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Rule

Angle Bisector Theorem

If a point lies on the bisector of an angle, then the point is equidistant from the two sides of the angle.

Based on the figure, the following conditional statement holds true.

Proof

Consider an angle and its bisector.

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

Since bisects by the definition of an angle bisector it can be said that and are congruent angles. Furthermore, and are both right angles. Therefore, they are also congruent angles.
By the Reflexive Property of Congruence, is congruent to itself.
Because two angles and a non-included side of are congruent to two angles and the corresponding non-included side of the triangles are congruent by the Angle-Angle-Side Congruence Theorem.
Corresponding parts of congruent triangles are congruent. Therefore, is congruent to Congruent segments have equal measures.
This means that is equidistant from the rays of