If two sides in a triangle are congruent, then the angles opposite them are congruent.

This theorem will be proven using congruent triangles.
$△ABC$ has two congruent sides.

Draw the angle bisector that bisects $\angle C$ and intersects $\overline{AB}$ at point $P.$

$△ACP$ and $△BCP$ share the following features.

• $\angle ACP\cong \angle PCB,$ since the angle bisector divides the angle in two equal parts.
• $\overline{AC}\cong \overline{BC}$ since they are the legs in an isosceles triangle.
• Both contain the side $\overline{CP}.$

For both triangles, two sides and the included angle are congruent. Thus, $$△ACP\cong △BCP$$ according to the Side-Angle-Side Congruence Theorem. Because $\angle PAC$ and $\angle PBC$ are corresponding angles in congruent triangles, they are congruent. Therefore, if two sides in a triangle are congruent, the angles opposite them are congruent.
This can be summarized in a two-column proof.