6–4 Isosceles Triangles - Weebly

6¨C4

What You¡¯ll Learn

You¡¯ll learn to identify

and use properties of

isosceles triangles.

Isosceles Triangles

Recall from Lesson 5¨C1 that an isosceles triangle has at least two

congruent sides. The congruent sides are called legs. The side opposite

the vertex angle is called the base. In an isosceles triangle, there are two

base angles, the vertices where the base intersects the congruent sides.

Why It¡¯s Important

vertex angle

Advertising Isosceles

triangles can be found

in business logos.

See Exercise 17.

leg

base angle

leg

base

base angle

You can use a TI¨C83/84 Plus graphing calculator to draw an isosceles

triangle and study its properties.

Graphing

Calculator Tutorial

See pp. 782¨C785.

Step 1

Draw a circle using the Circle tool on the F2 menu. Label the

center of the circle A.

Step 2

Use the Triangle tool on the F2 menu to draw a triangle that

has point A as one vertex and its other two vertices on the

circle. Label these vertices B and C.

Step 3

Use the Hide/Show tool on menu F5 to hide the circle. Press

the CLEAR key to quit the F7 menu. The figure that remains

on the screen is isosceles triangle ABC.

Try These

1. Tell how you can use the measurement tools on F5 to check that


ABC is isosceles. Use your method to be sure it works.

2. Use the Angle tool on F5 to measure B and C. What is the

relationship between B and C?

3. Use the Angle Bisector tool

on F3 to bisect A. Use

the Intersection Point tool on

F2 to mark the point where

the angle bisector intersects

B


C


. Label the point of

intersection D. What is point

D in relation to side B


C


?

246 Chapter 6 More About Triangles

4. Use the Angle tool on F5 to find the measures of ADB and

ADC.

5. Use the Distance & Length tool on F5 to measure


BD


and C


D


.

What is the relationship between the lengths of B


D


and C


D


?

BC

6. Is A


D


part of the perpendicular bisector of



? Explain.

The results you found in the activity are expressed in the following

theorems.

Theorem

Words

6¨C2

If two sides of a triangle are

congruent, then the angles

opposite those sides are

congruent.

Isosceles

Triangle

Theorem

6¨C3

Example

1

Models

Symbols

If A
B
 A
C
, then

C  B.

A

B

The median from the vertex

angle of an isosceles triangle

lies on the perpendicular

bisector of the base and the

angle bisector of the vertex

angle.

C

If A
B
 A
C
and

B
D
 C
D
, then

A
D


 B
C
and

BAD  CAD.

A

B

C

D

Find the value of each variable in isosceles

triangle DEF if


EG


is an angle bisector.

First, find the value of x.

Since
DEF is an isosceles triangle,

D  F. So, x
49.

E

D

x?

y?

G

Now find the value of y.

DF

By Theorem 6¨C3,


EG






. So, y
90.

49?

F

Your Turn

For each triangle, find the values of the variables.

N

a.

M

65?y

P

extra_examples

b. R

x?

50?

x?

O

?

y?

S

T

70?

Lesson 6¨C4 Isosceles Triangles 247

Suppose you draw two congruent acute angles on two pieces of patty

paper and then rotate one of the angles so that one pair of rays overlaps

and the other pair intersects.

X

Z

Y

Y

Z

What kind of triangle is formed?

What is true about angles Y and Z?

What is true about the sides opposite angles Y and Z?

Is the converse of Theorem 6¨C2 true?

Theorem 6¨C4

Converse of

Isosceles

Triangle

Theorem

Words: If two angles of a triangle are congruent, then the sides

opposite those angles are congruent.

Model:

B

Example

Algebra Link

2

Symbols: If B  C, then

A
C
 A
B
.

A

C

In
ABC, A  B and mA
48.

Find mC, AC, and BC.

A

4x

48?

First, find mC. You know that mA
48.

Since A  B, mB
48.

Algebra Review

Solving Multi-Step

Equations, p. 723

mA  mB  mC
180

48  48  mC
180

96  mC
180

96  96  mC
180  96

mC
84

C

6x  5

B

Angle Sum Theorem

Replace mA and mB with 48.

