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.
Angle Bisector of a Triangle
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.
How Many Angle Bisectors does a Triangle Have
Triangle Angle Bisector Theorem
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
Find the value of x. Given BD is the angle bisector of ∠ABC.
Find x. Given CD is the angle bisector of ∠ACB.