The isosceles triangle theorem states that if two sides of a triangle are

In an isosceles triangle, the base angles are congruent. In other words, iso-lateral (isosceles) triangles are iso-angular. This is sometimes called the isosceles triangle theorem, or pons asinorum.

If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.

∠ P ≅ ∠ Q

Proof:

Let S be the midpoint of P Q ¯ .

Join R and S .

Since S is the midpoint of  P Q ¯ , P S ¯ ≅ Q S ¯ .

By Reflexive Property ,

R S ¯ ≅ R S ¯

It is given that P R ¯ ≅ R Q ¯

Therefore, by SSS ,

Δ P R S ≅ Δ Q R S

Since corresponding parts of congruent triangles are congruent,

∠ P ≅ ∠ Q

The converse of the Isosceles Triangle Theorem is also true.

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

If ∠ A ≅ ∠ B , then A C ¯ ≅ B C ¯ .

In an isosceles triangle, two angles, and therefore sides (Base Angle Theorem), are congruent. This does not mean that all isosceles triangles are also right triangles - there is only one (45, 45, 90 triangle).