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ChapterChapter

How is it supported?

A skyway 41 stories above the groundconnects the Petronas Towers inMalaysia. The skyway is supported bybeams that make a triangular shape. Therigid structure of a triangle is very strong.

The spires of the Petronas Towers makethe building taller than Chicago’s SearsTower. However, the Sears Tower hasmore floors: 110 compared to the 88 inthe Petronas Towers.

Learn More About ItYou will learn more about the PetronasTowers in Exercise 26 on p. 196.

x

supportbeams

x

44TriangleRelationshipsTriangleRelationships

170

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Who usesTriangle Relationships?

How will you use these ideas?

• Learn more about basketball plays. (p. 177)• Understand how triangular structures are used to

prevent erosion. (p. 183)• See how rock climbers use a safety rope. (p. 189)• Analyze tile patterns. (p. 190)• Investigate a baseball’s path during a double play. (p. 205)

171

ROCK CLIMBERThe climber is using amethod of rock climbingcalled top roping. When theropes hanging down fromthe top of the rock are thesame length, the anglesthey form at the top of therock have the samemeasure. (p. 189)

WATER RESOURCE MANAGERWater resource managersgather information like rainfall data and water usageto study the effects of wateron the environment. They use triangular structures tominimize erosion. (p. 183)

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Visualize It!

172 Chapter 4 Triangle Relationships

Chapter Study GuideStudy Guide44

What’s the chapter about?

• Classifying triangles and finding their angle measures• Using the Distance Formula, the Pythagorean Theorem, and its converse• Showing relationships between a triangle’s sides and angles

Key Words

Take this quick quiz. If you are unsure of an answer, look at the

reference pages for help.

Vocabulary Check (refer to p. 61)

1. In the figure shown, BD&*( is the angle bisector of aABC. What is the value of x?

�A 10 �B 15 �C 20 �D 30

Skill Check (refer to pp. 30, 55)

2. What is the distance between P(2, 3) and Q(7, 3)?

�F 3 �G 4 �H 5 �J 7

3. What is the midpoint of a segment with endpoints A(0, 2) and B(6, 4)?

�A (3, 2) �B (3, 3) �C (4, 2) �D (0, 3)

CB

A D

30�x �

PREVIEWPREVIEW

Chapter Readiness QuizPREPAREPREPARE

• equilateral, isosceles, scalenetriangles, p. 173

• equiangular, acute, right, obtusetriangles, p. 174

• interior, exterior angles, p. 181• legs of an isosceles triangle,

p. 185

• base angles of an isoscelestriangle, p. 185

• hypotenuse, p. 192• Pythagorean Theorem, p. 192• Distance Formula, p. 194• median of a triangle, p. 207• centroid, p. 208

Drawing TrianglesVISUAL STRATEGYVISUAL STRATEGY

When you sketch a triangle, try to make the angles roughly the correct size. 80�

60�

40�50�

80�

50�

These angles are the samein an isosceles triangle.

This 80� angle is twiceas big as the 40� angle.

cgpe-04sg 07/13/2001 7:59 PM Page 172

4.1 Classifying Triangles 173

4.14.1 Classifying Triangles

A is a figure formed by three segments joining three noncollinear points. A triangle can be classified by its sides and by its angles.

triangleGoal

Classify triangles by theirsides and by their angles.

Key Words

• equilateral, isosceles,scalene triangles

• equiangular, acute, right,obtuse triangles

• vertex

Classify the triangle by its sides.

a. b. c.

Solution

a. Because this triangle has 3 congruent sides, it is equilateral.

b. Because this triangle has no congruent sides, it is scalene.

c. Because this triangle has 2 congruent sides, it is isosceles.

4

6643

2

4

4

4

EXAMPLE 111 Classify Triangles by Sides

VOCABULARY TIP

Equi- means “equal,”and -lateral means“side.” So, equilateralmeans equal sides.

Student Help

Classify Triangles by Sides


1. 2. 3.

6

710

2

2

28

9

8

CLASSIFICATION OF TRIANGLES BY SIDES

Equilateral Isosceles Scalene

Triangle Triangle Triangle

3 congruent sides At least 2 No congruent sidescongruent sides

SUMMARY

3 segments

3 noncollinear points

cgpe-0401 07/13/2001 7:23 PM Page 173


Classify the triangle by its angles and by its sides.

a. b. c.

Solution

a. Because this triangle has 3 angles with measures less than 90� and 2 congruent sides, it is an acute isosceles triangle.

b. Because this triangle has a right angle and no congruent sides, it is a right scalene triangle.

c. Because this triangle has one angle greater than 90� and nocongruent sides, it is an obtuse scalene triangle.

95�

40�

45�

9

5.8

6.43

4

5

40� 70�

70�

EXAMPLE 222 Classify Triangles by Angles and Sides

Classify Triangles by Angles and Sides

Classify the triangle by its angles and by its sides.

4. 5. 6. 30� 30�

120�50�70�

60�

7

7.66.272�

36�

72�

CLASSIFICATION OF TRIANGLES BY ANGLESSUMMARY

Equiangular Triangle

3 congruent angles

Acute Triangle

3 acute angles

Right Triangle

1 right angle

Obtuse Triangle

1 obtuse angle

MORE EXAMPLES

More examples atclasszone.com

Student HelpI C L A S S Z O N E . C O M

cgpe-0401 07/13/2001 7:23 PM Page 174


A of a triangle is apoint that joins two sides of thetriangle. The side across froman angle is the opposite side.

vertex

A C

B

Name the side that is opposite the angle.

a. aA b. aB c. aC

Solution

a. BC&* is the side that is opposite aA.

b. AC&* is the side that is opposite aB.

c. AB&* is the side that is opposite aC.

