Angle Bisector Theorem

Consider an angle and its bisector.

Let $D$ be a point lying on the bisector of the angle. Also, let $DB$ and $DC$ be the distances from $D$ to the sides of the angle. Recall that the distance from a point to a line is perpendicular to the line.

Since $\overrightarrow{AD}$ bisects $\angle BAC,$ by the definition of an angle bisector it can be said that $\angle BAD$ and $\angle CAD$ are congruent angles. Furthermore, $\angle ABD$ and $\angle ACD$ are both right angles. Therefore, they are also congruent angles. By the Reflexive Property of Congruence, $\overline{AD}$ is congruent to itself. Because two angles and a non-included side of $△ADB$ are congruent to two angles and the corresponding non-included side of $△ADC,$ the triangles are congruent by the Angle-Angle-Side Congruence Theorem. Corresponding parts of congruent triangles are congruent. Therefore, $\overline{DB}$ is congruent to $\overline{DC}.$ Congruent segments have equal measures. This means that $D$ is equidistant from the rays of $\angle BAC.$