Table of Contents
Last modified on August 3rd, 2023
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.
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.
Find the value of x. Given BD is the angle bisector of ∠ABC.
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.
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
Last modified on August 3rd, 2023