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4-8

Isosceles and Equilateral Triangles

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

- Warm Up
- 1. Find each angle measure.
- True or False. If false explain.
- 2. Every equilateral triangle is isosceles.
- 3. Every isosceles triangle is equilateral.

60 60 60

True

False an isosceles triangle can have only two

congruent sides.

Objectives

Prove theorems about isosceles and equilateral

triangles. Apply properties of isosceles and

equilateral triangles.

Vocabulary

legs of an isosceles triangle vertex

angle base base angles

Recall that an isosceles triangle has at least

two congruent sides. The congruent sides are

called the legs. The vertex angle is the angle

formed by the legs. The side opposite the vertex

angle is called the base, and the base angles are

the two angles that have the base as a side.

?3 is the vertex angle. ?1 and ?2 are the base

angles.

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Example 1 Astronomy Application

The m?YZX 180 140, so m?YZX 40.

Since ?YZX ? ?X, ?XYZ is isosceles by the

Converse of the Isosceles Triangle Theorem.

Thus YZ YX 20 ft.

Check It Out! Example 1

If the distance from Earth to a star in September

is 4.2 ? 1013 km, what is the distance from Earth

to the star in March? Explain.

4.2 ? 1013 since there are 6 months between

September and March, the angle measures will be

approximately the same between Earth and the

star. By the Converse of the Isosceles Triangle

Theorem, the triangles created are isosceles, and

the distance is the same.

Example 2A Finding the Measure of an Angle

Find m?F.

m?F m?D x

Isosc. ? Thm.

m?F m?D m?A 180

? Sum Thm.

Substitute the given values.

x x 22 180

Simplify and subtract 22 from both sides.

2x 158

Divide both sides by 2.

x 79?

Thus m?F 79

Example 2B Finding the Measure of an Angle

Find m?G.

m?J m?G

Isosc. ? Thm.

Substitute the given values.

(x 44)? 3x?

Simplify x from both sides.

44 2x

Divide both sides by 2.

x 22?

Thus m?G 22 44 66.

Check It Out! Example 2A

Find m?H.

m?H m?G x

Isosc. ? Thm.

m?H m?G m?F 180

? Sum Thm.

Substitute the given values.

x x 48 180

Simplify and subtract 48 from both sides.

2x 132

Divide both sides by 2.

x 66?

Thus m?H 66

Check It Out! Example 2B

Find m?N.

m?P m?N

Isosc. ? Thm.

Substitute the given values.

(8y 16)? 6y?

Subtract 6y and add 16 to both sides.

2y 16

Divide both sides by 2.

y 8?

Thus m?N 6(8) 48.

The following corollary and its converse show the

connection between equilateral triangles and

equiangular triangles.

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Example 3A Using Properties of Equilateral

Triangles

Find the value of x.

?LKM is equilateral.

Equilateral ? ? equiangular ?

The measure of each ? of an equiangular ? is 60.

(2x 32)? 60?

Subtract 32 both sides.

2x 28

Divide both sides by 2.

x 14

Example 3B Using Properties of Equilateral

Triangles

Find the value of y.

?NPO is equiangular.

Equiangular ? ? equilateral ?

Definition of equilateral ?.

5y 6 4y 12

Subtract 4y and add 6 to both sides.

y 18

Check It Out! Example 3

Find the value of JL.

?JKL is equiangular.

Equiangular ? ? equilateral ?

Definition of equilateral ?.

4t 8 2t 1

Subtract 4y and add 6 to both sides.

2t 9

t 4.5

Divide both sides by 2.

Thus JL 2(4.5) 1 10.

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Example 4 Using Coordinate Proof

Prove that the segment joining the midpoints of

two sides of an isosceles triangle is half the

base.

Given In isosceles ?ABC, X is the mdpt. of AB,

and Y is the mdpt. of AC.

Prove XY AC.

Example 4 Continued

Proof Draw a diagram and place the coordinates

as shown.

Check It Out! Example 4

What if...? The coordinates of isosceles ?ABC are

A(0, 2b), B(-2a, 0), and C(2a, 0). X is the

midpoint of AB, and Y is the midpoint of AC.

Prove ?XYZ is isosceles.

Proof Draw a diagram and place the coordinates

as shown.

Check It Out! Example 4 Continued

Lesson Quiz Part I

Find each angle measure. 1. m?R 2. m?P Find

each value. 3. x 4. y 5. x

28

124

6

20

26

Lesson Quiz Part II

6. The vertex angle of an isosceles triangle

measures (a 15), and one of the base angles

measures 7a. Find a and each angle measure.

a 11 26 77 77

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