CK12 - Geometry

AngleBisectorTheoremand its Converse

We can provethe followingpair of theoremsaboutanglebisectors.

AngleBisectorTheorem:If a pointis on the bisectorof an angle,thenthe pointis

equidistantfrom the sidesof the angle.

Beforewe proceedwith the proof,let’s recallthe definitionof the distancefrom a pointto a line. The distance
from a pointto a line is the lengthof the line segmentthat passesthroughthe pointand isperpendicularto
the originalline.

Proof. Consider with anglebisector , and segments and , perpendicularto each

side throughpoint as follows:

We will showthat.

1. Since is the bisectorof , then by the definitionof anglebisector. In

addition,since and are perpendicularto the sidesof , then and are

right anglesand thus congruent.Finally, by the reflexiveproperty.