Rule

Converse Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.
Isosceles triangle with a movable point that can be used to build triangles with different pairs of congruent angles.
Based on this diagram, the following relation holds true.


∠ B≅ ∠ C ⇒ AB≅ AC

This theorem is the converse theorem to the Isosceles Triangle Theorem. It is also known as the Converse Base Angles Theorem.

Proof

Consider a triangle ABC with two congruent angles.

An isosceles triangle ABC.

Let P be the point of intersection of BC and the angle bisector of ∠ A. Since AP is the angle bisector of ∠ A, then ∠ BAP ≅ ∠ CAP.

An isosceles triangle ABC.

By the Reflexive Property of Congruence, AP in △ ABP is congruent to AP in △ ACP. Because of the Angle-Angle-Side Congruence Theorem, both triangles are congruent. △ ABP ≅ △ ACP Since corresponding parts of congruent triangles are congruent, it follows that AB is congruent to AC. AB ≅ AC It has been proven that if two angles of a triangle are congruent, then the sides opposite them are congruent.

Exercises