What is one proof of the converse of the Isosceles Triangle Theorem?

See explanation.

The converse of the Isosceles Triangle Theorem states that if two angles ##hat A## and ##hat B## of a triangle ##ABC## are congruent, then the two sides ##BC## and ##AC## opposite to these angles are congruent.

The proof is very quick: if we trace the bisector of ##hat C## that meets the opposite side ##AB## in a point ##P##, we get that the angles ##hat(ACP)## and ##hat(BCP)## are congruent.

We can prove that the triangles ##ACP## and ##BCP## are congruent. In fact, the hypotheses of the AAS criterion are satisfied:

• ##hat A cong hat B## (hypotesis of the theorem)
• ##hat(ACP) cong hat(BCP)## since ##CP## lies on the bisector of ##hat C##
• ##CP## is a shared side between the two triangles

Since the triangles ##ACP## and ##BCP## are congruent, we conclude that ##BC cong AC##.