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

Steps

Statements

Reasons

1.

∠1 ≅ ∠1

An angle bisector is ray that forms two congruent angles

2.

An auxiliary line is drawn parallel to AD and extend line AC that meet at E

Through a point not on a line there is only one line parallel to the given line (Parallel Postulate)

3.

∠2 ≅ ∠3

If two parallel lines are cut by a transversal, the corresponding angles are congruent

4.

∠1 ≅ ∠4

If two parallel lines are cut by a transversal, the alternate interior angles are congruent

5.

∠3 ≅ ∠4

By substitution

6.

BD/CD = EA/CA

If a line is parallel to one side of a triangle and intersects the other two sides, it divides the sides proportionally (Side Splitter Theorem)

7.

BA ≅ EA

If two angles of a triangle are congruent, the sides opposite the angles are congruent (Isosceles triangle)

8.

BA = EA

Congruent segments have equal lengths

9.

BD/DC = AB/AC

By substitution

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