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,

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.

5x = 42

Divide both sides by 5.

The value of x is 8.4.