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236 Chapter 4 Congruent Triangles Isosceles, Equilateral, and Right Triangles USING PROPERTIES OF ISOSCELES TRIANGLES In Lesson 4.1, you learned that a triangle is isosceles if it has at least two congruent sides. If it has exactly two congruent sides, then they are the legs of the triangle and the noncongruent side is the base. The two angles adjacent to the base are the The angle opposite the base is the In the activity, you may have discovered the Base Angles Theorem, which is proved in Example 1. The converse of this theorem is also true. You are asked to prove the converse in Exercise 26. vertex angle. base angles. GOAL 1 Use properties of isosceles and equilateral triangles. Use properties of right triangles. Isosceles, equilateral, and right triangles are commonly used in the design of real-life objects, such as the exterior structure of the building in Exs. 29–32. Why you should learn it GOAL 2 GOAL 1 What you should learn 4.6 R E A L L I F E R E A L L I F E Investigating Isosceles Triangles Use a straightedge and a compass to construct an acute isosceles triangle. Then fold the triangle along a line that bisects the vertex angle, as shown. Repeat the procedure for an obtuse isosceles triangle. What observations can you make about the base angles of an isosceles triangle? Write your observations as a conjecture. 3 2 1 Developing Concepts ACTIVITY THEOREM 4.6 Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. If AB Æ £ AC Æ , then ™B £ ™C. THEOREM 4.7 Converse of the Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. If ™B £ ™C, then AB Æ £ AC Æ . THEOREMS B C A base base B C A base leg vertex angle base angles leg
Transcript
• 236 Chapter 4 Congruent Triangles

Isosceles, Equilateral, andRight Triangles

USING PROPERTIES OF ISOSCELES TRIANGLES

In Lesson 4.1, you learned that a triangle isisosceles if it has at least two congruent sides. If it has exactly two congruent sides, then theyare the legs of the triangle and the noncongruentside is the base. The two angles adjacent to thebase are the The angle opposite thebase is the

In the activity, you may have discovered the Base Angles Theorem, which isproved in Example 1. The converse of this theorem is also true. You are asked toprove the converse in Exercise 26.

vertex angle.base angles.

GOAL 1

Use properties ofisosceles and equilateraltriangles.

Use properties ofright triangles.

� Isosceles, equilateral, andright triangles are commonlyused in the design of real-lifeobjects, such as the exteriorstructure of the building in Exs. 29–32.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

4.6RE

AL LIFE

RE

AL LIFE

Investigating Isosceles Triangles

Use a straightedge and a compass to construct an acute isosceles triangle. Then fold the triangle along a line that bisects the vertex angle, as shown.

Repeat the procedure for an obtuse isosceles triangle.

What observations can you make about the base angles of an isosceles triangle? Write your observations as a conjecture.

3

2

1

DevelopingConcepts

ACTIVITY

THEOREM 4.6 Base Angles TheoremIf two sides of a triangle are congruent, then the angles opposite them are congruent.

If ABÆ

£ ACÆ

, then ™B £ ™C.

THEOREM 4.7 Converse of the Base Angles TheoremIf two angles of a triangle are congruent, then the sides opposite them are congruent.

If ™B £ ™C, then ABÆ

£ ACÆ

.

THEOREMS

B

C

A

base

base

B

C

A

base

leg

vertex angle

baseangles

leg

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• 4.6 Isosceles, Equilateral, and Right Triangles 237

Proof of the Base Angles Theorem

Use the diagram of ¤ABC to prove the Base Angles Theorem.

GIVEN � ¤ABC, ABÆ

£ ACÆ

PROVE � ™B £ ™C

Paragraph Proof Draw the bisector of ™CAB. By construction, ™CAD £ ™BAD.You are given that AB

Æ£ AC

Æ. Also, DA

Æ£ DA

Æby the Reflexive Property of

Congruence. Use the SAS Congruence Postulate to conclude that ¤ADB £ ¤ADC.Because corresponding parts of congruent triangles are congruent, it follows that™B £ ™C.

. . . . . . . . . .

Recall that an equilateral triangle is a special type of isosceles triangle. Thecorollaries below state that a triangle is equilateral if and only if it is equiangular.

Using Equilateral and Isosceles Triangles

a. Find the value of x.

b. Find the value of y.

SOLUTION

a. Notice that x represents the measure of an angle of an equilateral triangle.From the corollary above, this triangle is also equiangular.

3x° = 180° Apply the Triangle Sum Theorem.

x = 60 Solve for x.

b. Notice that y represents the measure of a base angle of an isosceles triangle. From the Base Angles Theorem, the other base angle has the same measure. The vertex angle forms a linear pair with a 60° angle, so itsmeasure is 120°.

120° + 2y° = 180° Apply the Triangle Sum Theorem.

y = 30 Solve for y.

