Triangle Angle Bisector Theorem

An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

By the Angle Bisector Theorem,


Extend to meet at point E.

By the Side-Splitter Theorem,


The angles are corresponding angles.


Since is a angle bisector of the angle

By the Alternate Interior Angle Theorem,

Therefore, by transitive property,

Since the angles are congruent, the triangle is an isosceles triangle with AE = AB.

Replacing AE by AB in equation (1),

Example :

Find the value of x.

By Triangle-Angle-Bisector Theorem,


Cross multiply.

5x = 42

Divide both sides by 5.

The value of x is 8.4.