Add.

Subtract 96 from each side.

Simplify.

AC

Next, find AC. Since A  B, Theorem 6¨C4 states that


BC






.

BC
AC

6x  5
4x

6x  5  6x
4x  6x

5
2x

5



2

2x






2

2.5
x

Definition of Congruent Segments

Replace AC with 4x and BC with 6x  5.

Subtract 6x from each side.

Simplify.

Divide each side by 2.

Simplify.

By replacing x with 2.5, you find that AC
4(2.5) or 10 and

BC
6(2.5)  5 or 10.

248 Chapter 6 More About Triangles

Equiangular:

Lesson 5¨C2;

Equilateral:

Lesson 5¨C1

In Chapter 5, the terms equiangular and equilateral were defined.

Using Theorem 6¨C4, we can now establish that equiangular triangles

are equilateral.


ABC is equiangular.

Since mA
mB
mC,

Theorem 6¨C4 implies that

BC
AC
AB.

A

C

B

Theorem 6¨C5

A triangle is equilateral if and only if it is equiangular.

Check for Understanding

Communicating

Mathematics

1. Draw an isosceles triangle. Label it
DEF with base


DF


. Then state

four facts about the triangle.

2. Explain why equilateral triangles are also equiangular and why

equiangular triangles are also equilateral.

Guided Practice

For each triangle, find the values of the variables.

Example 1

3.

D

x?

y?

75?

Example 2

E

V

4.

F

x

Exercises

? ? ? ? ?

?

?

?

W

6

5. Algebra In
MNP, M
P and

mM
37. Find mP, MQ, and PQ.

N

37?

3x  2

M

Practice

y?

45?

U

?

?

?

?

Q

?

2x  3

P

?

?

?

?

?

For each triangle, find the values of the variables.

6.

S

7.

B

60?

x?

Homework Help

For

Exercises

6-14, 17, 18

See

Examples

1

15, 16

2

A

5

R

y

46?

N

10.

I

x?

60?

x?

80?

Extra Practice

8

59?

T

O

x? G

F

11.

U

47?

15

y

See page 737.

y ?x ?

J

68?

K

M

60?

y?

P

H

E 59

?

52? C

y?

9.

y

8.

x

V ?

9

47?

W

Lesson 6¨C4 Isosceles Triangles 249

12. In
DEF,


DE

FE






. If mD
35,

what is the value of x?

13. Find the value of y if E

DF


N






.

MN

14. In
DMN,


DM






. Find mDMN.

E

y?

M

35?

D

x?

N

F

Exercises 12¨C14

Applications and

Problem Solving

AB

AC

15. Algebra In
ABC,







.

If mB
5x  7 and

mC
4x  2, find mB

and mC.

(4x  2)

16. Algebra In
RST, S  T,

mS
70, RT
3x  1, and

RS
7x  17. Find mT, RT,

and RS.

S

C

A

70?

7x  17

(5x  7)

B

T

R

3x  1

17. Advertising A business logo is shown.

a. What kind of triangle does the logo

contain?

b. If the measure of angle 1 is 110, what

are the measures of the two base

angles of that triangle?

1

18. Critical Thinking Find the measures of the angles of an isosceles

triangle such that, when an angle bisector is drawn, two more isosceles

triangles are formed.

Mixed Review

19. In
JKM,
JQ


bisects KJM. If

mKJM = 132, what is m1?

(Lesson 6¨C3)

J

1

K

TZ

20. In
RST,


SZ






. Name a

perpendicular bisector.

(Lesson 6¨C2)

R

Y

21. Graph and label point H at

(4, 3) on a coordinate plane.

(Lesson 2¨C4)

Standardized

Test Practice

22. Short Response Marcus used 37 feet

of fencing to enclose his triangular

garden. What is the length of each

side of the garden? (Lesson 1¨C6)

M

Q

S

X

T

Z

Exercise 20

r7

T

r2

L

r5

P

23. Short Response Write a sequence in which each term is 7 less than

the previous term. (Lesson 1¨C1)

250 Chapter 6 More About Triangles

self_check_quiz

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