Geometry Geometry: Measuring Area and Volume · Geometry Geometry: Measuring Area and Volume ... You will solve problems about angle relationships in Lesson 6-1. ... Foldable to help

252 Unit 3 Geometry and Measurement

Geometry

Geometry: MeasuringArea and Volume

Our world is made upof lines, angles, andshapes, both two- andthree-dimensional. Inthis unit, you willlearn about theproperties andmeasures ofgeometric figures.

Our world is made upof lines, angles, andshapes, both two- andthree-dimensional. Inthis unit, you willlearn about theproperties andmeasures ofgeometric figures.

252 Unit 3 Geometry and Measurement

Unit 3 Geometry and Measurement 253Unit 3 Geometry and Measurement 253

MATH andARCHITECTURE

UNDER CONSTRUCTIONCan you build it? Yes, you can! You’ve been selected to head the architectural andconstruction teams on a house of your own design. You’ll create the uniquely shapedfloor plan, research different floor coverings for the rooms in your house, and finallyresearch different loans to cover the cost of purchasing these floor coverings. So graba hammer and some nails, and don’t forget your geometry and measurement toolkits. You’re about to construct a cool adventure!

Log on to msmath3.net/webquest to begin your WebQuest.

http://msmath3.net/webquest

How is geometry used in the game of pool?A billiard ball is struck so that it bounces off the cushion of a pool table and headsfor a corner pocket. The three angles created by the path of the ball and thecushion together form a straight angle that measures 180°. Pool players usesuch angle relationships and the properties of reflections to make their shots.

You will solve problems about angle relationships in Lesson 6-1.

254 Chapter 6 Geometry

254–

255

Gar

y R

hijn

sbur

ger/

Mas

terf

ile

Geometry

C H A P T E R


Chapter 6 Getting Started 255

Take this quiz to see whether you are ready tobegin Chapter 6. Refer to the lesson number inparentheses if you need more review.

Vocabulary ReviewState whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.

1. For the right triangle shown, the Pythagorean Theorem states that

. (Lesson 3-4)

2. A rectangle is also a. (Lesson 4-5)

Prerequisite SkillsSolve each equation. (Lesson 1-8)

3. 49 � b � 45 � 180

4. t � 98 � 55 � 180

5. 15 � 67 � k � 180

Find the missing side length of each righttriangle. Round to the nearest tenth, ifnecessary. (Lesson 3-4)

6. a, 8 m; b, 6 m 7. b, 9 ft; a, 7 ft

8. a, 4 in.; c, 5 in. 9. c, 10 yd; a, 3 yd

Decide whether the figures are congruent. Write yes or no and explain your reasoning.(Lesson 4-5)

10. 11.

polygon

a2 � c2 � b2

Geometry Make thisFoldable to help youorganize your notes. Begin with a plain piece of 11" � 17" paper.

FoldFold the paper in fifths lengthwise.

Reading and Writing As you read and study the chapter,complete the table by writing down important definitionsand key concepts for each heading.

Open and FoldFold a 2�

12

�" tab along the short side. Then fold

the rest in half.

LabelDraw lines along folds and label each sectionas shown.

lines

polygons

symmetry

trans-formations

words model

a

b

c

3 cm3 cm

45˚60˚

Chapter 6 Getting Started 255

Identify special pairs ofangles and relationships of angles formed by twoparallel lines cut by atransversal.

acute angleright angleobtuse anglestraight anglevertical anglesadjacent anglescomplementary

anglessupplementary

angles perpendicular linesparallel linestransversal

Line and Angle Relationships

Angles can be classified by their measures.

• have measures less than 90°.

• have measures equal to 90°.

• have measures between 90° and 180°.

• have measures equal to 180°.

Pairs of angles can be classified by their relationship to each other.Recall that angles with the same measure are congruent.

Straight angles

Obtuse angles

Right angles

Acute angles


Work with a partner.

Draw two different pairs ofintersecting lines and label theangles formed as shown.

Find and record the measure of each angle.

Color angles that have thesame measure.

1. For each set of intersecting lines,identify the pairs of angles that have the same measure.

2. What is true about the sum of themeasures of the angles sharing a side?

• straightedge

• protractor

• coloredpencils

• notebookpaper

1 24 3

1 24 3

1017

020

160

3015

0

40

140

50

130

60

120

70110

80100 100

80

90110

70 12060

13050

14040

15030

16020

17010

1

2

3

4

5

6

in.

Special Pairs of Angles

are opposite have the same angles formed by intersecting lines. vertex, share a common side, and Vertical angles are congruent. do not overlap.

The sum of the measures of The sum of the measures of is 90°. is 180°.

�C and �D aresupplementary angles. m�C � m�D � 180°55˚

125˚

D

C�ABD and �DBC arecomplementary angles. m�ABD � m�DBC � 90°40˚

50˚B

A D

C

supplementary anglescomplementary angles

�1 and �2 are adjacent angles. m�ABC � m�1 � m�221A

CB

�1 and �2 arevertical angles. �1 � �2

3

421

Adjacent anglesVertical angles

Virginia SOL Standard 8.6 The student will verify by measuring and describe the relationships among vertical angles, supplementary angles, and complementary angles andwill measure and draw angles of less than 360°.

Lesson 6-1 Line and Angle Relationships 257

You can use the relationships between pairs of angles to find missing measures.

Lines that intersect at right angles are called . Twolines in a plane that never intersect or cross are called .parallel lines

perpendicular lines

Classify Angles and Angle Pairs

Classify each angle or angle pair using all names that apply.

m�1 is greater than 90°So, �1 is an obtuse angle.

�1 and �2 are adjacent angles since they havethe same vertex, share a common side, and donot overlap. Together, they form a straight anglemeasuring 180°. So, �1 and �2 are alsosupplementary angles.

Classify each angle or angle pair using all namesthat apply.

a. b. c.3

460˚

30˚

1 2

1

Find a Missing Angle Measure

In the figure, m�ABC � 90º. Find the value of x.

m�ABD � m�DBC � 90 Write an equation.

x � 65 � 90 m�ABD � x and m�DBC � 65

Subtract 65 from each side.

x � 25 Simplify.

Find the value of x in each figure.

d. e.

150˚x˚38˚x˚

� 65 � � 65

A

D

B

C

x˚65˚

Angle MeasureRead m�1 as themeasure of angle 1.

mq

n

p Red arrowheadsindicate thatlines p and q areparallel.

A red right anglesymbol indicatesthat lines m and nare perpendicular.

Symbols: m � n Symbols: p | | q

Parallel andPerpendicular LinesRead m � n as m isperpendicular to n.Read p � q as p isparallel to q.

msmath3.net/extra_examples/sol

http://msmath3.net/extra_examples/sol

258 Chapter 6 GeometryAaron Haupt

A line that intersects two or more otherlines is called a . When atransversal intersects two lines, eightangles are formed that have special names.

If the two lines cut by a transversal are parallel, then these special pairs ofangles are congruent.

transversal

You can use congruent angle relationships to solve real-life problems.

Parallel Lines

If two parallel lines are cut by a transversal, then the followingstatements are true.

• , those on oppositesides of the transversal and inside the othertwo lines, are congruent. Example: �2 � �8

• , those on oppositesides of the transversal and outside the othertwo lines, are congruent. Example: �4 � �6

• , those in the same position on the two lines in relation to the transversal, are congruent. Example: �3 � �7

Corresponding angles

Alternate exterior angles

Alternate interior angles

Find an Angle Measure

CARPENTRY You are building a bench for a picnic table. The top of the bench will be parallel to theground. If m�1 � 148°, find m�2 and m�3.

Since �1 and �2 are alternate interior angles, they are congruent. So, m�2 � 148°.

Since �2 and �3 are supplementary, the sum of their measures is180°. Therefore, m�3 � 180° � 148° or 32°.

For Exercises f–h, use the figure at the right.

f. Find m�2 if m�1 � 63°.

g. Find m�3 if m�8 � 100°.

h. Find m�4 if m�7 � 82°.

�

1 56

782

43

mp

23

1

End View

Interior and ExteriorAngles When twolines are cut by atransversal, the interiorangles lie inside thetwo lines and theexterior angles lieoutside the two lines.

1 2

4 3

5 6

8 7

1 24 3

56

87

transversal

How Does a CarpenterUse Math?Carpenters use anglerelationships when cuttinglumber to build anythingfrom furniture to houses.

Online ResearchFor information about a career as a carpenter, visit:msmath3.net/careers

http://msmath3.net/careers

Lesson 6-1 Line and Angle Relationships 259

1. OPEN ENDED Draw a pair of complementary angles.

2. Draw a pair of parallel lines and a third line intersecting them. Chooseone angle and mark it with a ✔ . Then mark all other angles that arecongruent to that angle with a ✔ . Explain.


3. 4. 5.


6. 7. 8.

For Exercises 9–12, use the figure at the right.

9. Find m�4 if m�5 � 43°.

10. Find m�1 if m�3 � 135°.

11. Find m�6 if m�8 � 126°.

12. Find m�A if m�B � 15° and �A and �Bare supplementary.

61 5

238 4

7

a b

c

x˚ 37˚

27˚

x˚60˚

x˚

34

63˚117˚

Exercise 2


13. 14. 15.

16. 17. 18.


19. 20. 21. 22.

23. 24. 25. 26.

x˚80˚x˚107˚

x˚20˚

x˚45˚

x˚

24˚x˚144˚

x˚

87˚

x˚140˚

7 85

634

21

Extra Practice See pages 629, 653.

For Exercises

13–18

19–29

30–38

See Examples

1, 2

3

4

msmath3.net/self_check_quiz/sol

http://msmath3.net/self_check_quiz/sol


27. ALGEBRA Angles P and Q are vertical angles. If m�P � 45° and m�Q � (x � 25)°, find the value of x.

28. ALGEBRA Angles A and B are supplementary. If m�A � 2x° and m�B � 80°, find the value of x.

29. POOL Aaron is trying a complicated pool shot.He wants to hit the number 8 ball into the corner pocket. If Aaron knows the angle measures shown in the diagram, what angle x must the path of the ball take to go into the corner pocket?


30. Find m�2 if m�3 � 108°. 31. Find m�6 if m�7 � 111°.




38. PARKING Engineers angled the parking spaces along a downtown street so that cars could park and back out easily. All of the lines marking the parking spaces are parallel. If m�1 � 55°, find m�2. Explain your reasoning.

39. CRITICAL THINKING Suppose two parallel lines are cut by a transversal. How are the interior angles on the same side of thetransversal related? Use a diagram to explain your reasoning.

k

h

6 8

3 15 7

j

4 2

55� 55�

x

40. SHORT RESPONSE If m�A � 81° and �A and �B are complementary,what is m�B?

41. MULTIPLE CHOICE Find the value of x in the figure at the right.

30 40 116 124

42. A savings account starts with $560. If the simple interest rate is 3%, findthe total amount after 18 months. (Lesson 5-8)

Find each percent of change. Round to the nearest tenth if necessary. Statewhether the percent of change is an increase or a decrease. (Lesson 5-7)

43. original: 20 44. original: 45 45. original: 620 46. original: 260new: 27 new: 18 new: 31 new: 299

DCBA

120˚

4x

PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 1-8)

47. n � 32 � 67 � 180 48. 45 � 89 � x � 180 49. 180 � 120 � a � 15

1

2

SOL Practice

Construct a line parallel to agiven line.

