Geom 3eTE.0405.X 228-233

4-5

1. Plan

Objectives

1 To use and apply properties of isosceles triangles

Examples

1 Using the Isosceles Triangle Theorems

2 Using Algebra 3 Real-World Connection

Math Background

Understanding the vocabulary, including legs, vertex angle, and base angles, is necessary for solving many exercises in this section. Help students remember the meanings of terms by having them describe everyday objects with the same words. Mention that isosceles derives from the Greek iso (same) and skelos (leg). Trapezoids, which will be studied in Chapter 4, can also be isosceles, and have two bases.

More Math Background: p. 196D

Lesson Planning and Resources

See p. 196E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to: Using Exterior Angles of Triangles Lesson 3-4: Example 3 Extra Skills, Word Problems, Proof

Practice, Ch. 3

4-5

Isosceles and Equilateral

Triangles

What You'll Learn

? To use and apply properties

of isosceles triangles

. . . And Why

To find the angles of a garden path, as in Example 3

Check Skills You'll Need

1. Name the angle opposite AB. lC 2. Name the angle opposite BC. lA 3. Name the side opposite &A. BC 4. Name the side opposite &C. BA x2 5. Algebra Find the value of x. 105

GO for Help Lesson 3-4

B x

75 A

30 C

New Vocabulary

? legs of an isosceles triangle ? base of an isosceles triangle ? vertex angle of an isosceles triangle ? base angles of an isosceles triangle ? corollary

1 The Isosceles Triangle Theorems

Vocabulary Tip

Isosceles is derived from the Greek isos for equal and skelos for leg.

Isosceles triangles are common in the real world. You can find them in structures such as bridges and buildings. The congruent sides of an isosceles triangle are its legs. The third side is the base. The two congruent sides form the vertex angle. The other two angles are the base angles.

Vertex Angle

Legs Base Base Angles

An isosceles triangle has a certain type of symmetry about a line through its vertex angle. The theorems below suggest this symmetry, which you will study in greater detail in Lesson 9-4.

Key Concepts

Theorem 4-3

Isosceles Triangle Theorem

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

&A > &B

Theorem 4-4

Converse of Isosceles Triangle Theorem

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

AC > BC

Theorem 4-5

The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

CD ' AB and CD bisects AB.

C

A

B

C

A

B

C

A DB

228 Chapter 4 Congruent Triangles

228

Special Needs L1 Have students collect other landscape designs from gardening and landscaping books and magazines. Have them select one design and describe it using geometric terms.

learning style: tactile

Below Level L2 Discuss whether the equilateral-equiangular relationship holds for polygons with more than 3 sides. Ask students to support their reasoning with examples.

learning style: verbal

In the following proof of the Isosceles Triangle Theorem, you use a special segment, the bisector of the vertex angle. You will prove Theorems 4-4 and 4-5 in the Exercises.

Proof Proof of the Isosceles Triangle Theorem

Begin with isosceles #XYZ with XY > XZ.

X

Draw XB, the bisector of the vertex angle &YXZ.

12

Given: XY > XZ, XB bisects &YXZ.

Prove: &Y > &Z

Proof: You are given that XY > XZ. By the definition

Y

B

Z

of angle bisector, &1 > &2. By the Reflexive Property

of Congruence, XB > XB. Therefore, by the SAS

Postulate, #XYB > #XZB, and &Y > &Z by CPCTC.

Real-World Connection

This A-shaped roof has congruent legs and congruent base angles.

1 EXAMPLE Using the Isosceles Triangle Theorems

Explain why each statement is true.

a. &WVS > &S

WV > WS so &WVS > &S by the Isosceles Triangle Theorem.

R b. TR > TS

&R > &WVS and &WVS > &S, so &R > &S by the Transitive Property of >. TR > TS by the Converse of the Isosceles Triangle Theorem.

Quick Check 1 Can you deduce that #RUV is isosceles? Explain.

No, point U could be anywhere between R and T.

T U

W

V

S

E

D

C

B 1 A

B 2 A

3 A

B

4 A

B

5 A

B

E D C

D E C

D E C

E D C

E D C

B

Test-Taking Tip

Remember that the acute angles of a right triangle are complementary.

2 EXAMPLE Using Algebra

Multiple Choice Find the value of y.

17

27

54

90

By Theorem 4-5, you know that MO ' LN, so &MON = 90. #MLN is isosceles, so &L > &N and m&N = 63.

m&N + 90 + y = 180 Triangle Angle-Sum Theorem 63 + 90 + y = 180 Substitute for mlN and x. y = 27 Subtract 153 from each side.

