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,

Proof:

Extend to meet at point *E*.

By the Side-Splitter Theorem,

---------(1)

The angles are corresponding angles.

So,

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,

Substitute.

Cross multiply.

5*x* = 42

Divide both sides by 5.

The value of *x* is 8.4.