• compass• straightedge• paper

Lesson 6-1b Hands-On Lab: Constructing Parallel Lines 261

Constructing Parallel LinesIn this lab, you will construct a line parallel to a given line.

A Follow-Up of Lesson 6-1

Work with a partner. Use the information in the activity above.

1. Classify �DBC and �FAE in relationship to lines p, q, andtransversal AB.

2. Explain why you should expect �ABC to be congruent to �FAE.

Draw a line and label it p. Then drawand label a point A not on line p.

Draw a line through point A so that itintersects line p. Label the point ofintersection point B.

Place the compass at point B anddraw a large arc. Label the pointwhere the arc crosses line p as pointC, and label where it crosses line ABas point D.

With the same compass opening, place the compass at point A and draw a large arc. Label the point of intersection with line AB as point E.

Use your compass to measure thedistance between points D and C.

With the compass opened the same amount, place the compass at pointE and draw an arc to intersect the arc already drawn. Label this point F.

Draw a line through points A and F. Label this line q. You have drawn q � p.

Draw a line. Then construct aline parallel to it.

p

A

B

Steps 1–2

p

A

B C

D

F

E

Steps 5–6

p

qF

A

B C

D

E

Step 7

E

A

B

D

Cp

Steps 3–4

Find missing anglemeasures in triangles andclassify triangles by theirangles and sides.

triangleacute triangleobtuse triangleright trianglescalene triangleisosceles triangleequilateral triangle

Triangles and Angles

A is a figure formed by three line segments that intersectonly at their endpoints. Recall thattriangles are named by the letters attheir vertices. Triangle LMN is written �LMN.

vertex

anglesideM

L N

triangle


Find the value of x in �RST.

m�R � m�S � m�T � 180

x � 72 � 74 � 180

x � 146 � 180 Simplify.


x � 34 The value of x is 34.

� 146 � � 146

The sum of themeasures is 180.



Investigate the relationship among themeasures of the angles of a triangle.

Use a straightedge to draw a triangleon your paper. Then shade each angleof the triangle using a different colorand cut out the triangle.

Cut off each angle and arrange thepieces as shown so that the threeangles are adjacent.

Repeat the steps above with several other triangles.

1. What do you think is the sum of the measures of the three angles of any triangle? Explain your reasoning.

• paper

• straightedge

• coloredpencils

• scissors

Angles of a Triangle

Words The sum of the measures of Modelthe angles of a triangle is 180º.

Symbols x � y � z � 180x˚ z˚

y˚

x˚ 72˚

74˚

T

SR

Replace m�R with x, m�Swith 72, and m�T with 74.

Naming TrianglesRead �RST astriangle RST.

Lesson 6-2 Triangles and Angles 263

All triangles have at least two acute angles. Triangles can be classifiedby the measure of the third angle.

Triangles can also be classified by the number of congruent sides.Congruent sides are often marked with tick marks.

Classify Triangles by Angles

three acute angles one obtuse angle one right angle

70˚

60̊

50̊ 40̊ 65˚25˚

110˚ 30˚

Right TriangleObtuse TriangleAcute Triangle

Classify Triangles by Sides

no congruent sides at least two sides congruent three sides congruent

Equilateral TriangleIsosceles TriangleScalene Triangle

Classify Triangles

Classify each triangle by its angles and by its sides.

Angles �ABC has all acute angles.

Sides �ABC has two congruent sides.

So, �ABC is an acute isosceles triangle.

Angles �XYZ has one right angle.

Sides �XYZ has no congruent sides.

So, �XYZ is a right scalene triangle.


a. b. c.60˚

60˚60˚35˚

35̊110˚

57˚ 45˚78˚

60˚30˚

X

YZ

71˚38˚

71˚C

A

B

Base Angles InExample 2, noticethat the anglesopposite thecongruent sides arecongruent. Thecongruent angles inan isosceles triangleare called the baseangles.

In an equiangular triangle, all angles have the same measure, 60º.



75�

75� 30�


1. OPEN ENDED Name a real-life object that is shaped like an isoscelestriangle. Explain.

2. Describe the types of angles that are in a right triangle.

Find the value of x in each triangle.

3. 4. 5.


6. 7. 8. 9.

38� 38�THE THIRSTY

WHALE

THE THIRSTY

WHALE56�

Albus

Stillwater

Lawton

OKLAHOMA48�

55� 77�

29˚

x˚117˚

x˚

30˚

38˚

68˚

x˚

Exercises 1 & 2

Find the value of x in each triangle.

10. 11. 12.

13. 14. 15.


16. 17. 18. 19.

20. 21. 22. 23. 20� 20�80�45�

45�

YIELD

75�

15�100�

ALABAMA

Huntsville

Tuscaloosa

Dothan

60� 60�

60�

65�

65�

36˚

2x˚

53˚

72˚

(x � 5)˚

24˚

22˚x˚

25˚

122˚ x˚

50˚

x˚

64˚ 60˚

x˚Extra Practice

See pages 629, 653.

For Exercises

10–15

16–32

See Examples

1

2, 3

Lesson 6-2 Triangles and Angles 265AP/Wide World Photos

24. BRIDGE BUILDING At a Science Olympiad tournament, your team is to design and construct a bridge that will hold the most weight for a given span. Your team knows that triangles add stability to bridges. Below is a side view of your team’s design. Name and classify three differently-shaped triangles in your design.

Draw each triangle. If it is not possible to draw the triangle, write not possible.

25. three acute angles 26. two obtuse angles

27. obtuse isosceles with two acute angles 28. obtuse equilateral

29. right equilateral 30. right scalene

Determine whether each statement is sometimes, always, or never true.

31. Isosceles triangles are equilateral. 32. Equilateral triangles are isosceles.

33. CRITICAL THINKING Explain why all triangles have at least two acute angles.

C D E

B

A P N M L K J H G

F

34. SHORT RESPONSE Triangle ABC is isosceles. What is the value of x?

35. MULTIPLE CHOICE Which term describes the relationship between the two acute angles of a right triangle?

adjacent complementary

vertical supplementary

Find the measure of each angle in the figure if m � n and m�7 � 95°.(Lesson 6-1)

36. �4 37. �3 38. �1 39. �2

40. SAVINGS Shala’s savings account earned $4.56 in 6 months at a simple interest rate of 4.75%. How much was in her account at thebeginning of that 6-month period? (Lesson 5-8)

m

n

p

123 8

745 6

DC

BA

20˚5 cm

11 cmx

xB

A

C

PREREQUISITE SKILL Find the missing side length of each right triangle. Round to the nearest tenth if necessary. (Lesson 3-4)

41. a, 5 ft; b, 8 ft 42. b, 10 m; c, 12 m 43. a, 6 in.; c, 13 in. 44. a, 7 yd; b, 7 yd

msmath3.net/self_check_quiz

SOL Practice

/sol


Bisect an angle.

• compass• straightedge• paper

Bisecting AnglesIn this lab, you will learn to bisect an angle.



1. Describe what is true about the measures of �JKH and �HKL.

2. Explain why we say that KH�� is the bisector of �JKL.

3. The point where the bisectors of all three angles of a trianglemeet is called the incenter. Draw a triangle. Then locate itsincenter using only a compass and straightedge.


Draw �JKL.

Place the compass at point K anddraw an arc that intersects bothsides of the angle. Label theintersections X and Y.

With the compass at point X, draw an arc in the interior of �JKL.

Using this setting, place thecompass at point Y. Draw another arc.

Label the intersection of these arcs H. Then draw KH��. KH�� is the bisector of �JKL.

Draw each kind of angle.Then bisect it.

a. acute b. obtuse

J

X

Y LK

Steps 1–2

J

X

LK

YSteps 3–4

J

XH

Y LK

Step 5

Everyday Meaning ofBisect: to divide into twoequal parts

SymbolsKH�� is read ray KH.A ray is a path thatextends infinitelyfrom one point in acertain direction.

Find missing measures in30°-60° right triangles and45°-45° right triangles.

Lesson 6-3 Special Right Triangles 267

Special Right Triangles

The sides of a triangle whose angles measure 30°, 60°, and 90° have a special relationship.The hypotenuse is always twice as long as the side opposite the 30° angle.

Find Lengths of a 30°-60° Right Triangle

Find each missing length.

Step 1 Find a.

a � �12

�c Write the equation.

a � �12

�(10) or 5 Replace c with 10.(continued on the next page)


Trace the equilateral triangle andsquare below and cut them out.

Measure each angle.

Fold the triangle so that one half matches the other. Foldthe square in half along a diagonal.

1. What type of triangles have you formed?

2. What are the measures of the angles of the folded triangle?

3. Measure and describe the relationship between the shortest andlongest sides of this triangle.

4. What are the measures of the angles of the triangle formed byfolding the square?

5. Measure and describe the relationship between the legs of this triangle.

• pencil

• paper

• scissors

• ruler

ac

b

60˚

30˚

side opposite30° angle c � 2a or a � c1

2

10 ftb

a60˚

30˚

msmath3.net/extra_examples

Pythagorean Theorem:in a right triangle, thesquare of the length ofthe hypotenuse is equalto the sum of the squaresof the lengths of the legs(Lesson 3-4)

/sol


268 Chapter 6 Geometry(l)Art Resource, NY, (r)Courtesy Greece Cultural Minister

A 45°-45° right triangle is also an isosceles triangle because two angle measures are the same. Thus, the legs are always congruent.

Step 2 Find b.

c2 � a2 � b2 Pythagorean Theorem

102 � 52 � b2 Replace c with 10 and a with 5.

100 � 25 � b2 Evaluate 102 and 52.

100 � 25 � 25 � b2 � 25 Subtract 25 from each side.

75 � b2 Simplify.

�75� � �b2� Take the square root of each side.

8.7 � b Use a calculator.

The length of a is 5 feet, and the length of b is about 8.7 feet.

Find each missing length. Round to the nearesttenth if necessary.

a. b. c.

7 m

b

c 60˚

30˚

15 cm

b

a 60˚30˚12 in. b

a60˚

30˚

Find Lengths of a 45°-45° Right Triangle

ART The ancient Greeks sometimesused 45°-45° right triangles in their art. The sculpture on the right is based on such a triangle. Suppose the base of a reproductionof the sculpture shown is 15 feetlong. Find each missing length.

Step 1 Find a.

Sides a and b are the samelength. Since b � 15 feet, a � 15 feet.