E X A M P L E 2

E X A M P L E 1

COROLLARY TO THEOREM 4.6

If a triangle is equilateral, then it is equiangular.

COROLLARY TO THEOREM 4.7

If a triangle is equiangular, then it is equilateral.

COROLLARIES

C

B

A D

A

B C

60� y �120�

y �

x � y �

www.mcdougallittell.comfor extra examples.

INTE

RNET

STUDENT HELP

UsingAlgebra

xyxy

Proof

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• 238 Chapter 4 Congruent Triangles

USING PROPERTIES OF RIGHT TRIANGLES

You have learned four ways to prove that triangles are congruent.

• Side-Side-Side (SSS) Congruence Postulate (p. 212)• Side-Angle-Side (SAS) Congruence Postulate (p. 213)• Angle-Side-Angle (ASA) Congruence Postulate (p. 220)• Angle-Angle-Side (AAS) Congruence Theorem (p. 220)

The Hypotenuse-Leg Congruence Theorem below can be used to prove that tworight triangles are congruent. A proof of this theorem appears on page 837.

Proving Right Triangles Congruent

The television antenna is perpendicular to the plane containing the points B, C, D, and E. Each of the stays running from the top of the antenna to B, C, and D uses the same length of cable. Prove that ¤AEB, ¤AEC, and ¤AED are congruent.

GIVEN � AEÆ

fi EBÆ

, AEÆ

fi ECÆ

, AEÆ

fi EDÆ

, ABÆ

£ ACÆ

PROVE � ¤AEB £ ¤AEC £ ¤AED

SOLUTION

Paragraph Proof You are given that AEÆ

fi EBÆ

and AEÆ

fi ECÆ

, which implies that™AEB and ™AEC are right angles. By definition, ¤AEB and ¤AEC are right triangles. You are given that the hypotenuses of these two triangles, AB

Æand AC

Æ,

are congruent. Also, AEÆ

is a leg for both triangles, and AEÆ

£ AEÆ

by the ReflexiveProperty of Congruence. Thus, by the Hypotenuse-Leg Congruence Theorem,¤AEB £ ¤AEC.

� Similar reasoning can be used to prove that ¤AEC £ ¤AED. So, by theTransitive Property of Congruent Triangles, ¤AEB £ ¤AEC £ ¤AED.

E X A M P L E 3

GOAL 2

THEOREM 4.8 Hypotenuse-Leg (HL) Congruence TheoremIf the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

If BCÆ

£ EFÆ

and ACÆ

£ DFÆ

, then ¤ABC £ ¤DEF.

THEOREM

A

B C

D

E F

A

B

CD E

Proof

STUDENT HELP

Study TipBefore you use the HL Congruence Theoremin a proof, you need toprove that the trianglesare right triangles.

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• 4.6 Isosceles, Equilateral, and Right Triangles 239

1. Describe the meaning of equilateral and equiangular.

Find the unknown measure(s). Tell what theorems you used.

2. 3. 4.

Determine whether you are given enough information to prove that thetriangles are congruent. Explain your answer.

5. 6. 7.

USING ALGEBRA Solve for x and y.

8. 9. 10.

LOGICAL REASONING Decide whether enough information is given toprove that the triangles are congruent. Explain your answer.

11. 12. 13.

14. 15. 16.

B

A

CE

F

DK

J L M

PT

R

S

q

G

HE

FB

C

D

A

T

U W

V

y �

40�

x �

46�

y �

x �

xyxy

PRACTICE AND APPLICATIONS

B

A C

E

D F

S

R T

W

U VM P

N q

H

G J

?

F

D

?

5 cmEB

A C

?

?50�

GUIDED PRACTICEVocabulary Check ✓

Concept Check ✓

Skill Check ✓

STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 26–28Example 2: Exs. 8–10,

17–25Example 3: Exs. 31, 33,

34, 39

63�

y �

x �

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• 240 Chapter 4 Congruent Triangles

USING ALGEBRA Find the value of x.

17. 18. 19.

USING ALGEBRA Find the values of x and y.

20. 21. 22.

23. 24. 25.

PROOF In Exercises 26–28, use the diagrams that accompany thetheorems on pages 236 and 237.

26. The Converse of the Base Angles Theorem on page 236 states, “If two anglesof a triangle are congruent, then the sides opposite them are congruent.”Write a proof of this theorem.

27. The Corollary to Theorem 4.6 on page 237 states, “If a triangle is equilateral,then it is equiangular.” Write a proof of this corollary.

28. The Corollary to Theorem 4.7 on page 237 states, “If a triangle is equiangular,then it is equilateral.” Write a proof of this corollary.

ARCHITECTURE The diagram represents part of the exterior of the building in thephotograph. In the diagram, ¤ABD and ¤CBDare congruent equilateral triangles.

