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

See explanation.

#### Explanation:

The converse of the Isosceles Triangle Theorem states that if two angles $\hat{A}$ and $\hat{B}$ of a triangle $A B C$ are congruent, then the two sides $B C$ and $A C$ opposite to these angles are congruent.

The proof is very quick: if we trace the bisector of $\hat{C}$ that meets the opposite side $A B$ in a point $P$, we get that the angles $\hat{A C P}$ and $\hat{B C P}$ are congruent.

We can prove that the triangles $A C P$ and $B C P$ are congruent. In fact, the hypotheses of the AAS criterion are satisfied:

• $\hat{A} \cong \hat{B}$ (hypotesis of the theorem)
• $\hat{A C P} \cong \hat{B C P}$ since $C P$ lies on the bisector of $\hat{C}$
• $C P$ is a shared side between the two triangles

Since the triangles $A C P$ and $B C P$ are congruent, we conclude that $B C \cong A C$.