The correct answer is B.

Quick Check 2 Find the value of x. 6

M

y

N 63 O L

A corollary is a statement that follows immediately from a theorem. Corollaries to the Isosceles Triangle Theorem and its converse appear on the next page.

Lesson 4-5 Isosceles and Equilateral Triangles 229

Advanced Learners L4 Have students explain why Theorems 4-3, 4-4, and 4-5 do or do not apply to equilateral triangles.

learning style: verbal

English Language Learners ELL Point out that equilateral and equiangular can be used interchangeably for triangles, but not for other polygons. For example, squares and rectangles are both equiangular but only the square is equilateral.

learning style: verbal

2. Teach

Guided Instruction

Connection to Chemistry

Ask: What are different forms of a chemical element with the same atomic number called? isotopes Point out that the prefix iso-, meaning "equal," is a variation of the prefix isos- in isosceles.

Visual Learners

Students may think that the base of an isosceles triangle is always at the bottom. Illustrate by rotating a physical model that the base can be in any orientation, as in Exercise 30.

Alternative Method

Draw identical copies of XYZ side by side. Then label congruent parts, asking the class to justify each step. ? Use a single tick mark to show

XY on the first copy XZ on the second copy. ? Use an arc to show &X on the first copy &X on the second copy. ? Use double tick marks to show XZ on the first copy XY on the second copy. Ask: Which triangles are congruent? By what postulate? kXYZ O kXZY; SAS Which angles are congruent by CPCTC? lY and lZ

1 EXAMPLE Math Tip

Point out that every corollary is a theorem that can be proved.

3 EXAMPLE Teaching Tip

Have students calculate the total measure of the four angles formed by the x-axis and the y-axis at the origin. Discuss the fact that there are 360? about any point. Ask students to suggest other ways to show that there are 360? about any point, such as drawing a straight angle and adding the measures on each side.

229

PowerPoint

Additional Examples

1 Explain why ABC is isosceles.

X

A

B

C

Because X* A) n B* C),

lABC O lXAB. By the angles marked O and the Transitive Prop., lABC O lACB. kABC is isosceles by Converse of the Isosceles Triangle Thm.

2 Use the diagram for Example 2. Suppose that m&L 2 63 and m&L = y. Find the values of x and y. x 90, y 45

3 In the drawing for Example 3, suppose that a segment is drawn between the endpoints of the segments that determine the angle marked x?. Find the angle measures of the triangle that is formed. 120, 30, 30

Resources

? Daily Notetaking Guide 4-5 L3

? Daily Notetaking Guide 4-5--

Adapted Instruction

L1

Closure

An angle exterior to the vertex of an isosceles triangle measures 80. Find the angle measures of the triangle. 100, 40, 40

230

Key Concepts

Vocabulary Tip

Equilateral: Congruent sides Equiangular: Congruent angles

Corollary

Corollary to Theorem 4-3

If a triangle is equilateral, then the triangle is equiangular.

&X > &Y > &Z

Corollary

Corollary to Theorem 4-4

If a triangle is equiangular, then the triangle is equilateral.

XY > YZ > ZX

Y

X

Z

Y

X

Z

3 EXAMPLE Real-World Connection

For: Isosceles Triangle Activity Use: Interactive Textbook, 4-5

Quick Check

Landscaping A landscaper uses rectangles and

equilateral triangles for the path around the

x?

hexagonal garden. Find the value of x.

In a rectangle, an angle measure is 90; in an equilateral triangle, it is 60.

x + 90 + 60 + 90 = 360 x = 120

3 What is the measure of the angle at each outside corner of the path? 150

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

GO

for Help

Example 1 (page 229)

Example 2 (page 229)

Example 3 (page 230)

Complete each statement. Explain why it is true.

1. VT > 9

V

2. UT > 9 > YX

3. VU > 9

U

4. &VYU > 9

T

W

x2 Algebra Find the values of x and y.

1. VX ; Converse of the

Isosc. kThm.

2. UW ; Converse of the

Isosc. kThm.

3. VY ; VT VX (Ex. 1)

and UT YX (Ex. 2), so

Y

VU VY by the Subtr.

Prop. of .

X 4. Answers may vary. Sample: lVUY; ' opposite O sides are O.

5.

6.

