# Angle Bisector of a Triangle

## What is an Angle Bisector of a Triangle

An angle bisector of a triangle is a line segment that bisects a vertex angle of a triangle and meets the opposite side of the triangle when extended. They are also called the internal bisector of an angle.

Shown below is a ΔABC, with angle bisector AD of ∠BAC.

How Many Angle Bisectors does a Triangle Have

Every triangle has 3 angle bisectors.

As we know, the point where three or more lines intersect is called the point of concurrency, thus the three angle bisectors of the internal angles of a triangle are concurrent.

## Triangle Angle Bisector Theorem

### Proof of Triangle Angle Bisector Theorem

To prove: BD/DC = AB/AC

Proof:

Given: AD is the bisector of ∠BAC

Let us solve some examples to understand the concept better.

## How to Find the Angle Bisector of a Triangle with Examples

Find the value of x. Given BD is the angle bisector of ∠ABC.

Solution:

Given that BD is the angle bisector of ∠ABC
By Triangle Angle Bisector Theorem, we know,
AB/BC = AD/DC, here AB = 4 cm, BC = 12 cm, AD = 2.5 cm, and DC = x cm
=> 4/12 = 2.5/x
=> x = 12 × 2.5/4 = 7.5 cm

Find x. Given CD is the angle bisector of ∠ACB.

Solution:

Given that CD is the angle bisector of ∠ACB
By Triangle Angle Bisector Theorem, we know,
AD/AC = DB/BC, here AD = 12 cm, AC = 16 cm, DB = x cm, and BC = 24 cm
=> 12/16 = x/24
=> x = 12 × 24/16 = 18 cm