Converse Angle Bisector Theorem

Consider $\angle BAC$ and an interior point $D$ that is equidistant from the two sides of the angle.

By the Reflexive Property of Congruence, $\overline{AD}$ is congruent to itself.

It is given that $△ADB$ and $△ADC$ are both right triangles that share a hypotenuse, $\overline{AD}.$ Also, $\overline{DB}$ is one of the legs of $△ADB$ and $\overline{DC}$ is one of the legs of $△ADC.$

Since the hypotenuse and a leg of $△ADB$ are congruent to the hypotenuse and a leg of $△ADC,$ the triangles are congruent by the Hypotenuse-Leg Congruence Theorem. Corresponding parts of congruent triangles are congruent. Therefore, $\angle BAD$ and $\angle CAD$ are congruent angles. By the definition of an angle bisector, $\overrightarrow{AD}$ bisects $\angle BAC.$