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).