
tan
ce F
orm
ula
. Sh
e pl
aces
th
e br
acke
t on
a c
oord
inat
e gr
id w
ith
th
e ri
ght
angl
e at
th
e or
igin
. Th
e le
ngt
h o
f ea
ch l
eg i
s a.
Wh
at a
re t
he
coor
din
ates
of
th
e ve
rtic
es m
akin
g th
e tw
o ac
ute
an
gles
?(a
, 0)
an
d (
0,
a)
2. F
LAG
S A
fla
g is
sh
aped
lik
e an
iso
scel
es
tria
ngl
e. A
des
ign
er w
ould
li
ke t
o m
ake
a dr
awin
g of
th
e fl
ag o
n a
coo
rdin
ate
plan
e.
Sh
e po
siti
ons
it s
o th
at t
he
base
of
the
tria
ngl
e is
on
th
e y-
axis
wit
h o
ne
endp
oin
t lo
cate
d at
(0,
0).
Sh
e lo
cate
sth
e ti
p of
th
e fl
ag a
t (a
, b − 2
).W
hat
are
th
e co
ordi
nat
es o
f th
e th
ird
vert
ex?
(0,
b)
3. B
ILLI
AR
DS
Th
e fi
gure
sh
ows
a si
tuat
ion
on
a b
illi
ard
tabl
e.(–
b, ?
)y
xO
( 0, 0
)
Wh
at a
re t
he
coor
din
ates
of
the
cue
ball
be
fore
it
is h
it a
nd
the
poin
t w
her
e th
e cu
e ba
ll h
its
the
edge
of
the
tabl
e?
Sam
ple
an
sw
er:
(-
2b
, 0),
(-
b,
a)
4. T
ENTS
Th
e en
tran
ce t
o M
att’s
ten
t m
akes
an
iso
scel
es t
rian
gle.
If
plac
ed o
n
a co
ordi
nat
e gr
id w
ith
th
e ba
se o
n t
he
x–ax
is a
nd
the
left
cor
ner
at
the
orig
in,
the
righ
t co
rner
wou
ld b
e at
(6,
0)
and
the
vert
ex a
ngl
e w
ould
be
at (
3, 4
).
Pro
ve t
hat
it
is a
n i
sosc
eles
tri
angl
e.
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
FTIN
GA
n e
ngi
nee
r is
des
ign
ing
a ro
adw
ay. T
hre
e ro
ads
inte
rsec
t to
fo
rm a
tri
angl
e. T
he
engi
nee
r m
arks
tw
o po
ints
of
the
tria
ngl
e at
(-
5, 0
) an
d (5
, 0)
on a
coo
rdin
ate
plan
e.
a. D
escr
ibe
the
set
of p
oin
ts i
n t
he
coor
din
ate
plan
e th
at c
ould
not
be
use
d as
th
e th
ird
vert
ex o
f th
e tr
ian
gle.
the x
-axis
b.
Des
crib
e th
e se
t of
poi
nts
in
th
e co
ordi
nat
e pl
ane
that
wou
ld m
ake
the
vert
ex o
f an
iso
scel
es t
rian
gle
toge
ther
wit
h t
he
two
con
gru
ent
side
s.
the y
-axis
excep
t fo
r th
e o
rig
in
c. D
escr
ibe
the
set
of p
oin
ts i
n t
he
coor
din
ate
plan
e th
at w
ould
mak
e a
righ
t tr
ian
gle
wit
h t
he
oth
er t
wo
poin
ts i
f th
e ri
ght
angl
e is
loc
ated
at
(-5,
0).
the p
oin
ts w
ith
x-c
oo
rdin
ate
-
5,
excep
t fo
r (-
5,
0)
Wor
d Pr
oble
m P
ract
ice
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
4-8
037_
054_
GE
OC
RM
C04
_890
513.
indd
534/
12/0
812
:37:
37A
M
Answers (Lesson 4-8)
Chapter 4 A25 Glencoe Geometry
A01_A26_GEOCRM04_890513.indd 25A01_A26_GEOCRM04_890513.indd 25 5/23/08 11:41:26 PM5/23/08 11:41:26 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
Pdf Pass
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pte
r 4
54
G
lenc
oe G
eom
etry
Recta
ng
le P
ara
do
xU
se g
rid
pape
r to
dra
w t
he
two
rect
angl
es b
elow
. Cu
t th
em i
nto
pie
ces
alon
g th
e da
shed
lin
es. T
hen
rea
rran
ge t
he
piec
es t
o fo
rm a
new
rec
tan
gle.
1. M
ake
a sk
etch
of
the
new
rec
tan
gle.
Lab
el t
he
wh
ole
nu
mbe
r le
ngt
hs
2. W
hat
is
the
com
bin
ed a
rea
of t
he
two
orig
inal
rec
tan
gles
?
3
9 u
nit
s2
3. W
hat
is
the
area
of
the
new
rec
tan
gle?
4
0 u
nit
s2
4. M
ake
a co
nje
ctu
re a
s to
wh
y th
ere
is a
dis
crep
ancy
bet
wee
n t
he
answ
ers
to
Exe
rcis
es 2
an
d 3.
S
ee s
tud
en
ts’
an
sw
ers
.
5. U
se a
str
aigh
t ed
ge t
o pr
ecis
ely
draw
th
e fo
ur
shap
es t
hat
m
ake
up
the
larg
er r
ecta
ngl
e. W
hat
doe
s yo
u d
raw
ing
show
?
I
t sh
ow
s a
gap
. S
o,
the r
ecta
ng
les d
on
’t q
uit
e
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me t
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er
like t
hey l
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ike t
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ill.
6. H
ow c
ould
you
use
slo
pe t
o sh
ow t
hat
you
dra
win
g is
acc
ura
te?
T
he s
lop
e o
f th
e b
ott
om
ed
ge o
f th
e t
op
left
pie
ce a
nd
th
e s
lop
e o
f th
e
bo
tto
m e
dg
e o
f th
e a
dja
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t p
iece a
re n
ot
the s
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pie
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geth
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to m
ake a
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ou
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53
5
5
3
3
3
3
5
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35
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037_
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42A
M
Answers (Lesson 4-8)
Chapter 4 A26 Glencoe Geometry
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Chapter 4 A27 Glencoe Geometry
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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 TestPage 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 A28 Glencoe Geometry
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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 A29 Glencoe Geometry
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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 A30 Glencoe Geometry
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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
yC(b–
2, c)
B(b, 0)A(0, 0)
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Chapter 4 A31 Glencoe Geometry
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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 A32 Glencoe Geometry
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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 A33 Glencoe Geometry
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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 A34 Glencoe Geometry
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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 A35 Glencoe Geometry
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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 A36 Glencoe Geometry
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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
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