Isosceles Triangle Theorems
An isosceles triangle is a triangle with two congruent sides.
Theorem:
If two sides of a triangle are congruent, the angles opposite them are congruent.
Theorem: (converse)
If two angles of a triangle are congruent, the sides opposite them are congruent.
More Info:
When the altitude is drawn in an isosceles triangle, two congruent triangles are formed, proven by Hypotenuse-Leg.
(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:
- The altitude to the base of an isosceles triangle bisects the vertex angle.
- The altitude to the base of an isosceles triangle bisects the base.
Examples:
- 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. - 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 
Solution:
