Recall from Lesson 5–1 that an isosceles triangle has at least twocongruent sides. The congruent sides are called legs. The side opposite the vertex angle is called the base. In an isosceles triangle, there are twobase angles, the vertices where the base intersects the congruent sides.

You can use a TI–83/84 Plus graphing calculator to draw an isoscelestriangle and study its properties.

Step 1 Draw a circle using the Circle tool on the menu. Label thecenter of the circle A.

Step 2 Use the Triangle tool on the menu to draw a triangle thathas point A as one vertex and its other two vertices on thecircle. Label these vertices B and C.

Step 3 Use the Hide/Show tool on menu to hide the circle. Press the key to quit the menu. The figure that remainson the screen is isosceles triangle ABC.

Try These1. Tell how you can use the measurement tools on to check that

�ABC is isosceles. Use your method to be sure it works.2. Use the Angle tool on to measure �B and �C. What is the

relationship between �B and �C?

3. Use the Angle Bisector tool on to bisect �A. Use the Intersection Point tool on

to mark the point wherethe angle bisector intersectsB�C�. Label the point ofintersection D. What is point D in relation to side B�C�?

F2

F3

F5

F5

F7CLEAR

F5

F2

F2

legleg

basebase angle base angle

vertex angle

246 Chapter 6 More About Triangles

What You’ll LearnYou’ll learn to identifyand use properties ofisosceles triangles.

Why It’s ImportantAdvertising Isoscelestriangles can be foundin business logos. See Exercise 17.

Isosceles Triangles6–46–4

See pp. 782–785.

GraphingCalculator Tutorial

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

5. Use the Distance & Length tool on to measure B�D� and C�D�. What is the relationship between the lengths of B�D� and C�D�?

6. Is A�D� part of the perpendicular bisector of B�C�? Explain.

The results you found in the activity are expressed in the followingtheorems.

Find the value of each variable in isosceles triangle DEF if E�G� is an angle bisector.

First, find the value of x.Since �DEF is an isosceles triangle, �D � �F. So, x � 49.

Now find the value of y.By Theorem 6–3, E�G� � D�F�. So, y � 90.

For each triangle, find the values of the variables.

a. b. R

S

T

70˚

x˚

y˚

O

P

N

M

65˚

50˚

y˚

x˚

GF

E

D x˚y˚

49˚

F5

F5

Lesson 6–4 Isosceles Triangles 247

Words Models Symbols

If two sides of a triangle are If A�B� � A�C�, then congruent, then the angles �C � �B.opposite those sides arecongruent.

The median from the vertex If A�B� � A�C� andangle of an isosceles triangle B�D� � C�D�, thenlies on the perpendicular A�D� � B�C� andbisector of the base and the �BAD � �CAD.angle bisector of the vertex angle.

A

B CD

A

B C

Theorem

6–2

IsoscelesTriangleTheorem

6–3

Example

Your Turn

1

www.geomconcepts.com/extra_examples

Suppose you draw two congruent acute angles on two pieces of pattypaper and then rotate one of the angles so that one pair of rays overlapsand the other pair intersects.

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 6–2 true?

In �ABC, �A � �B and m�A � 48. Find m�C, AC, and BC.

First, find m�C. You know that m�A � 48. Since �A � �B, m�B � 48.

m�A � m�B � m�C � 180 Angle Sum Theorem48 � 48 � m�C � 180 Replace m�A and m�B with 48.

96 � m�C � 180 Add.96 � 96 � m�C � 180 � 96 Subtract 96 from each side.

m�C � 84 Simplify.

Next, find AC. Since �A � �B, Theorem 6–4 states that B�C� � A�C�.

BC � AC Definition of Congruent Segments6x � 5 � 4x Replace AC with 4x and BC with 6x � 5.

6x � 5 � 6x � 4x � 6x Subtract 6x from each side.�5 � �2x Simplify.

��

�

52� � �

�

�

22x

� Divide each side by �2.