Step 2 Find c.

c2 � a2 � b2 Pythagorean Theorem

c2 � 152 � 152 Replace a with 15 and b with 15.

c2 � 225 � 225 Evaluate 152.

c2 � 450 Add 225 and 225.

�c2� � �450� Take the square root of each side.

c � 21.2 Simplify.

60�

30�

60�

30�

ART The Greek reliefentitled The Grave Stele ofHegeso makes use of a30°-60° right triangle.

b

c a45˚

45˚

a � b

15 ft15 ft15 ft

cccaa

Achilles Wounded

Lesson 6-3 Special Right Triangles 269

1. Write a sentence describing the relationship between the hypotenuse of a 30°-60° right triangle and the leg opposite the 30° angle.

2. OPEN ENDED Give a real-life example of a 45°-45° right triangle.

Exercise 1

Find each missing length. Round to the nearest tenth if necessary.

6. 7. 8.

9. 10. 11.

12. The length of the hypotenuse of a 30°-60° right triangle is 7.5 meters.Find the length of the side opposite the 30° angle.

13. In a 30°-60° right triangle, the length of the side opposite the 30° angle is5.8 centimeters. What is the length of the hypotenuse?

14. The length of one of the legs of a 45°-45° right triangle is 6.5 inches. Findthe lengths of the other sides.

15. In a 45°-45° right triangle, the length of one leg is 7.5 feet. What are thelengths of the other sides?

16. HISTORY Redan lines such as the one below were used at manybattlefields in the Civil War. A redan is a triangular shape that goes outfrom the main line of defense. What is the distance h from the base ofeach redan to its farthest point?

60�

30�60 yd

redan

line of defenseh

Aerial View of Battlefield

b

c 45˚ 18 yd

45˚c

a

45˚14 cm

45˚c

b

60˚25 ft

30˚

c

b60˚

15 m30˚

b

a 60˚ 22 in.

30˚

b

a60˚18 cm

30˚


For Exercises

6–9, 12–13,16, 18

10–11, 14–15,17

See Examples

1

2

Find each missing length. Round to the nearest tenth if necessary.

3. 4. 5.

bc

45˚9 m

45˚c

b60˚

12 ft

30˚b

a 60˚ 20 in.

30˚



270 Chapter 6 GeometryAaron Haupt

17. QUILTING Refer to the photograph at the right. Each triangle in the Flying Geese pattern is a 45°-45° right triangle. If the length of a leg is 2�

12

� inches, find the length of each hypotenuse.

18. SKIING A ski jump is constructed so that the length of the board necessary for the surface of the ramp is twice as long as the ramp is high h. If the ramp forms a right triangle, what is the measure of �1? Explain your reasoning.

19. WRITE A PROBLEM Write a real-life problem involving a 30°-60° righttriangle or a 45°-45° right triangle. Then solve the problem.

20. CRITICAL THINKING Find the length of each leg of a 45°-45° righttriangle whose hypotenuse measures �162� centimeters.

h2h

1

21. MULTIPLE CHOICE The midpoints of the sides of the square at theright are joined to form a smaller square. What is the area of thesmaller square?

196 in2 98 in2 49 in2 9.9 in2

22. MULTIPLE CHOICE Which values represent the sides of a 30°-60°right triangle?

3 cm, 4 cm, 5 cm 8 cm, 8 cm, �128� cm

4 cm, 8 cm, �48� cm 5 cm, 12 cm, 13 cm

23. Two sides of a triangle are congruent. The angles opposite these sides each measure 70°. Classify the triangle by its angles and by its sides. (Lesson 6-2)

Classify each angle or angle pair using all names that apply. (Lesson 6-1)

24. 25. 26. 27.

Multiply. Write in simplest form. (Lesson 2-3)

28. �23

� � �58

� 29. ��25

� � �34

� 30. �1�12

� ��2�13

�� 31. 2�23

� ��2�14

��

56

3 412

IH

GF

DCBA

PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 1-8)

32. x � 90 � 50 � 100 � 360 33. 45 � 150 � x � 85 � 360

45˚ 45˚

45˚45˚45˚

45˚

45˚ 45˚14 in.

SOL Practice

Construct a perpendicularbisector of a segment.

• compass• straightedge• protractor• paper

Lesson 6-3b Hands-On Lab: Constructing Perpendicular Bisectors 271

Constructing Perpendicular BisectorsIn this lab, you will learn to construct a line perpendicular to asegment so that it bisects that segment.



1. Describe what is true about the measures of A�M� and M�B�.

2. Find m�XMB. Then describe the relationship between A�B�and X�Y�.

3. Explain how to construct a 45°-45° right triangle with legs half aslong as the segment below. Then construct the triangle.

DC

Draw A�B�. Then place the compassat point A. Using a setting greaterthan one half the length of A�B�,draw an arc above and below A�B�.

Using this setting, place thecompass at point B. Draw anotherset of arcs above and below A�B�as shown.

Label the intersection of these arcs X and Y as shown. Then draw X�Y�. X�Y� is the perpendicularbisector of A�B�. Label theintersection of A�B� and this newline segment M.

Draw a line segment. Then construct the perpendicular bisector ofthe segment.

BA

BA

BA M

X

Y

Find missing anglemeasures in quadrilateralsand classify quadrilaterals.

quadrilateraltrapezoidparallelogramrectanglerhombussquare

Everyday meaning ofprefix quadri-: four

Classifying Quadrilaterals

The angles of a quadrilateral have a special relationship.


Find the value of w in quadrilateral WXYZ.

m�W � m�X � m�Y � m�Z � 360 The sum of the measures is 360.

w � 45 � 110 � 65 � 360 Replace m�W with w, m�X with 45,m�Y with 110, and m�Z with 65.

w � 220 � 360 Simplify.


w � 140 Simplify.

� 220 � � 220

w˚ZW

X

Y

65˚45˚

110˚



The polygon at the right is a ,since it has four sides and four angles.

Draw a quadrilateral.

Pick one vertex and draw the diagonal to the opposite vertex.

1. Name the shape of the figures formed when you drew the diagonal. How many figures were formed?

2. You know that the sum of the angle measures of a triangle is 180°. Use this fact to find the sum of the angle measures in aquadrilateral. Explain your reasoning.

3. Find the measure of each angle of your quadrilateral. Comparethe sum of these measures to the sum you found in Exercise 2.

quadrilateral

• paper

• straightedge

• protractor

Angles of a Quadrilateral

Words The sum of the measures of the angles of a quadrilateral is 360°.

Model Symbols w � x � y � z � 360w˚ x˚

z˚ y˚

Lesson 6-4 Classifying Quadrilaterals 273

The concept map below shows how quadrilaterals are classified. Noticethat the diagram goes from the most general type of quadrilateral tothe most specific.

The best description of a quadrilateral is the one that is the most specific.

Classify Quadrilaterals

Classify each quadrilateral using the name that best describes it.

The quadrilateral has one pair of parallel sides. Itis a trapezoid.

The quadrilateral is a parallelogram with fourcongruent sides. It is a rhombus.

Classify each quadrilateral using the name that bestdescribes it.

a. b.

ClassifyingQuadrilateralsWhen classifying aquadrilateral, beginby counting thenumber of parallellines. Then countthe number of rightangles and thenumber ofcongruent sides.

Isosceles TrapezoidA trapezoid with onepair of oppositecongruent sides isclassified as anisosceles trapezoid.

Trapezoidquadrilateral withone pair of parallelopposite sides

Rectangleparallelogram with4 right angles

Parallelogramquadrilateral with bothpairs of opposite sidesparallel and congruent

Squareparallelogram with 4 congruent sides and 4 right angles

Rhombusparallelogram with 4 congruent sides

Quadrilateral




1. Explain why a square is a type of rhombus.

2. OPEN ENDED Give a real-life example of a parallelogram.

3. Which One Doesn’t Belong? Identify the quadrilateral that does notbelong with the other three. Explain your reasoning.

trapezoidsquarerectanglerhombus

Exercises 1 & 3

Find the value of x in each quadrilateral.

10. 11. 12.

13. 14. 15.


16. 17. 18. 19.

20. 21. 22. 23.

x˚ 67˚52˚

99˚

55˚45˚145˚ x˚

84̊61˚

103˚

x˚

58˚

95˚

112˚

x˚

120˚

60˚106˚

x˚

35˚

90˚

170˚x˚


For Exercises

10–15, 25–26

16–24, 27–30

See Examples

1

2, 3

Find the value of x in each quadrilateral.

4. 5. 6.


7. 8. 9.

x˚50˚

x˚150˚

30˚

150˚

x˚

110˚

80˚

125˚

Lesson 6-4 Classifying Quadrilaterals 275Art Resource, NY

35. MULTIPLE CHOICE Which of the following does not describe the quadrilateral at the right?

parallelogram square

trapezoid rhombus

36. SHORT RESPONSE In rhombus WXYZ, m�Z � 70°, m�X � 70°, andm�Y � 110°. Find the measure of �W.

37. The length of the hypotenuse of a 30°-60° right triangle is 16 feet. Find the length of the side opposite the 60° angle. Round to the nearest tenth. (Lesson 6-3)

38. The length of one of the legs of a 45°-45° right triangle is 8 meters. Find the length of the hypotenuse. Round to the nearest tenth. (Lesson 6-3)

Classify each triangle by its angles and by its sides. (Lesson 6-2)

39. 40. 41.

23˚

28˚

129˚54̊50˚

76˚

DC

BA

PREREQUISITE SKILL Decide whether the figures are congruent. Write yesor no and explain your reasoning. (Lesson 4-5)

42. 43. 44.8 mm

4 mm130˚ 130˚

5 in.5 in.

24. INTERIOR DESIGN The stained glass window shown is an example of how geometric figures can be used in decorating.Identify all of the quadrilaterals within the print.

25. ALGEBRA In parallelogram WXYZ, m�W � 45°, m�X � 135°, m�Y � 45°, and m�Z � (x � 15)°. Find the value of x.

26. ALGEBRA In trapezoid ABCD, m�A � 2a°, m�B � 40°, m�C � 110°, and m�D � 70°. Find the value of a.

Name all quadrilaterals with the given characteristic.

27. only one pair of parallel sides 28. opposite sides congruent

29. all sides congruent 30. all angles are right angles

CRITICAL THINKING Determine whether each statement is trueor false. If false, draw a counterexample.

31. All trapezoids are quadrilaterals.

32. All squares are rectangles.

33. All rhombi (plural of rhombus) are squares.

34. A trapezoid can have only one right angle.


SOL Practice

/sol


(l)Aaron Haupt, (r)John Evans

What You’ll LearnSolve problems usingthe logical reasoningstrategy.

6-4b Problem-Solving StrategyA Follow-Up of Lesson 6-4


1. Deductive reasoning uses an existing rule to make a decision. Determinewhere Zach and Jacy used deductive reasoning. Explain.

2. Inductive reasoning is the process of making a rule after observing severalexamples and using that rule to make a decision. Determine where Zach and Jacy used inductive reasoning. Explain.

