Warmup

Which is the correct definition of isosceles?a.) a triangle with at least 2 congruent

sides

b.) a triangle with exactly 2 congruent sides

4.5 Isosceles TrianglesLEQ: How can we use properties of isosceles triangles to construct proofs?

Word BankLeg

Leg

Vertex

Base

Base Angles

Given ABC is isosceles. Prove m<A=m<B.Proof:

1.) Definition of Isosceles

2.)Defn. of bisect

1.) AC=BC

*Construct CD so that it bisects <C*2.)m<ACD=m<BCD3.) CD=CD

3.) reflexive property

4.) ACD BCD 4.) SAS

What can you conclude about <A and <B?5.) <A <B

Statements

Reasons

5.) CPCTC

A B

C

D

Isosceles Triangle Theorem:If 2 sides of a triangle are congruent, then

the base angles are congruent as well.

Converse:If the base angles of a triangle are

congruent, then the 2 opposite legs of the triangle are congruent as well.

What kind of statement is this?

Given BC=AC, answer the following: a.) If m<1=140, m<2=_____, m<3=______ and m<4=______ 

b.) If m<4=65, m<2=_____, m<3=______and m<1=______.  

A

12

3 4

B

C

Corollaries (small theorem that follow a big theorem)•An equilateral triangle is also equiangular.

•An equiangular triangle has all 60 angles.

Perpendicular bisector:

•Draw an isosceles triangle•Bisect the vertex•What do you notice about the intersection of the bisector and the base?

Corollary 3

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

(divides the base in half and is perpendicular to the base)

Try examples 1-3.

Ex. 1

An exterior angle of an isosceles triangle has the measure 11o degrees. What are the 2 possible sets of measures for the angles of the triangle?

110

110

7070

4011070

70+x+x=180 x=55

40,70,70 and 70,55,55

x x

Ex. 2: Find the values of m and n:

45

m

n

60

m=

m+n=