2.5 � x Simplify.

By replacing x with 2.5, you find that AC � 4(2.5) or 10 and BC � 6(2.5) � 5 or 10.

Y YZ

Z

X

248 Chapter 6 More About Triangles

ExampleAlgebra Link

2

Theorem 6–4Converse ofIsoscelesTriangleTheorem

Words: If two angles of a triangle are congruent, then the sidesopposite those angles are congruent.

Model: Symbols: If �B � �C, thenA�C� � A�B�.

A

B C

Solving Multi-StepEquations, p. 723

Algebra Review

A

B

C48˚

6x � 5

4x

Check for Understanding

• • • • • • • • • • • • • • • • • •Exercises

CommunicatingMathematics

Guided Practice

Practice

In Chapter 5, the terms equiangular and equilateral were defined. Using Theorem 6–4, we can now establish that equiangular triangles are equilateral.

�ABC is equiangular.Since m�A � m�B � m�C, Theorem 6–4 implies that BC � AC � AB.

1. Draw an isosceles triangle. Label it �DEF with base D�F�. Then statefour facts about the triangle.

2. Explain why equilateral triangles are also equiangular and whyequiangular triangles are also equilateral.

For each triangle, find the values of the variables.

3. 4.

5. Algebra In �MNP, �M � �P and m�M � 37. Find m�P, MQ, and PQ.

For each triangle, find the values of the variables.

6. 7. 8.

9. 10. 11. U

47˚

47˚x˚9

15

WV

y

N

M P

O

60˚

80˚

y˚

x˚I

J K68˚y˚x˚

F

EH

G

59˚

59˚46˚

8

x˚

y

R

S

T

60˚

60˚

5

x˚

y

B

A

C52˚y˚

x˚

N

M P37˚3x � 2 2x � 3

Q

V

U W45˚ y˚

x 6

E

D

F75˚

x˚y˚

A

B C

Lesson 6–4 Isosceles Triangles 249

Equiangular:Lesson 5–2;Equilateral: Lesson 5–1

Theorem 6–5 A triangle is equilateral if and only if it is equiangular.

6-14, 17, 18 1

15, 16 2

See page 737.Extra Practice

ForExercises

SeeExamples

Homework Help

Example 1

Example 2

Applications andProblem Solving

Mixed Review

12. In �DEF, D�E� � F�E�. If m�D � 35,what is the value of x?

13. Find the value of y if E�N� � D�F�.14. In �DMN, D�M� � M�N�. Find m�DMN.

15. Algebra In �ABC, A�B� � A�C�. 16. Algebra In �RST, �S � �T,If m�B � 5x � 7 and m�S � 70, RT � 3x � 1, andm�C � 4x � 2, find m�B RS � 7x � 17. Find m�T, RT,and m�C. and RS.

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 baseangles of that triangle?

18. Critical Thinking Find the measures of the angles of an isoscelestriangle such that, when an angle bisector is drawn, two more isoscelestriangles are formed.

19. In �JKM, J�Q� bisects �KJM. If m�KJM = 132, what is m�1?(Lesson 6–3)

20. In �RST, S�Z� � T�Z�. Name a perpendicular bisector. (Lesson 6–2)

21. Graph and label point H at (�4, 3) on a coordinate plane.(Lesson 2–4)

22. Short Response Marcus used 37 feet of fencing to enclose his triangulargarden. What is the length of eachside of the garden? (Lesson 1–6)

23. Short Response Write a sequence in which each term is 7 less thanthe previous term. (Lesson 1–1)

T L

P

r � 5r � 2

r � 7

R

T

Y

X ZS

Exercise 20

1J

MQ

K

1

S

T R

7x � 17

3x � 1

70˚A

B

C(4x � 2)�

(5x � 7)�

E

FD

M

N35˚ x˚

y˚

Exercises 12–14

250 Chapter 6 More About Triangles

Standardized Test Practice

www.geomconcepts.com/self_check_quiz