**Definition : **A Line which cuts an angle into two equal halves is called an Angle Bisector.

We already know bisecting means to cutting it into two equal parts. In an angle bisector, it is a line passing through the vertex of the angle that cuts it into two equal smaller angles.

__THEOREM__:

Consider a triangle *ABC*. Let the angle bisector of angle *A* intersect side *BC* at a point *D* between *B* and *C*. The angle bisector theorem states that the ratio of the length of the line segment *BD* to the length of segment *DC* is equal to the ratio of the length of side *AB* to the length of side *AC*:

And conversely, if a point *D* on the side *BC* of triangle *ABC* divides *BC* in the same ratio as the sides *AB* and *AC*, then *AD* is the angle bisector of angle *∠** A*. The generalized angle bisector theorem states that if *D* lies on the line *BC*, then.

This reduces to the previous version if *AD* is the bisector of *∠** BAC*. When *D* is external to the segment *BC*, directed line segments and directed angles must be used in the calculation.

The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.

An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.