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Isosceles Triangle Theorem

If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.

$\angle P\cong \angle Q$

Proof:

Let $S$ be the midpoint of $\overline{PQ}$ .

Join $R$ and $S$ .

Since $S$ is the midpoint of  $\overline{PQ}$ , $\overline{PS}\cong \overline{QS}$ .

By Reflexive Property ,

$\overline{RS}\cong \overline{RS}$

It is given that $\overline{PR}\cong \overline{RQ}$

Therefore, by SSS ,

$\Delta PRS\cong \Delta QRS$

Since corresponding parts of congruent triangles are congruent,

$\angle P\cong \angle Q$

The converse of the Isosceles Triangle Theorem is also true.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If $\angle A\cong \angle B$ , then $\overline{AC}\cong \overline{BC}$ .