3. Write about a situation in which you use inductive reasoning to solve aproblem. Then solve the problem.

Use Logical Reasoning

Jacy, how can we be sure this playing field we’ve marked out is a rectangle? We don’t haveanything we can use to measure its angles.

Someone told me that there is somethingspecial about the diagonals of a rectangle.Zach, let’s see if we can use logicalreasoning to figure out what that is.

ExploreThe playing field is a parallelogram because its opposite sides are the samelength. Our math teacher said that means they are also parallel. We need to see what the relationship is between the diagonals of a rectangle.

PlanLet’s draw several different rectangles, measure the diagonals, and see ifthere is a pattern.

Solve

It appears that the diagonals of a rectangle are congruent. If the diagonals of our field are congruent, then we can reason that it is a rectangle.

Examine

Do all parallelograms, not just rectangles, have congruent diagonals? The counterexample at the right suggests that this statement is false.

A B

D CAC � BD

A

DAC � BD AC � BD AC � BD

B

C

A

D

B

C

A

D

B

C

Lesson 6-4b Problem-Solving Strategy: Use Logical Reasoning 277

Solve. Use any strategy.

6. GEOMETRY If the sides of the pentagonsshown are 1 unit long, find the perimeter of8 pentagons arranged according to thepattern below.

7. MONEY After a trip to the mall, Alex andMarcus counted their money to see howmuch they had left. Alex said, “If I had $4 more, I would have as much as you.”Marcus replied, “If I had $4 more, I wouldhave twice as much as you.” How muchdoes each boy have?

8. WEATHER Based on the data shown, what isa reasonable estimate for the difference inthe July high and low temperatures inStatesboro?

9. MEASUREMENT You have a largecontainer of pineapple juice, an empty 4-pint container, and an empty 5-pintcontainer. Explain how you can use thesecontainers to measure 2 pints of juice for a punch recipe.

10. LAUNDRY You need two clothespins tohang one towel on a clothesline. Oneclothespin can be used on a corner of onetowel and a corner of the towel next to it.What is the least number of clothespinsyou need to hang 8 towels?

11. STANDARDIZED TEST PRACTICEVanessa and Ashley varied the length of apendulum and measured the time it tookfor the pendulum to complete one swingback and forth. Based on their data, howlong do you think a pendulum with aswing of 5 seconds is?

21 ft 23 ft

24 ft 25 ftDC

BA

5 pt 4 pt

Tem

pera

ture

( °F)

100

80

60

40

20

0

Year’98’97 ’99 ’00 ’01

Statesboro July Temperatures

Time (s)

Length (ft)

1

1

2

4

3

9

4

16

Solve. Use logical reasoning.

4. GEOMETRY Draw several parallelogramsand measure their angles. What can youconclude about opposite angles ofparallelograms? Did you use deductive orinductive reasoning?

5. SPORTS Noah, Brianna, Mackenzie,Antoine, and Bianca were the first fivefinishers of a race. From the given clues,give the order in which they finished.

• Noah passed Mackenzie just before thefinish line.

• Bianca finished 5 seconds ahead ofNoah.

• Brianna crossed the finish line afterMackenzie.

• Antoine was fifth at the finish line.

oppositeangles

SOL Practice

Find the sum of the anglemeasures of polygons.

• paper• straightedge

Angles of PolygonsIn this lab, you will use the fact that the sum of the angle measures of atriangle is 180° to find the sum of the angle measures of any polygon.

INVESTIGATE Work with a partner.

Copy and complete the table below.

A Preview of Lesson 6-5

1. Predict the number of triangles in an octagon and the sum of itsangle measures. Check your prediction by drawing a figure.

2. Write an algebraic expression that tells the number of triangles inan n-sided polygon. Then write an expression for the sum of theangle measures in an n-sided polygon.

REGULAR POLYGONS A regular polygon is one that is equilateral(all sides congruent) and equiangular (all angles congruent).Polygons that are not regular are said to be irregular.

3. Use your results from Exercise 2 to find the measure of each anglein the four regular polygons shown above. Check your results byusing a protractor to measure one angle of each polygon.

4. Write an algebraic expression that tells the measure of each anglein an n-sided regular polygon. Use it to predict the measure ofeach angle in a regular octagon.

regularhexagon

regularpentagon

squareequilateraltriangle


Number of Sketch Number of Sum ofSides of Figure Triangles Angle Measures

3 1 1(180°) � 180°

4 2 2(180°) � 360°

5

6

7polygon: a simple closedfigure in a plane formedby three or more linesegments (Lesson 4-5)

Identify congruentpolygons.

congruent polygons

Lesson 6-5 Congruent Polygons 279Aaron Haupt

Congruent Polygons

QUILTING A template, or pattern, for a quilt block contains the minimum number of shapes needed to create the pattern.

1. How many different kinds of triangles are shown in the Winter Stars quilt at the right? Explain your reasoning and draw each triangle.

2. Copy the quilt and label all matching triangles with the same number, starting with 1.

am I ever going to use this?

Polygons that have the same size and shape are called . Recall that the parts of polygons that “match” are called

corresponding parts.polygons

congruent

In a congruence statement, the letters identifying each polygon arewritten so that corresponding vertices appear in the same order. Forexample, for the diagram below, write �CBD � �PQR.

�CBD � �PQR

Vertex C corresponds to vertex P.Vertex B corresponds to vertex Q.Vertex D corresponds to vertex R.

Two polygons are congruent if all pairs of corresponding angles arecongruent and all pairs of corresponding sides are congruent.

D

C

B

R

P

Q

Congruent Polygons

Words If two polygons are congruent, their corresponding sides arecongruent and their corresponding angles are congruent.

Model

Symbols Congruent angles: �A � �F, �B � �G, �C � �H

Congruent sides: B�C� � G�H�, A�C� � F�H�, A�B� � F�G�

A C

B

F H

G

← ←← ←← ←


You can use corresponding parts to find the measures of an angle orside in a figure that is congruent to a figure with known measures.

Identify Congruent Polygons

Determine whether the triangles shown are congruent. If so, namethe corresponding parts and write a congruence statement.

Angles The arcs indicate that �X � �M, �Y � �N, and �Z � �L.

Sides The side measures indicate that X�Y� � M�N�, Y�Z� � N�L�, and X�Z� � M�L�.

Since all pairs of corresponding angles and sides are congruent, the two triangles are congruent. One congruence statement is �XYZ � �MNL.

Determine whether the polygons shown arecongruent. If so, name the corresponding parts and write a congruence statement.

a. b. 8 ft

4 ft

4 ft

7 ft

F T V

X W

E

GH

D

C

B50˚

40˚

P

R

Q

50˚

40˚

Y

ZX 4 cm

9 cm6 cm

N

L M4 cm

9 cm6 cm

Recall that symbols like NR refer to themeasure of thesegment with thoseendpoints.

Find Missing Measures

In the figure, �AFH � �QRN.

Find m�Q.

According to the congruence statement, �A and �Q are corresponding angles. So, �A � �Q. Since m�A � 40°, m�Q � 40°.

Find NR.

F�H� corresponds to N�R�. So, F�H� � N�R�. Since FH � 9 inches, NR � 9 inches.

In the figure, quadrilateral ABCD is congruent toquadrilateral WXYZ. Find each measure.

c. m�X

d. YX

e. m�YA

B

W

X

C

D

Y

Z

4 m

3 m145˚

70˚

A

Q

N R

F

H

40˚65˚ 9 in.

13 in.

CongruenceStatements Otherpossible congruencestatements forExample 1 are�YZX � �NLM,�ZXY � �LMN,�YXZ � �NML,�XZY � �MLN,and �ZYX � �LNM.

msmath3.net/extra_examples


A EB D

C F

Lesson 6-5 Congruent Polygons 281Michael & Patricia Fogden/CORBIS

1. OPEN ENDED Draw and label a pair of congruent polygons. Be sure toindicate congruent angles and sides on your drawing.

2. FIND THE ERROR Justin and Amanda are writing a congruence statement for the triangles at the right. Who is correct? Explain.

Amanda�ABC � �YXZ

Justin

�ABC � �XYZ

A Y

C Z

B X

Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement.

3. 4.

In the figure, �PQR � �YWX. Find each measure.

5. m�X 6. YW

7. XY 8. m�W

RW Y

XQP10 yd

7 yd

61˚

73˚

KJ L

N M

12 in.

15 in.9 in.

15 in.

12 in.

AF

G

H

C

E

85˚

85˚45˚

45˚

50˚50˚

Exercise 2

Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement.

9. 10.

11. 12.

13. BIRDS The wings of a hummingbird are shaped like triangles. Determine whether these triangles are congruent. If so, name the corresponding parts and write a congruence statement.

F

N

J

C

15 ft

15 ft 18 ft

12.5 ft

WQ


For Exercises

9–13

14–23

See Examples

1

2, 3

H J

K

MQ

P

6 cm6 cm3 cm

3 cm5 cm

5 cm

9 m

6 m

AB

H E

G FDC

V T

WP

SZ

QY



282 Chapter 6 GeometryDoug Martin

In the figure, �JKL � �PNM. Find In the figure, quadrilateral ABCD �each measure. quadrilateral HEFG. Find each measure.

14. PN 15. PM 18. AD 19. m�H

16. m�P 17. m�N 20. m�G 21. CD

22. ALGEBRA Find the value of x in the two congruent triangles.

23. TRAVEL An overhead sign on an interstate highway is shown at the right. In the scaffolding, �ABC � �DCB, AC � 2.5 meters, BC � 1 meter, and AB � 2.7 meters. What is the length of B�D�?

24. CRITICAL THINKING Tell whether the following statement issometimes, always or never true. Explain your reasoning.

If the perimeters of two triangles are equal, then the triangles are congruent.

A C

B D

3x˚

39˚

A

D

B F

GH

E

C

76˚

81˚ 13 in.

11 in.

18 in.J

N

M PL

K70˚

8 m

6 m65˚

25. SHORT RESPONSE Which of the following polygons appear congruent?

a. b. c. d.

26. MULTIPLE CHOICE If �AFG � �PQR, which statement is not true?

�G � �R A�G� � P�Q� �P � �A A�G� � P�R�

Classify each quadrilateral using the name that best describes it. (Lesson 6-4)

27. 28. 29.

30. The length of each leg of a 45°-45° right triangle is 14 feet. Find thelength of the hypotenuse. (Lesson 6-3)

DCBA

BASIC SKILL Which figure cannot be folded so one half matches the other?

31. 32.

A B C DA

B C D

SOL Practice

Construct congruenttriangles.

• compass• straightedge• protractor• paper

Use a straightedge todraw a line. Put apoint on it labeled X.

Open your compass tothe same width as thelength of A�B�. Put thecompass point at X.Draw an arc thatintersects the line.Label this point ofintersection Y.

Open your compass tothe same width as thelength of A�C�. Placeyour compass point atX and draw an arcabove the line.

