Isosceles Triangle Theorem

Consider a triangle $ABC$ with two congruent sides, or an isosceles triangle.

In this triangle, let $P$ be the point of intersection of $\overline{BC}$ and the angle bisector of $\angle A.$ From the diagram, the following facts about $△BAP$ and $△CAP$ can be observed.

Statement	Reason
$\angle BAP\cong \angle CAP$	Definition of an angle bisector
$\overline{BA}\cong \overline{CA}$	Given
$\overline{AP}\cong \overline{AP}$	Reflexive Property of Congruence

Therefore, $△BAP$ and $△CAP$ have two pairs of corresponding congruent sides and one pair of congruent included angles. By the Side-Angle-Side Congruence Theorem, $△BAP$ and $△CAP$ are congruent triangles. Corresponding parts of congruent figures are congruent. Therefore, $\angle B$ and $\angle C$ are congruent. It has been proven that if two sides of a triangle are congruent, then the angles opposite them are congruent.

Consider an isosceles triangle $△ABC.$

A line passing through $A$ and the midpoint of $\overline{BC}$ will be drawn. Let $P$ be the midpoint.

Since $\overline{BP}$ and $\overline{PC}$ are congruent, the distance between $B$ and $P$ is equal to the distance between $C$ and $P.$ Therefore, $B$ is the image of $C$ after a reflection across $\overleftrightarrow{AP}.$ Also, because $A$ lies on $\overleftrightarrow{AP},$ a reflection across $\overleftrightarrow{AP}$ maps $A$ onto itself. The same is true for $P.$

Reflection Across $\overleftrightarrow{AP}$
Preimage	Image
$C$	$B$
$A$	$A$
$P$	$P$

The table shows that the images of the vertices of $△CAP$ are the vertices of $△BAP.$ It can be concluded that $△BAP$ is the image of $△CAP$ after a reflection across $\overleftrightarrow{AP}.$ Since a reflection is a rigid motion, this proves that the triangles are congruent. Corresponding parts of congruent figures are congruent, so $\angle B$ and $\angle C$ are congruent.