An
isosceles triangle is a
triangle
with two congruent sides.



If two sides of a triangle are congruent, the angles
opposite them are congruent. 

If two angles of a
triangle are congruent, the sides opposite them are
congruent. 
When the
altitude is drawn in an isosceles triangle, two congruent triangles
are formed, proven by HypotenuseLeg. 

(The congruent legs of the
isosceles triangle become the congruent hypotenuses and the
altitude becomes a shared leg.) 
These congruent triangles make it possible, by use of CPCTC, to conclude that
the following statements are true regarding an isosceles
triangle: 
1. The altitude to the base of
an isosceles triangle bisects the vertex angle. 



2. The altitude to the base of an isosceles triangle bisects the base.




Examples:
1. 
Find
x.

Solution:
If two angles of a triangle are congruent,
the sides opposite them are congruent.
Set: 6x
 8 = 4x + 2
2x
= 10
x = 5
Note: The side
labeled 2x + 2 is a distracter and is not used in
finding x. 
2. 
Find the
measures of angles 1, 2, 3, 4.

Solution:
If two sides of a triangle are congruent, the
angles opposite them are congruent.
So m<1 =
m<2 and
m<3 = 40 degrees.
180  50 = 130 180
 (40 + 40) = 100
m<1 = 65 degrees
m <4 = 100 degrees
m<2 = 65 degrees 
3. 