Open your compass tothe same width as thelength of B�C�. Place thecompass point at Y anddraw an arc above theline so that it intersects the arc drawn in Step 3. Label this point Z.

Draw Y�Z� and X�Z�. �ABC � �XYZ.

Lesson 6-5b Hands-On Lab: Constructing Congruent Triangles 283

Constructing Congruent Triangles


1. Explain why the corresponding sides of �ABC and �XYZare congruent.

2. Draw three different triangles. Then construct a triangle that iscongruent to each one.

X Y

Z

A B

C

X Y

X Y


XX

XX

CORBIS

CHAPTER

1. Describe three ways to classify triangles by their sides. (Lesson 6-2)

2. List and define five types of quadrilaterals. (Lesson 6-4)

For Exercises 3–5, use the figure at the right. (Lesson 6-1)

3. Find m�6 if m�7 � 84°.

4. Find m�5 if m�1 � 35°.

Find the value of x in each figure. (Lessons 6-2 and 6-4)

5. 6. 7.

8. FLAGS The “Union Jack”, a common name for the flag of the United Kingdom, is shown at the right. The blue portions of the flag aretriangular. Determine whether the trianglesindicated are congruent. If so, write acongruence statement. (Lesson 6-5)

(x � 2)˚ 35˚

105˚

2 13 4

6 57 8

�

m

x˚25˚ 88˚

105˚

9. MULTIPLE CHOICE How manypairs of congruent triangles areformed by the diagonals of arectangle? (Lesson 6-5)

2 3

4 5

10. GRID IN Find the value of a and b.(Lesson 6-1)

b

a

26 m

60˚

30˚

DC

BA

A B

C

G H

F

x˚

25̊

SOL Practice

The Game Zone: Classifying Polygons 285John Evans

Players: twoMaterials: 10 counters, 1 number cube, marker, 1 large red cube, 1 large blue cube, 2 square sheets of paper

• Write quadrilateral, trapezoid, parallelogram,rectangle, rhombus, and square on differentfaces of the red cube.

• In the same manner, write scalene, isosceles, equilateral, acute, right, and obtuse on different faces of the blue cube.

• Create two boards like the one shown bydrawing a different polygon in each square.Use no shape more than once.

• The starting player rolls the number cube. If an even number is rolled,the player rolls the red cube. If an odd number is rolled, the playerrolls the blue cube.

• The player covers with a counter any shape that matches theinformation on the top face of the cube. If a player cannot find a figure matching the information, he or she loses a turn.

• Who Wins? The first player to get three counters in a row wins.

Polygon Bingo

Identify line symmetry androtational symmetry.

line symmetryline of symmetryrotational symmetryangle of rotation

Symmetry

A figure has if it can be folded over a line so that onehalf of the figure matches the other half. This fold line is called the

.

vertical line horizontal line no line of symmetry of symmetry of symmetry

Some figures, such as the starfish in the Mini Lab above, have more than one line of symmetry. The figure at the right has one vertical, one horizontal, and two diagonal lines of symmetry.

line of symmetry

line symmetry



Trace the outline of the starfish shown onto both a piece of tracing paper and atransparency.

1. Draw a line down the center of your starfish outline. Thenfold your paper across this line. What do you notice about the two halves?

2. Are there other lines you can draw on your outline that will produce the same result? If so, how many?

3. Place the transparency over the outline on your tracing paper. Use your pencil point at the centers of the starfish to holdthe transparency in place. How many times can you rotate thetransparency from its original position so that the two figuresmatch? Do not count the original position.

4. Find the first angle of rotation by dividing 360° by the number of times the figures matched.

5. List the other angles of rotation by adding the first angle ofrotation to the previous angle. Stop when you reach 360°.

• tracing paper

• transparency

• pencil

• overheadmarkers

Lesson 6-6 Symmetry 287Doug Martin

A figure has if it can be rotated or turned lessthan 360° about its center so that the figure looks exactly as it does inits original position. The degree measure of the angle through whichthe figure is rotated is called the . Some figures havejust one angle of rotation, while others, like the starfish, have several.

angle of rotation

rotational symmetry

Identify Rotational Symmetry

LOGOS Determine whether each figure has rotational symmetry.Write yes or no. If yes, name its angle(s) of rotation.

Yes, this figure has rotational symmetry. It will matchitself after being rotated 180°.

Yes, this figure has rotational symmetry. It will matchitself after being rotated 120° and 240°.

0˚60˚

120˚

0˚

180˚90˚

LOGOS Many companiesand nonprofit groups use alogo so people can easilyidentify their products orservices. They often designtheir logo to have line orrotational symmetry.

Identify Line Symmetry

NATURE Determine whether the figure has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none.

This figure has one vertical line of symmetry.

Determine whether each figure has line symmetry.If it does, trace the figure and draw all lines ofsymmetry. If not, write none.

a. b. c.



288 Chapter 6 GeometryCourtesy Boston Bruins

1. OPEN ENDED Draw a figure that has rotational symmetry.

2. Which One Doesn’t Belong? Identify the capital letter that does nothave the type of symmetry as the other three. Explain your reasoning.

A M SB

SPORTS For Exercises 3–6, complete parts a and b for each figure.

a. Determine whether the logo has line symmetry. If it does, trace thefigure and draw all lines of symmetry. If not, write none.

b. Determine whether the logo has rotational symmetry. Write yes or no.If yes, name its angle(s) of rotation.

3. 4. 5. 6.

Exercise 2

JAPANESE FAMILY CRESTS For Exercises 7–14, complete parts a and b for each figure.

a. Determine whether the figure has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none.

b. Determine whether the figure has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation.

7. 8. 9. 10.

11. 12. 13. 14.

15. TRIANGLES Which types of triangles—scalene, isosceles, equilateral—haveline symmetry? Which have rotational symmetry?

16. ALPHABET What capital letters of the alphabet produce the same letterafter being rotated 180°?


For Exercises

7–15, 17, 21

7–15, 16, 18

See Examples

1

2, 3

Lesson 6-6 Symmetry 289(t)Doug Martin, (c)Scott Kim, (b)National Council of Teachers of Mathematics

23. MULTIPLE CHOICE Which shape has only two lines of symmetry?

24. SHORT RESPONSE Copy the figure at the right. Then shade two squares so that the figure has rotational symmetry.

25. DESIGN The former symbol for the National Council of Teachers of Mathematics is shown at the right. Which triangles in the symbol appear to be congruent? (Lesson 6-5)

26. In parallelogram ABCD, m�A � 55°, m�B � 125°, m�C � x°, and m�D � 125°. Find the value of x. (Lesson 6-4)

DCBA

PREREQUISITE SKILL Graph each point on a coordinate plane. (Page 614)

27. A(3, 2) 28. B(�1, 4) 29. C(�2, �1) 30. D(0, 3)

ROAD SIGNS For Exercises 17 and 18, use the diagrams below.

a. b. c. d.

17. Determine whether each sign has line symmetry. If it does, trace the sign and draw all lines of symmetry. If not, write none.

18. Which of the signs above could be rotated and still look the same?

19. RESEARCH Use the Internet or other resource to find other examples ofroad signs that have line and/or rotational symmetry.

20. ART Artist Scott Kim uses reflections of words or names as part of his art. Patricia’s reflected name is at the right. Create a reflection designfor your name using tracing paper.

CRITICAL THINKING Determine whether each statement is true or false. If false, give a counterexample.

21. If a figure has one horizontal and one vertical line of symmetry, then it also has rotational symmetry.

22. If a figure has rotational symmetry, it also has line symmetry.


SOL Practice


Graph reflections on acoordinate plane.

reflectionline of reflectiontransformation

Everyday Meaning ofreflection: theproduction of an imageby or as if by a mirror

Darrell Gulin/CORBIS

Reflections

PHOTOGRAPHY The undisturbed surface of a pond acts like a mirror and can provide the subject for beautiful photographs.

1. Compare the shape and size of the bird to its image in the water.

2. Compare the perpendicular distance from the water line to each of the points shown. What do you observe?

3. The points A, B, and C appear counterclockwise on the bird. How are these points oriented on the bird’s image?


The mirror image produced by flipping a figure over a line is called a. This line is called the . A reflection is one

type of or mapping of a geometric figure.transformationline of reflectionreflection

Draw a Reflection

Copy �JKL at the right on graph paper. Then draw the image of the figure after a reflection over the given line.

Step 1 Count the number of units between each vertex and the line of reflection.

Step 2 Plot a point for each vertex the same distance away from the line on the other side.

Step 3 Connect the new vertices to form the image of �JKL, �J�K�L�.

JJ'K' K

LL'

11

4 4

2 2

JK

L


Notation Read P� asP prime. It is the imageof point P.

Properties of Reflections

1. Every point on a reflection is the Modelsame distance from the line of reflection as the corresponding point on the original figure.

2. The image is congruent to the original figure, but the orientation of the image is different from that of the original figure. image

original

line ofreflection

X'

X

Z'

ZY

Y'

B C

A

C'B'

A'

Virginia SOL Standard 8.8 The student will apply transformations (rotate or turn, reflect orflip, translate or slide, and dilate or scale) to geometric figures represented on graph paper. The student will identify applications of transformations, such as tiling, fabric design, art, and scaling.

Lesson 6-7 Reflections 291

If a figure touches the line of reflection as it does in Example 3, thenthe figure and its image form a new figure that has line symmetry. Theline of reflection is a line of symmetry.

Reflect a Figure over the x-axis

Graph �PQR with vertices P(�3, 4), Q(4, 2), and R(�1, 1). Thengraph the image of �PQR after a reflection over the x-axis, andwrite the coordinates of its vertices.

The coordinates of the vertices of the imageare P�(�3, �4), Q�(4, �2), and R�(�1, 1).Examine the relationship between thecoordinates of each figure.

same

opposites

P(�3, 4) P�(�3, �4)

Q(4, 2) Q�(4, �2)

R(�1, 1) R�(�1, �1)

Notice that the y-coordinate of a point reflected over the x-axis is theopposite of the y-coordinate of the original point.

y

xO

P

P'

RR'

Q'

Q

Reflect a Figure over the y-axis

Graph quadrilateral ABCD with vertices A(�4, 1), B(�2, 3), C(0, �3), and D(�3, �2). Then graph the image of ABCD after areflection over the y-axis, and write the coordinates of its vertices.

The coordinates of the vertices of the imageare A�(4, 1), B�(2, 3), C�(0, �3), and D�(3, �2).Examine the relationship between thecoordinates of each figure.

opposites

same

A(�4, 1) A�(4, 1)

B(�2, 3) B�(2, 3)

C(0, �3) C�(0, �3)

D(�3, �2) D�(3, �2)

Notice that the x-coordinate of a point reflected over the y-axis is theopposite of the x-coordinate of the original point.

Graph �FGH with vertices F(1, �1), G(5, �3), andH(2, �4). Then graph the image of �FGH after a reflection overthe given axis, and write the coordinates of its vertices.

a. x-axis b. y-axis

y

xO

D

B'B

A'A

D'C'C

Points on Line ofReflection Noticethat if a point lieson the line ofreflection, theimage of that pointhas the samecoordinates asthose of the pointon the originalfigure.

