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

1 Answer

See explanation.

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.