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.

Honors Geometry 1/14/15

Hot Potato!

Prove theorems about isosceles and equilateral triangles.

Apply properties of isosceles and equilateral triangles.

Objectives

legs of an isosceles triangle

vertex angle

base

base angles

Vocabulary

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.

The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”

Reading Math

Example 1: Astronomy Application

The length of YX is 20 feet.

Explain why the length of YZ is the same.

Since YZX X, ∆XYZ is isosceles by the Converse of the Isosceles TriangleTheorem.

The mYZX = 180 – 140, so mYZX = 40°.

Thus YZ = YX = 20 ft.

The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.

Example 3A: Using Properties of Equilateral Triangles

Find the value of x.

∆LKM is equilateral.

(2x + 32) = 60The measure of each of an equiangular ∆ is 60°.

2x = 28 Subtract 32 both sides.

x = 14 Divide both sides by 2.

Equilateral ∆ equiangular ∆

Example 3B: Using Properties of Equilateral Triangles

Find the value of y.

∆NPO is equiangular.

Equiangular ∆ equilateral ∆

5y – 6 = 4y + 12Definition of equilateral ∆.

y = 18Subtract 4y and add 6 to both sides.

Check It Out! Example 3

Find the value of JL.

∆JKL is equiangular.

Equiangular ∆ equilateral ∆

4t – 8 = 2t + 1Definition of equilateral ∆.

2t = 9Subtract 4y and add 6 to both sides.

t = 4.5 Divide both sides by 2.

Thus JL = 2(4.5) + 1 = 10.

ISOSCELES TRIANGLE

vocabulary

legs

vertex angle

base

base angles

C

B A

C

Isosceles Triangle Theorem

If two sides of a triangle are congruent,

then the angles opposite those sides

are congruent.

B A

???????????????????????

Is the converse true?

Converse Isosceles

Triangle Theorem

If two angles of a triangle are

congruent, then the sides

opposite those angles are

congruent.

CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

·Corresponding

·Parts of

·Congruent

·Triangles are

·Congruent

CPCTC

SSS, SAS, ASA, AAS, and HL use corresponding parts

to prove triangles congruent. CPCTC uses congruent

triangles to prove corresponding parts congruent.

Remember!

Example 3: Using CPCTC in a Proof

Prove: MN || OPGiven: NO || MP, N P

5. CPCTC5. NMO POM

6. Conv. Of Alt. Int. s Thm.

4. AAS4. ∆MNO ∆OPM

3. Reflex. Prop. of

2. Alt. Int. s Thm.2. NOM PMO

1. Given

ReasonsStatements

3. MO MO

6. MN || OP

1. N P; NO || MP

Example 3 Continued

Check It Out! Example 3

Prove: KL || MN

Given: J is the midpoint of KM and NL.

Check It Out! Example 3 Continued

5. CPCTC5. LKJ NMJ

6. Conv. Of Alt. Int. s Thm.

4. SAS Steps 2, 34. ∆KJL ∆MJN

3. Vert. s Thm.3. KJL MJN

2. Def. of mdpt.

1. Given

ReasonsStatements

6. KL || MN

1. J is the midpoint of KM and NL.

2. KJ MJ, NJ LJ

Eyes on Isosceles

Triangles

R

O

N

CW

Given:

CO=RW

CR=WO

S

T

RM

C H

Given: CR=SH

m<RCH=m<SHC

E

A

B4

D C

3

2 1

E

A

B4

D C

3

2 1

E

A

B4

D C

3

2 1

E

A

B4

D C

3

2 1

Lesson Quiz: Part I

Find each angle measure.

1. mR

2. mP

Find each value.

3. x 4. y

5. x

124°

28°

20 6

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°

STW review all types of

triangles & triangle congruency

to prepare for Chapter test