←← ←← ←

←

←

←← ←← ←

←

←

←




1. OPEN ENDED Draw a triangle on grid paper. Then draw a horizontal line below the triangle. Finally, draw the image of the triangle after it isreflected over the horizontal line.

2. Explain how a reflection and line symmetry are related.

3. Which One Doesn’t Belong? Identify the transformation that is not thesame as the other three. Explain your reasoning.

4. Copy the figure at the right on graph paper. Then draw the image of the figure after a reflection over the given line.

Graph the figure with the given vertices. Then graph its image after a reflection over the given axis, and write the coordinates of its vertices.

5. parallelogram QRST with vertices Q(�3, 3), R(2, 4), S(3, 2), and T(�2, 1); x-axis

6. triangle JKL with vertices J(�2, 3), K(�1, �4), and L(�4, �2); y-axis

B C

DA

Exercises 2 & 3

Use a Reflection

MASKS Copy and complete the mask shown so that the completed figure has a vertical line of symmetry.

You can reflect the half of the mask shown over the indicated vertical line.Find the distance from each vertex on the figure to the line of reflection. Then plot a point that same distanceaway on the opposite side of the line. Connect vertices as appropriate.

Lesson 6-7 Reflections 293Art Resource, NY

Copy each figure onto graph paper. Then draw the image of the figure after a reflection over the given line.

7. 8. 9.

10. 11. 12.

For Exercises 13–16, determine whether the figure in green is a reflectionof the figure in blue over the line n. Write yes or no. Explain.

13. 14. 15. 16.

Graph the figure with the given vertices. Then graph its image after a reflection over the given axis, and write the coordinates of its vertices.

17. triangle ABC with vertices A(�1, �1), B(�2, �4), and C(�4, �1); x-axis

18. triangle FGH with vertices F(3, 3), G(4, �3), and H(2, 1); y-axis

19. square JKLM with vertices J(�2, 0), K(�1, �2), L(�3, �3), and M(�4, �1); y-axis

20. quadrilateral PQRS with vertices P(1, 3), Q(3, 5), R(5, 2), and S(3, 1); x-axis

Name the line of reflection for each pair of figures.

21. 22. 23. 24.

25. DESIGN Does the rug below have 26. DESIGN Copy and complete the rug line symmetry? If so, sketch the rug pattern shown so that the completed and draw the line(s) of symmetry. figure has line symmetry.

O

y

xO

y

xO

y

x

y

xO

nn

n

n

F D

A

CB

V

Q R ST

UF

J

G

H

L

M

K

JD

C

B

Z

YX


For Exercises

7–16, 27–28

17–24

25–26

See Examples

1

2, 3

4




ALPHABET For Exercises 27 and 28, use the figure at the right. It shows that the capital letter A looks the same after a reflection over a vertical line. It does not look the same after a reflection over a horizontal line.

27. What other capital letters look the same after a reflection over a vertical line?

28. Which capital letters look the same after a reflection over a horizontal line?

29. CRITICAL THINKING Suppose a point P with coordinates (�4, 5) is reflected so that the coordinates of its image are (�4, �5). Without graphing, which axis was this point reflected over? Explain.

SHORT RESPONSE For Exercises 30 and 31, use the drawingat the right.

30. The drawing shows the pattern for the left half of the frontof the shirt. Copy the pattern onto grid paper. Then drawthe outline of the pattern after it has been flipped over avertical line. Label it “Right Front.”

31. Use two geometric terms to explain the relationshipbetween the left and right fronts of the shirt.

32. MULTIPLE CHOICE Which of the following is the reflection of �ABCwith vertices A(1, �1), B(4, �1), and C(2, �4) over the x-axis?

CARDS Determine whether each card has rotational symmetry. Write yesor no. If yes, name its angle(s) of rotation. (Lesson 6-6)

33. 34. 35. 36.

37. Find the value of x if the triangles at the right are congruent. (Lesson 6-5)

B D

A E

C12 ft

16 ftx ft20 ft

Q

Q

5

53

3

A

A

y

xO

Dy

xO

Cy

xO

By

xO

A

PREREQUISITE SKILL Add. (Lesson 1-4)

38. �4 � (�1) 39. �5 � 3 40. �1 � 4 41. 2 � (�2)

LeftFront

RightFront

SOL Practice

Study Skill: How to Study Math Vocabulary 295

SKILL PRACTICEMake a definition map for each term.

1. complementary angles (Page 256)

3. isosceles triangle (Page 263)

2. perpendicular lines (Page 257)

4. square (Page 273)

Study Math Vocabulary

A definition map can help you visualize the parts of a good definition. Ask yourself these questions about the vocabulary terms.

• What is it? (Category)

• What can it be compared to? (Comparisons)

• What is it like? (Properties)

• What are some examples? (Illustrations)

Here’s a definition map for reflection.

ComparisonsWhat can it becompared to?

CategoryWhat is it?

PropertiesWhat is it like?dilation

translation

rotation

Reflection

image of

self in mirror

left and

right hands

a flip

mirror image

can produce

figure with line

symmetry

transformation ormapping of figure

IllustrationsWhat are some examples?

Understanding a

math term requires

more than just

memorizing a

definition. Try

completing a

definition map

to expand your

understanding

of a geometry

vocabulary word.

DEFINITION MAPDEFINITION MAP

Translations

CHESS In chess, there are rulesgoverning how many spaces and inwhat direction each game piece can bemoved during a player’s turn. Thediagram at the right shows one legalmove of a knight.

1. Describe the motion involved inmoving the knight.

2. Compare the shape, size, and orientation of the knight in itsoriginal position to that of the knight in its new position.


A (sometimes called a slide) is the movement of a figurefrom one position to another without turning it.

translation

Draw a Translation

Copy trapezoid WXYZ at the right on graph paper. Then draw the image of the figure after a translation 4 units left and 2 units down.

Step 1 Move each vertex of the trapezoid 4 units left and 2 units down.

Step 2 Connect the new vertices to form the image.

X

YW

W'

ZY'Z'

X'

X

YW

W'ZY'

Z'

X'

X

YW

Z


Properties of Translations

1. Every point on the original figure Modelis moved the same distance and in the same direction.

2. The image is congruent to the original figure, and the orientation of the image is the same as that of the original figure.

X'

X

Y

Y'

Z'

Z

original

image

Graph translations on acoordinate plane.

translation


Lesson 6-8 Translations 297

Translation in the Coordinate Plane

Graph �JKL with vertices J(�3, 4), K(1, 3), and L(�4, 1). Thengraph the image of �JKL after a translation 2 units right and 5 units down. Write the coordinates of its vertices.

The coordinates of the vertices of the image are J�(�1, �1), K�(3, �2), and L�(�2, �4). Notice that these vertices can also be found by adding 2 to the x-coordinates and �5 to the y-coordinates, or (2, �5).

Original Add (2, �5). Image

J(�3, 4) → (�3 � 2, 4 � (�5)) → J�(�1, �1)

K(1, 3) → (1 � 2, 3 � (�5)) → K�(3, �2)

L(�4, 1) → (�4 � 2, 1 � (�5)) → L�(�2, �4)

Graph �ABC with vertices A(4, �3), B(0, 2), andC(5, 1). Then graph its image after each translation, and write thecoordinates of its vertices.

a. 2 units down b. 4 units left and 3 units up

y

xO

J

J'

K

L

L'

K'

y

xO

JK

LJ'

L'K'

Use a Translation

MULTIPLE-CHOICE TEST ITEM Point N is movedto a new location, N�. Which white shapeshows where the shaded figure would be if it was translated in the same way?

A B C D

Read the Test Item

You are asked to determine which figure has been moved according to the same translation as Point N.

Solve the Test Item

Point N is translated 4 units left and 1 unit up. Identify the figurethat is a translation of the shaded figure 4 units left and 1 unit up.

Figure A: 2 units left and 2 units up

Figure B: represents a turn, not a translation

Figure C: 4 units left and 1 unit up

The answer is C.

DCBA

Translations In thecoordinate plane, atranslation can bedescribed using anordered pair. Atranslation up or tothe right is positive.A translation downor to the left isnegative.(2, �5) means atranslation 2 unitsright and 5 unitsdown.

Checking AnswersTo check your answer,compare the coordinatesof corresponding points oneach figure.

y

xO

D CA

B

NN'


SOLPractice



1. Which One Doesn’t Belong? Identify the transformation that is not thesame as the other three. Explain your reasoning.

2. OPEN ENDED Draw a rectangle on grid paper. Then draw the image of the rectangle after a translation 2 units right and 3 units down.

3. Copy the figure at the right on graph paper. Then draw the image of the figure after a translation 4 units left and 1 unit up.

Graph the figure with the given vertices. Then graph the image of the figure after the indicated translation, and write the coordinates of its vertices.

4. triangle XYZ with vertices X(�4, �4), Y(�3, �1), and Z(2, �2) translated 3 units right and 4 units up

5. trapezoid EFGH with vertices E(0, 3), F(3, 3), G(4, 1), and H(�2, 1)translated 2 units left and 3 units down

B

C

A

Exercise 1

Copy each figure onto graph paper. Then draw the image of the figure after the indicated translation.

6. 5 units right and 3 units up 7. 3 units right and 4 units down

Graph the figure with the given vertices. Then graph the image of thefigure after the indicated translation, and write the coordinates of its vertices.

8. �ABC with vertices A(1, 2), B(3, 1), and C(3, 4) translated 2 units left and1 unit up

9. �RST with vertices R(�5, �2), S(�2, 3), and T(2, �3) translated 1 unitleft and 3 units down

10. rectangle JKLM with vertices J(�3, 2), K(3, 5), L(4, 3), and M(�2, 0)translated by 1 unit right and 4 units down

11. parallelogram ABCD with vertices A(6, 3), B(4, 0), C(6, �2), and D(8, 1)translated 3 units left and 2 units up

Q

R

PG

E

FH


For Exercises

6–7

8–11

13–14

See Examples

1

2

3

Lesson 6-8 Translations 299(t)Burstein Collection/CORBIS, (b)Robert Brons/BPS/Getty Images

16. MULTIPLE CHOICE Which of the following is a vertex of the figure shown at the right after a translation 4 units down?

(1, �5) (�6, �2) (1, �3) (�2, �2)

17. SHORT RESPONSE What are the coordinates of W(–6, 3) after it is translated 2 units right and 1 unit down?

18. Graph polygon ABCDE with vertices A(�5, �3), B(�2, 1), C(�3, 4), D(0, 2), and E(0, �3). Then graph the image of the figure after a reflection over the y-axis, and write the coordinates of its vertices. (Lesson 6-7)

LIFE SCIENCE For Exercises 19 and 20, use the diagram of the diatom at the right. (Lesson 6-6)

19. Does the diatom have line symmetry? If so, trace the figure and draw any lines of symmetry. If not, write none.

20. Does the diatom have rotational symmetry?Write yes or no. If yes, name its angle(s) of rotation.

DCBA

y

xO

PREREQUISITE SKILL Determine whether each figure has rotational symmetry. Write yes or no. If yes, name its angles of rotation. (Lesson 6-6)

21. 22. 23. 24.