EXAMPLE 333 Identify the Parts of a Triangle

Point B is a vertex.

BC&* is opposite aA.

Exercises4.14.1

1. What is the difference between an obtuse triangle and an acute triangle?

In Exercises 2–4, use the diagram.

2. Name the side opposite aP.

3. Name the side opposite aQ.

4. Classify the triangle by its sides.


5. 6. 7.

Classify the triangle by its angles.

8. 9. 10.

40�

70�

70�

6

457

7

7

Skill Check

P

P

R

Vocabulary Check

Guided Practice

VOCABULARY TIP

The plural of vertex isvertices.

Student Help

A B

C

cgpe-0401 07/13/2001 7:23 PM Page 175


Classifying Triangles Classify the triangle by its sides.

11. 12. 13.

14. 15. 16.

Classifying Triangles Classify the triangle by its angles.

17. 18. 19.

20. 21. 22.

23. Error Analysis A student claimsthat the triangle is both obtuse and acute because it has an obtuse angle and an acute angle. What is wrong with his reasoning?

Classifying Triangles Classify the triangle by its angles and by

its sides.

24. 25. 26.

27. 28. 29. J

KL

85�

50�45�

T

V

U42� 42�

96�

P P

R

L M

N

120�

D

E

FA

B

C

59�

59� 62�

60�

65�

55�

22�68�

90�

43�

82�

55�30�

28�

122�

5

7 7

5

4

35

5

5

5

53

8

7 3

Practice and Applications

Extra Practice

See p. 681.

Example 1: Exs. 11–16Example 2: Exs. 17–41Example 3: Exs. 42–47

Homework Help

30�

130� 20�

This triangle isacute and obtuse.

cgpe-0401 02/04/2002 5:02 PM Page 176


Matching Triangles In Exercises 30–36, use the example above to

match the triangle description with the most specific name.

30. Side lengths: 2 cm, 3 cm, 4 cm A. Equilateral

31. Side lengths: 3 cm, 2 cm, 3 cm B. Scalene

32. Side lengths: 4 cm, 4 cm, 4 cm C. Obtuse

33. Angle measures: 60�, 60�, 60� D. Equiangular

34. Angle measures: 30�, 60�, 90� E. Isosceles

35. Angle measures: 20�, 145�, 15� F. Acute

36. Angle measures: 50�, 55�, 75� G. Right

Basketball The diagram shows the position and spacing of five

basketball players running the “triangle offense.”

37. What type of triangle is formed by players A, B, and C?

38. What type of triangle is formed by players C, D, and E?

39. What type of triangle is formedby players B, D, and E?

40. Which three players appear to form an obtuse triangle?

41. Which three players appear to form a scalene triangle?

BASKETBALL The triangleoffense is used by manyprofessional teams. Playersare usually spaced 15 feet to18 feet apart from each other.This provides many optionsfor passing so a player canmake a basket.

Sports

A

B CD

E

Classify the triangle described.

a. Side lengths: 6, 8, 9 b. Angle measures: 50�, 60�, 70�

Solution

You may want to sketch the triangle.

a. b.

Because the triangle has three Because the triangle has threesides with different lengths, angles with measures less thanthe triangle is scalene. 90�, the triangle is acute.

60�

70�

50�9

68

EXAMPLE Classify Triangles

VISUAL STRATEGY

In Exs. 30–36 draw a sketch with sidelengths or anglemeasures that areroughly correct, asshown on p. 172.

Student Help

cgpe-0401 07/27/2001 9:23 AM Page 177


To help you determinethe side opposite avertex, you can drawan arrow from thevertex.

FG& is opposite aE.E

F

G

Visualize It!

A C

B

Identifying Parts of Triangles Identify which side is opposite

each angle.

42. 43. 44.

45. 46. 47.

Draw an example of the triangle.

48. obtuse scalene 49. right isosceles 50. acute scalene

51. right scalene 52. acute isosceles 53. obtuse isosceles

54. Multiple Choice Which of the following terms can be used to describea triangle with angle measures of 17�, 17 �, and 146�?

�A acute �B right �C obtuse �D equiangular

55. Multiple Choice What side is opposite aC?

�F AB&* �G BC&*

�H AC&* �J both BC&* and AC&*

Complements and Supplements Find the value of each variable.

(Lesson 2.3)

56. 57. 58.

Translations Find the image of the point using the translation

(x, y) → (x � 2, y � 4). (Lesson 3.7)

59. (2, 5) 60. (1, �3) 61. (�1, 2) 62. (0, �5)

63. (�4, �2) 64. (0, 0) 65. (�6, 4) 66. (�3, �1)

Solving Equations Solve the equation. (Skills Review, p. 673)

67. 5x � 15 � 180 68. x � 2x � 36 � 180

69. 3x � 5x � 20 � 180 70. �3x � (x � 8) � 180

71. 2(x � 1) � 3x � 7 � 180 72. 4(3x � 1) � 9x � 10 � 180

Algebra Skills

50� 2y �

8x �

(5x � 3)�(11x � 7)�

4x �

(6x � 10)�

Mixed Review

Standardized Test

Practice

Visualize It!

S

R T

P

N P

L

K

M

G

H J

D

E

F

A B

C

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