Converse Isosceles Triangle Theorem

Consider a triangle $ABC$ with two congruent angles.

Let $P$ be the point of intersection of $\overline{BC}$ and the angle bisector of $\angle A.$ Since $\overline{AP}$ is the angle bisector of $\angle A,$ then $\angle BAP\cong \angle CAP.$

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