12. ART Explain why Andy Warhol’s 1962 Self Portrait, shown atthe right, is an example of an artist’s use of translations.

MUSIC For Exercises 13 and 14, use the following information.The sound wave of a tuning fork is given below.

13. Look for a pattern in the sound wave. Then copy the sound wave and indicate where this pattern repeats or is translated.

14. How many translations of the original pattern are shown?

15. CRITICAL THINKING Triangle RST has vertices R(4, 2), S(�8, 0), and T(6, 7). When translated, R� has coordinates (�2, 4). Find the coordinatesof S� and T�.


SOL Practice

/sol


Graph rotations on acoordinate plane.

rotationcenter of rotation

Rotations

The Mini Lab suggests the following properties of rotations.

A is a transformation involving theturning or spinning of a figure around a fixedpoint called the .

Draw a polygon, placing a dot at one vertex. Place asecond dot, the center of rotation, in a nearby corner.

Form an angle of rotation by connecting the first dot, thecenter of rotation, and a point on the edge of the paper.

Place a second paper over the first and trace the figure, the dots, and the ray passing through the figure.

With your pencil on the center of rotation, turn the toppaper until its ray lines up with the ray passing through the edge of the first paper. Tape the papers together.

1. Measure the distances from points on the original figure and corresponding points on the image to the center of rotation. What do you observe?

2. Measure the angles formed by connecting the center of rotation to pairs of corresponding points. What do you observe?

center ofrotation

angle ofrotation

Step 4Step 3Step 1 Step 2

center of rotation

rotation• tracing paper

• straightedge

• tape

• protractor

Properties of Rotations

1. Corresponding points are the same distance from R. The angles formed by connecting R to corresponding points are congruent.

2. The image is congruent to the original figure, and their orientations are the same.


Model

YY'

X

X'

Z

Z'

Rm�XRX' � m�YRY'

original

image

center ofrotation

angle of rotation: thedegree measure of theangle through which afigure is rotated(Lesson 6-6)


Lesson 6-9 Rotations 301Courtesy Ramona Maston/FolkArt.com

If a figure touches its center of rotation, then one or more rotations of the figure can be used to create a new figure that has rotationalsymmetry.

Rotations in the Coordinate Plane

Graph �XYZ with vertices X(2, 2), Y(4, 3), and Z(3, 0). Then graphthe image of �XYZ after a rotation 90° counterclockwise about theorigin, and write the coordinates of its vertices.

Step 1 Lightly draw a line connecting point X to the origin.

Step 2 Lightly draw O�X�� so that m�X�OX � 90° and OX� � OX.

Step 3 Repeat steps 1–2 for points Y and Z. Then erase all lightlydrawn lines and connect the vertices to form �X�Y�Z�.

Triangle X�Y�Z� has vertices X�(�2, 2), Y�(�3, 4), and Z�(0, 3).

Graph �ABC with vertices A(1, �2), B(4, 1), andC(3, �4). Then graph the image of �ABC after the indicatedrotation about the origin, and write the coordinates of its vertices.

a. 90° counterclockwise b. 180° counterclockwise

y

xO

XY

Z

X'

y

xO

XY

Y'

Z

X'

y

xO

XY

Y'

Z

Z'

X'

Use a Rotation

FOLK ART Copy and complete the barn sign shown so that the completed figure has rotational symmetry with 90°, 180°, and 270°as its angles of rotation.

Use the procedure described above and the points indicated to rotate the figure 90°, 180°, and 270°counterclockwise. Use a 90° rotation clockwise to produce the same rotation as a 270° rotation counterclockwise.

90° counterclockwise 180° counterclockwise 90° clockwise

Rotations Aboutthe Origin Thecenter of rotation inExample 1 is theorigin and the angleof rotation is 90°counterclockwise.

FOLK ART ThePennsylvania Dutch, orPennsylvania Germans,created signs that werepainted on the sides ofbarns or houses. Manyfeature designs that haverotational symmetry.Source: www.folkart.com




1. OPEN ENDED Give three examples of rotating objects you see every day.

2. FIND THE ERROR Anita and Manuel are graphing �MNP with verticesM(�3, 2), N(�1, �1), and P(�4, �2) and its image after a rotation 90° counterclockwise about the origin. Who is correct? Explain.

Graph the figure with the given vertices. Then graph the image of thefigure after the indicated rotation about the origin, and write thecoordinates of its vertices.

3. triangle ABC with vertices A(�2, �4), B(2, �1), and C(4, �3); 90° counterclockwise

4. quadrilateral DFGH with vertices D(�3, 2), F(�1, 0), G(�3, �4), andH(�4, �2); 180°

Ani ta

y

xO

N

M

M' P'P

N'

Manuel

y

xON

MM'

P'

P

N'

Exercises 1 & 2

Graph the figure with the given vertices. Then graph the image of the figure after the indicated rotation about the origin, and write thecoordinates of its vertices.

5. triangle VWX with vertices V(�4, 2), W(�2, 4), and X(2, 1); 180º

6. triangle BCD with vertices B(�5, 3), C(�2, 5), and D(�3, 2); 90º counterclockwise

7. trapezoid LMNP with vertices L(0, 3), M(4, 3), N(1, �3), and P(�1, 1); 90º counterclockwise

8. quadrilateral FGHJ with vertices F(�5, 4), G(�3, 4), H(0, �1), and J(�5, 2); 180º

Determine whether the figure in green is a rotation of the figure in blue about the origin. Write yes or no. Explain.

9. 10. 11. 12. y

xO

y

xO

y

xO

y

xO


For Exercises

5–12

13

See Examples

1

2

Lesson 6-9 Rotations 303

13. FABRIC DESIGN Copy and complete the handkerchief design at theright so that it has rotational symmetry. Rotate the figure 90°, 180°, and 270° counterclockwise about point C.

14. CRITICAL THINKING What are the new coordinates of a point at (x, y) after the point is rotated 90° counterclockwise? 180°?

15. SHORT RESPONSE Draw a rectangle. Then draw the image of therectangle after it has been translated 1.5 inches to the right and thenrotated 90º counterclockwise about the bottom left vertex. Label thisrectangle I.

16. MULTIPLE CHOICE Which illustration shows the figure at the right rotated 180°?

Identify each transformation as a reflection, a translation, or a rotation.(Lessons 6-7, 6-8, and 6-9)

17. 18. 19. 20.

For Exercises 21–25, use the graphic at the right.

21. Describe a translation used in this graphic.(Lesson 6-8)

22. Trace at least two examples of figures orparts of figures in the graphic that appear to have line symmetry. Then draw all lines of symmetry. (Lesson 6-6)

23. Trace a figure or part of a figure used in thegraphic that appears to have rotationalsymmetry. (Lesson 6-6)

24. Trace and then classify the quadrilateral thatmakes up the top portion of the collar on theshirt. (Lesson 6-4)

25. Trace the two triangles that make up thecollar on the shirt. Classify each triangle by its angles and by its sides and thendetermine whether the two triangles are congruent. (Lessons 6-2 and 6-5)

y

xO

y

xO

y

xO

y

xO

DCBA

Crafts are tops on this gift list51%

USA TODAY Snapshots®

By Cindy Hall and Suzy Parker, USA TODAY

Source: WGBH in conjunction with AppliedResearch & Consulting LLC for ZOOM

19%15%

7%

If they were a father, whatkids say they would like toreceive for Father’s Day:

Somethingyour child

made

A night ofpeace and

quiet

A nightout on

the town

Offer tohelp aroundthe house

C


SOL Practice

/sol


Create Escher-like drawingsusing translations androtations.

• index cards• scissors• tape• paper

304 Chapter 6 GeometryCordon Art-Baarn-Holland. All rights reserved.

TessellationsMaurits Cornelis Escher (1898–1972) was a Dutch artist whose work usedtessellations. A tessellation is a tilingmade up of copies of the same shape or shapes that fit together without gapsand without overlapping. The sum ofthe angle measures where vertices meetin a tessellation must equal 360°. Forthis reason, equilateral triangles and squares will tessellate a plane.

90˚ 90˚90˚ 90˚

4 � 90° = 360°60˚

60˚ 60˚60˚ 60˚

60˚

6 � 60° = 360°


Create a tessellation using a translation.

Draw a square on the back of an index card. Then draw atriangle inside the top of the square as shown below.

Cut out the square. Then cut out the triangle and translateit from the top to the bottom of the square.

Tape the triangle and square together to form a pattern.

Trace this pattern onto a sheet of paper as shown tocreate a tessellation.

Make an Escher-like drawing using each pattern.

a. b. c.

Step 1 Step 2 Step 3

Symmetry drawing E70 by M.C. Escher. © 2002 Cordon Art-Baarn-Holland. All rights reserved.

Lesson 6-9b Hands-On Lab: Tessellations 305

1. Design and draw a pattern for an Escher-like drawing.

2. Describe how to use your pattern to create a pattern unit for yourtessellation. Then create a tessellation using your pattern.

3. Name another regular polygon other than an equilateral triangle orsquare that will tessellate a plane. Explain your reasoning.

Determine whether each of the following figures will tessellate aplane. Explain your reasoning.

4. 5. 6.

30˚

60˚

115˚115˚

65˚65˚

70˚ 110˚

70˚110˚

Create a tessellation using a rotation.

Draw an equilateral triangle on the back of an index card.Then draw a right triangle inside the left side of thetriangle as shown below.

Cut out the equilateral triangle. Then cut out the righttriangle and rotate it so that the right triangle is on theright side as indicated.

Tape the right triangle and equilateral triangle together toform a pattern unit.

Trace this pattern onto a sheet of paper as shown to create a tessellation.

Make an Escher-like drawing using each pattern.

d. e. f.