7.

x

x

4

x

100

50 y

x 80; y 40

110

y

x 40; y 70

52 y x 38; y 4

8. A square and a regular hexagon are placed so that they have

a common side. Find m&SHA and m&HAS. 150; 15 9. Five fences meet at a point to form angles with measures x, 2x, S H

3x, 4x, and 5x around the point. Find the measure of each angle.

A

24, 48, 72, 96, 120

230 Chapter 4 Congruent Triangles

18. Answers may vary.

Sample: Corollary to

Thm. 4-3: Since

XY O YZ, lX O lZ by Thm. 4-3. YZ O ZX, so lY O lX by Thm. 4-3 also. By the Trans.

Prop., lY O lZ, so lX O lY O lZ. Corollary to Thm. 4-4:

Since lX O lZ, XY O YZ by Thm. 4-4. lY O lX, so YZ O ZX by Thm. 4.4 also. By the Trans

Prop., XY O ZX, so XY O YZ O ZX.

26. No; the k can be positioned in ways such that the base is not on the bottom.

B Apply Your Skills

Find each value.

10. If m&L = 58, then m&LKJ = 7. 64

11.

If JL

=

5, then ML

=

7.

2

1 2

12. If m&JKM = 48, then m&J = 7. 42

13. If m&J = 55, then m&JKM = 7. 35

K J ML

Exercise 14

14. Architecture Seventeen spires, pictured at the left, cover the Cadet Chapel

at the Air Force Academy in Colorado Springs, Colorado. Each spire is an

isosceles triangle with a 40? vertex angle. Find the measure of each base angle. 70

15. Developing Proof Here is another way to prove the

Isosceles Triangle Theorem. Supply the missing parts.

K

Begin with isosceles #HKJ with KH > KJ. Draw a. 9, a bisector of the base HJ. KM

Given: KH > KJ, b. 9 bisects HJ. KM Prove: &H > &J

H

M

J

Statements

Reasons

1. KM bisects HJ. 2. HM > JM 3. KH > KJ 4. KM > KM 5. #KHM > #KJM 6. &H > &J

c. 9 By construction d. 9 Def. of segment bisector 3. Given e. 9 Reflexive Prop. of O f. 9 SSS g. 9 CPCTC

17a.

30, 30, 120

GO nline

Homework Help

Visit: Web Code: aue-0405

Proof 16. Supply the missing parts in this statement of the Converse of the Isosceles Triangle Theorem. Then write a proof. R Begin with #PRQ with &P > &Q. Draw a. 9, the bisector of &PRQ. RS

Given: &P > &Q, b. 9 bisects &PRQ. RS

Prove: PR > QR See back of book.

P

S

Q

17. Graphic Arts A former logo for

GPS the National Council of Teachers

of Mathematics is shown at the

right. Trace the logo onto paper.

a. Highlight an obtuse isosceles

The triangles in the logo

triangle in the design. Then

have these congruent

find its angle measures. See left.

sides and angles.

b. How many different sizes of angles

can you find in the logo? What are their measures? 5; 30, 60, 90, 120, 150

18. Writing Explain how each corollary on page 230

follows from its theorem. First, write one explanation

x

and then write the second similar to the first. See margin.

19. Multiple Choice The perimeter of the triangle at the

right is 20. Find x. C

3

5

6

7

2x 5

Lesson 4-5 Isosceles and Equilateral Triangles 231

3. Practice

Assignment Guide

1 A B 1-32 C Challenge

33-40

Test Prep Mixed Review

41-44 45-48

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 6, 8, 17, 18, 28.

Connection to Architecture Exercise 14 The world's tallest

cathedral spire, measuring 528 ft, is on the Protestant Cathedral of Ulm in Germany. Construction began in 1377, but the spire was not finished until 1890.

Exercise 19 To avoid careless errors, encourage students to show the solution steps for solving their algebraic equation.

GPS Guided Problem Solving

L3

Enrichment

L4

Reteaching

L2

Adapted Practice

L1

PraNcamte ice

Class

Date

L3

Practice 4-5

Find the values of the variables.

1.

2.

x

110 y

10

x y

4. s r

7. x 2x 6

5. y

125 x

z

8.

c

a

30

b

Isosceles and Equilateral Triangles 3.

t

6. WXYZV is a

X

regular polygon. W

b

Y

Z c

V

a

9.

z 60

Complete each statement. Explain why it is true.

10. AF 9 11. CA 9

12. KI 9 13. EC 9 14. JA 9 15. HB 9

A

K

J G

F

B

I

C

H

L

E

D

Given mlD 25, find the measure of each angle.

