Question

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

Answer 1

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`.