29. Explain why ¤ABC is isosceles.

30. Explain why ™BAE £ ™BCE.

31. PROOF Prove that ¤ABEand ¤CBE are congruent righttriangles.

32. Find the measure of ™BAE.

60�

x �

y �40�

x �

y �

x �

y �

140�x �

y �

75�

x �

x �

y �y � x �

xyxy

56 ft

8x ft

12 in.

2x in.(x + 13) ft 24 ft

xyxy

A

C

B DE

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• 4.6 Isosceles, Equilateral, and Right Triangles 241

PROOF Write a two-column proof or a paragraph proof.

33. GIVEN � D is the midpoint of CEÆ

, 34. GIVEN � VWÆ

∞ ZYÆ

,™BCD and ™FED are UV

Æ£ XW

Æ, UZÆ

£ XYÆ

,right angles, and BD

Æ£ FD

Æ. VW

Æfi VZ

Æ, VWÆ

fi WYÆ

PROVE � ¤BCD £ ¤FED PROVE � ™U £ ™X

COLOR WHEEL Artists use a color wheel to show relationshipsbetween colors. The 12 triangles in the diagram are isosceles triangles with congruent vertex angles.

35. Complementary colors lie directlyopposite each other on the colorwheel. Explain how you know thatthe yellow triangle is congruent tothe purple triangle.

36. The measure of the vertex angle ofthe yellow triangle is 30°. Find themeasures of the base angles.

37. Trace the color wheel. Then form a triangle whose vertices are themidpoints of the bases of the red,yellow, and blue triangles. (Thesecolors are the primary colors.) What type of triangle is this?

38. Form other triangles that are congruent to the triangle in Exercise 37. The colors of the vertices are called triads. What are the possible triads?

PHYSICS Use the information below. When a light ray from an object meets a mirror, it is reflected back to your eye. For example, in the diagram, a light ray from point C is reflected at point D and travels back to point A. The law of reflection states that the angle of incidence ™CDB is equal to the angle of reflection ™ADB.

39. GIVEN � ™CDB £ ™ADBDBÆ

fi ACÆ

PROVE � ¤ABD £ ¤CBD

40. Verify that ¤ACD is isosceles.

41. Does moving away from the mirror have any effect on the amount of his or herreflection the person sees?

U Z Y X

V WB

C D E

F

For a person to see his or hercomplete reflection, the mirrormust be at least one half theperson’s height.

yellow yellow-orange

orange

red-orange

red

red-purplepurple

blue-purple

blue

blue-green

green

yellow-green

A

B

C

D

ISAAC NEWTONThe English scientist

Isaac Newton (1642–1727)observed that light is madeup of a spectrum of colors.Newton was the first personto arrange the colors of thespectrum in a “color wheel.”

INTE

RNET

RE

AL LIFE

RE

AL LIFE

FOCUS ONPEOPLE

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• 242 Chapter 4 Congruent Triangles

QUANTITATIVE COMPARISON In Exercises 42 and 43, refer to the figuresbelow. Choose the statement that is true about the given values.

¡A The value in column A is greater.¡B The value in column B is greater.¡C The two values are equal.¡D The relationship cannot be determined

from the given information.

42.

43.

44. LOGICAL REASONING A regular hexagon has six congruent sides and six congruent interior angles.It can be divided into six equilateral triangles.Explain how the series of diagrams below suggests aproof that when a triangle is formed by connectingevery other vertex of a regular hexagon, the result isan equilateral triangle.

CONGRUENCE Use the Distance Formula to decide whether ABÆ

£ ACÆ

. (Review 1.3 for 4.7)

45. A(0, º4) 46. A(0, 0) 47. A(1, º1)B(5, 8) B(º6, º10) B(º8, 7)C(º12, 1) C(6, 10) C(8, 7)

FINDING THE MIDPOINT Find the coordinates of the midpoint of a segmentwith the given endpoints. (Review 1.5 for 4.7)

48. C(4, 9), D(10, 7) 49. G(0, 11), H(8, º3) 50. L(1, 7), M(º5, º5)

51. C(º2, 3), D(5, 6) 52. G(0, º13), H(2, º1) 53. L(º3, º5), M(0, º20)

WRITING EQUATIONS Line j is perpendicular to the line with the givenequation and line j passes through point P. Write an equation of line j.(Review 3.7)

54. y = º3x º 4; P(1, 1) 55. y = x º 7; P(0, 0)

56. y = º�190�x + 3; P(5, º12) 57. y = �23�x + 4; P(º3, 4)

MIXED REVIEW

TestPreparation

★★ Challenge

EXTRA CHALLENGE

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Regular hexagon

Column A Column B

™D ™EFD

™B ™EFD

B

A CE

D F G120�

60�

60�60�60�

60�

60�

120�

120� 120�

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