16. JAB 17. FAL 18. JKI 19. DLA

Find the values of x and y.

20.

y

21.

x 110

22.

y

x

y

55

x

? Pearson Education, Inc. All rights reserved.

231

4. Assess & Reteach

PowerPoint

Lesson Quiz

Use the diagram for Exercises 1?3.

A

B

C

M

1. If m&BAC = 38, find m&C. 71

2. If m&BAM = m&CAM = 23, find m&BMA. 90

3. If m&B = 3x and m&BAC = 2x - 20, find x. 25

4.

18 y

60? 60?

60? x?

Find the values of x and y.

x 60; y 9

5. ABCDEF is a regular hexagon.

Find m&BAC.

A

B

F

C

E

D

30

Alternative Assessment

Draw the diagram below on the board.

The diagram illustrates the Isosceles Triangle Theorem. Have the class draw similar diagrams to illustrate the Converse of the Isosceles Triangle Theorem and the two corollaries from this lesson.

29. AC O CB and lACD O lDCB are given. CD O CD by the Refl. Prop. of

232 O, so kACD O kBCD by

x2 Algebra Find the values of x and y.

20.

21.

D

22.

23. Two sides of a k are O if and only if the ' opp. those sides are O.

60 x y x 60; y 30

E

x

C

y

A

B

ABCDE is a regular pentagon.

y 30 x

x 30; y 120

x 36; y 36 23. Write the Isosceles Triangle Theorem and its converse as a biconditional.

24. Critical Thinking An exterior angle of an isosceles triangle has measure 100.

Find two possible sets of measures for the angles of the triangle. 80, 80, 20; 80, 50, 50 25. a. Communications In the diagram isosc. >

at the right, what type of triangles

are formed by the cables of the 1000

same height and the ground?

ft

Radio Tower 1009 ft tall

b. What are the two different base 800

lengths of the triangles? 900 ft; 1100 ft

c. How is the tower related to

600

each of the triangles? The tower

is the ' bis. of the base of each k.

26. Critical Thinking Curtis defines

400

Cables

the base of an isosceles triangle as

Real-World Connection

Careers Radio broadcasters must respond to opinions given by "call-in" listeners.

its "bottom side." Is his definition a 200 good one? Explain. See margin.

27. Reasoning What are the measures

0

450 ft

550 ft

of the base angles of an isosceles

Tower cables extend to both widths.

right triangle? Explain. 45; they are and have sum 90.

28. lA O lD by the Isos. k Thm.

Proof 28. Given: AE > DE, AB > DC

E

kABE O kDCE

Prove: #ABE > #DCE

by SAS.

Proof 29. Prove Theorem 4-5. Use the diagram

next to it on page 228. See margin. A B

CD

x2 Algebra Find the values of m and n.

C Challenge

30.

126

31.

n

m

32.

n

50

m 36; n 27

m

n m

m 60; n 30

m 20; n 45

Coordinate Geometry For each pair of points, there are six points that could be

the third vertex of an isosceles right triangle. Find the coordinates of each point.

33. (0, 0), (4, 4), (?4, 0), (0, ?4), (8, 4), (4, 8)

34. (5, 0); (0, 5); (?5, 5); (5, ?5); (0, 10); (10, 0)

35. (5, 3); (2, 6); (2, 9); (8, 3); (?1, 6); (5, 0))

33. (4, 0) and (0, 4)

34. (0, 0) and (5, 5)

35. (2, 3) and (5, 6)

x2 36. Algebra A triangle has angle measures x + 15, 3x - 35, and 4x.

a. Find the value of x. 25

b. Find the measure of each angle.

c. What type of triangle is it? Why?

40; 40; 100

Obtuse isosc. > : 2 of the ' are O and one l is obtuse.

37. State the converse of Theorem 4-5. If the converse is true, write a paragraph

proof. If the converse is false, give a counterexample. See margin.

232 Chapter 4 Congruent Triangles

SAS. So AD O DB by CPCTC, and CD bisects

AB. Also lADC O lBDC by CPCTC, mlADC ? mlBDC 180 by l Add. Post., so mlADC mlBDC

90 by the Subst. Prop. So CD is the # bis. of AB.

37. The # bis. of the base of an isosc. k is the bis. of the vertex l; given isosc. kABC with # bis. CD,

lADC O lCDB and AD O DB by def. of # bis.

Since CD O CD by Refl. Prop., kACD O kBCD by SAS. So lACD O lBCD by CPCTC, and CD bisects lACB.

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