Step 1 Step 2 Step 3

Line and Angle Relationships (pp. 256–260)6-16-1

CHAPTER

acute angle (p. 256)

acute triangle (p. 263)

adjacent angles (p. 256)

alternate exterior angles (p. 258)

alternate interior angles (p. 258)

angle of rotation (p. 287)

center of rotation (p. 300)

complementary angles (p. 256)

congruent polygons (p. 279)

corresponding angles (p. 258)

equilateral triangle (p. 263)

isosceles triangle (p. 263)

line of reflection (p. 290)

line of symmetry (p. 286)

line symmetry (p. 286)

obtuse angle (p. 256)

obtuse triangle (p. 263)

parallel lines (p. 257)

parallelogram (p. 273)

perpendicular lines (p. 257)

quadrilateral (p. 272)

rectangle (p. 273)

reflection (p. 290)

rhombus (p. 273)

right angle (p. 256)

right triangle (p. 263)

rotation (p. 300)

rotational symmetry (p. 287)

scalene triangle (p. 263)

square (p. 273)

straight angle (p. 256)

supplementary angles (p. 256)

transformation (p. 290)

translation (p. 296)

transversal (p. 258)

trapezoid (p. 273)

triangle (p. 262)

vertical angles (p. 256)

Lesson-by-Lesson Exercises and Examples

State whether each sentence is true or false. If false, replace the underlined wordto make a true sentence.1. A(n) angle has a measure greater than 90° and less than 180°.2. The sum of the measures of angles is 180°.3. lines intersect at a right angle.4. In a(n) triangle, all three sides are congruent.5. A(n) is a parallelogram with four congruent sides.6. An isosceles trapezoid has symmetry.7. The orientations of a figure and its reflected image are .different

rotationalrhombus

scaleneParallel

supplementaryacute

Vocabulary and Concept Check

msmath3.net/vocabulary_review

Find the value of x in each figure.8. 9.

For Exercises 10 and 11, use the figure at the right.10. Find m�8 if

m�4 � 118°.11. Find m�6 if

m�2 � 135°.

43˚ x˚125˚x˚

8 71 2

6 53 4

cb

a

Example 1If m�1 � 105°, find m�3, m�5, and m�8.Since �1 and �3 are vertical angles �1 � �3. So, m�3 � 105°.

Since �1 and �5 are correspondingangles, �1 � �5. Therefore, m�5 � 105°.

Since �5 and �8 are supplementary, m�8 � 180° � 105° or 75°.

1 24 3

5 68 7

x

y

z


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19. In quadrilateral JKLM, m�J � 123º, m�K � 90º, and m�M � 45º. Find m�L.

20. Classify the quadrilateral shown using the name that best describes it.

Example 4 Find the value of x in quadrilateral ABCD.x � 93 � 90 � 105 � 360

x � 228 � 360�228 � �228

x � 72

A

CD

x˚

105˚

93˚B

Classifying Quadrilaterals (pp. 272–275)6-46-4

Find the value of x in each triangle.12. 13.

14. Classify the triangle in Exercise 13 byits angles and by its sides.

x˚67˚

46˚ x˚54̊

Triangles and Angles (pp. 262–265)

Find each missing length. Round to thenearest tenth if necessary.15. 16.

17. 18. b

c 4.5 in.60̊30˚

c

b10 ft

45˚45˚

a

c3 m 45˚

45˚b

a4 cm60̊

30˚

Special Right Triangles (pp. 267–270)

Example 3 Find eachmissing length.

a � �12

�(6) or 3 m

To find b, use the Pythagorean Theorem.62 � 32 � b2 c2 � a2 � b2

36 � 9 � b2 Evaluate 62 and 92.

27 � b2 Subtract 9 from each side.

5.2 � b Take the square root of each side.

b

a

6 m

60̊

30˚

In the figure, FGHJ � YXWZ. Find each measure.21. m�X22. WZ23. YX24. m�Z

WX

YZ

J H

GF

124˚75˚

58˚8 cm

11 cm

10 cm

Congruent Polygons (pp. 279–282)

Example 5In the figure, �ABC � �RPQ. Find PQ.P�Q� corresponds to B�C�. Since BC �

5 feet, PQ � 5 feet.

65˚

45˚

5 ft4 ft

A

C

B

P

Q

R

6-26-2

6-36-3

6-56-5

Example 2Find the value of x in �JKL.

x � 20 � 20 � 180x � 180 � (20 � 20) or 140

x˚

20˚

20˚J

K

L

Chapter 6 Study Guide and Review 307

BOATING Determine whether each signalflag has line symmetry. If it does, trace the figure and draw all lines of symmetry.If not, write none.25. 26. 27.

28. Which of the figures above hasrotational symmetry? Name theangle(s) of rotation.

Symmetry (pp. 286–289)

Example 6 Determine whether the logo at the right has rotational symmetry. If itdoes, name its angles of rotation.

The logo has rotational symmetry. Itsangles of rotation are 90º, 180º, and 270º.

0˚45˚

90˚

Graph �JKL with vertices J(�1, 3), K(1, 1), and L(3, 4). Then graph its imageafter the indicated rotation about theorigin, and write the coordinates of its vertices. 33. 90° counterclockwise34. 180° counterclockwise

Rotations (pp. 300–303)

Example 9 Graph �PQR with vertices P(1, 3), Q(2, 1), and R(4, 2) and its imageafter a rotation of 90º counterclockwise about the origin.

y

xO

PR'

Q'

RP'

Q

6-66-6

Graph parallelogram QRST with verticesQ(2, 5), R(4, 5), S(3, 1), and T(1, 1). Thengraph its image after a reflection over the given axis, and write the coordinatesof its vertices.29. x-axis 30. y-axis

Reflections (pp. 290–294)

Example 7 Graph �FGH with vertices F(1, �1), G(3, 1), and H(2, �3) and its image after a reflection over the y-axis.

y

xO

H'

G'

H

G

F' F

6-76-7

Graph �ABC with vertices A(2, 2), B(3, 5), and C(5, 3). Then graph its image after the indicated translation, and write the coordinates of its vertices.31. 6 units down32. 2 units left and 4 units down

Translations (pp. 296–299)

Example 8 Graph �XYZ with vertices X(�3, �1), Y(�1, 0),and Z(�2, �3) and its image after a translation 4 units right and 1 unit up.

y

xO

Z

XZ'

Y'Y X'

6-86-8

6-96-9


CHAPTER

Chapter 6 Practice Test 309

1. Draw a pair of complementary angles. Label the angles �1 and �2.

2. OPEN ENDED Draw an obtuse isosceles triangle.



5. Name a pair of corresponding angles.

Find each missing measure. Round to the nearest tenth.

6. 7.

DESIGN Identify each quadrilateral in the stained glass window using the name that best describes it.

8. A 9. B 10. C

In the figure at the right, �MNP � �ZYX. Find each measure.

11. ZY 12. �Z

MUSIC Determine whether each figure has line symmetry. If it does,trace the figure and draw all lines of symmetry. If not, write none.

13. 14. 15.

16. Which of the figures in Exercises 13–15 has rotational symmetry?

Graph �JKL with vertices J(2, 3), K(�1, 4), and L(�3, �5). Then graph itsimage, and write the coordinates of its vertices after each transformation.

17. reflection over the x-axis 18. translation by (�2, 5) 19. rotation 180°

45˚45˚

6 in.

b

c

60˚30˚

32 cm

b

a

20. MULTIPLE CHOICE W�Y� is a diagonal of rectangle WXYZ. Which angle is congruent to �WYZ?

�WXY �WYX �ZWY �XWYDCBA

m

n4

1

3

2

7

5

8

6

�

N

X Z

YM

P

28˚5.7 m

6.3 m

35˚

XW

YZ

AB

C

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SOL Practice

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CHAPTER

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

1. One week Alexandria ran 500 meters, 600 meters, 800 meters, and 1,100 meters.How many kilometers did she run thatweek? (Prerequisite Skill, pp. 606–607)

1 2 3 4

2. The graph shows the winning times inseconds of the women’s 4 � 100-meterfreestyle relay for several Olympic games.

What is a reasonable prediction for thewinning time in 2008? (Lesson 1-1)

212 s 215 s 218 s 221 s

3. Which expression is equivalent to xy2z�1?(Lesson 2-2)

�x � y

1� y � z� x � y � y � z

x � y � y � z �x � y

z� y

�

4. Aleta went to the grocery store and paid$19.71 for her purchases. A portion of herreceipt is shown below.

About how much did the beef cost perpound? (Lesson 4-1)

$2.60 $3.34 $3.80 $4.25

5. Which of the stores represented in thecircle graph is preferred by �

275� of the

students? (Lesson 5-1)

Ultimate Jeans All That!

Terrific Trends Formal For You

6. Which of the following could not be the measure of �M? (Lesson 6-4)

35° 50° 45° 116°

7. Which of the following figures is not a rotation of the figure at the right? (Lesson 6-9)

DC

BA

IHGF

L

M

NP115˚ 130˚

DC

BA

44%Ultimate Jeans

28%All That!

10%Formal For

You

18%Terrific Trends

Preferred Clothing Stores

IHGF

2.3 lbs grapes............ $2.75 1.6 lbs cheddar.......... $4.23 3.1 lbs beef.............. $11.71 1.2 lbs tomatoes........ $1.02

DC

BA

IHGF

Tim

es (s

)

Year’84’76

0

216

212

220

224

’92 ’00 ’08

y

x

DCBA

Question 3 Answer every question whenthere is no penalty for guessing. If you must guess, eliminate answers you know are incorrect. For Question 3, eliminateChoice B since xy2 � x � y � y.

SOL Practice

Chapters 1–6 Standardized Test Practice 311

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

8. Ms. Neville has 26 students in herhomeroom. All of her students take at leastone foreign language. Thirteen studentstake Spanish, 11 students take French, and5 students take Japanese. No student takesall three languages. How many students takemore than one language? (Lesson 1-1)

9. Write �8�, �6�, 3.2, and �13

� in order fromleast to greatest. (Lesson 3-3)

10. An area of 2,500 square feet of grassproduces enough oxygen for a family of 4.What is the area of grass needed to supply afamily of 5 with oxygen? (Lesson 4-4)

11. You buy a sweater on sale for $29.96. Youpaid 25% less than the original price. Whatwas the original price of the sweater?(Lesson 5-7)

12. If a � b, find the value of x. (Lesson 6-1)

13. Name a quadrilateral with one pair ofparallel sides and one pair of sides that arenot parallel. (Lesson 6-4)

14. How many lines of symmetry does the figure at the right have?(Lesson 6-6)

15. If �JKL � �MNP, name the segment in�MNP that is congruent to L�J�. (Lesson 6-5)

16. If �ABC is reflected about the y-axis, what are the coordinates of point A�?(Lesson 6-7)

Record your answers on a sheet of paper. Show your work.

17. The graph below shows Arm 1 of thedesign for a company logo. (Lesson 6-9)

a. To create Arm 2 of the logo, graph theimage of figure ABCDE after a rotation90° counterclockwise about the origin.Write the coordinates of the vertices ofArm 2.

b. To create Arm 3 of the logo, graph thecoordinates of figure ABCDE after arotation 180° about the origin. Write thecoordinates of the vertices of Arm 3.

c. To create Arm 4 of the logo, graph thecoordinates of Arm 2 after a rotation180° about the origin. Write thecoordinates of the vertices of Arm 4.

d. Does the completed logo have rotationalsymmetry? If so, name its angle(s) ofrotation.

(With teacher red)

y

xO

C

D

EB

A

Arm 1

y

xOC

A

B

a

b

135˚

3x˚

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Geometry Geometry: Measuring Area and Volume · Geometry Geometry: Measuring Area and Volume ... You will solve problems about angle relationships in Lesson 6-1. ... Foldable to help