The angle bisector theorem states that the angle bisector of any angle meeting at any side divides it into a ratio equal to the ratio of the opposite sides of the triangle. The line that divides the angle into two equal parts is called an angle bisector. The angle bisector theorem is useful for triangles, quadrilaterals, etc.
For the given triangle ABC,
The formula for the angle bisector theorem is given by:
\frac{AB}{AC} = \frac{BD}{BC}
Types of Angle Bisector Theorem
The two types of angle bisector theorems are
1. Interior Angle Bisector Theorem
The interior angle bisector theorem states that the angle bisector divides the interior angle formed by two sides in the ratio of the third side divided by the bisector. If PS is the angle bisector of angle P in triangle PQR with sides PQ, PR, and QS, then the interior angle theorem formula is given by
QS / SR = PQ / PR
2. Exterior Angle Bisector Theorem
The exterior angle bisector theorem states that the bisector divides the exterior angle and divides the third side externally in the ratio of the other two sides forming the angle. If PR is the exterior angle bisector of angle P in triangle PQR with sides PQ, PR, and QS, then the exterior angle theorem formula is given by:
PQ /PS = QR / SR
Angle Bisector Theorem Proof
The proofs of both types of angle bisector theorem are discussed below.
Interior Angle Bisector Proof
Consider the below figure triangle PQR, with the interior angle bisector PS
To Prove: QS / SR = PQ / PR
Draw line RT parallel to PS and extend T so that it meets P as shown in the figure.
PR is the traversal of RT || PS. So, by the property, alternate interior angles are equal.
∠ SPR = ∠ PRT [interior angles] -------(1)
∠QPS = ∠PTR [corresponding angles] -------(2)
In the figure, PS is the angle bisector of P.
∠QPS = ∠ SPR ------(3)
From (2) and (3)
∠PTR = ∠ SPR ------(4)
From (1) and (4)
∠PRT = ∠PTR ------(5)
Equation (5) implies that triangle PQR is an isosceles triangle and PS = PR.
Now, by the proportionality theorem:
QS / SR = PQ / PS
Since, PS = PR
QS / SR = PQ / PR
Hence Proved
Exterior Angle Bisector Proof
Consider the below figure, triangle PQR, with PR as the external angle bisector of angle QPS that meets QS at R.
To Prove: QS / SR = PQ / PS
Draw ST || PR such that S intersects line PQ at point T.
PS be the traversal of ST || PR
∠ TSP = ∠ SPR [Alternate interior angles are equal] -------(1)
∠ STP = ∠ RPA [Corresponding Angles] --------(2)
PR is the angle bisector of ∠SPA
∠SPR = ∠RPA --------(3)
From (2) and (3)
∠STP = ∠SPR --------(4)
From (1) and (4)
∠TSP = ∠ STP ---------(5)
From (5) we can conclude that triangle TPS is an isosceles triangle and PT = PS.
Now by proportionality theorem
QT / TP = QS / SR
Adding 1 on both sides
(QT / TP) + 1 = (QS / SR) + 1
(QT + TP) / TP = (QS + SR) / SR
PQ / TP = QR / SR
Since, TP = PS
PQ PS = QR / SR
Hence Proved
Converse of Angle Bisector Theorem
If the line divides a vertex of a triangle and divides the opposite sides into two so that it is proportional to the other two sides of the triangle, then the point intersecting the opposite side of the angle lies on its angle bisector.
In a triangle ABC, if AB and AC are the two sides of triangle and AD intersects BC at D, then, by converse angle bisector theorem,
If AB /AC = BD / CD then, AD is the angle bisector of the angle A
Converse of Angle Bisector Theorem Proof
Consider the below figure triangle PQR, with the interior angle bisector PS
To Prove: PS is angle bisector of P
Since, PS divides QR into two parts
PQ / PR = QS / QR --------(1)
Draw RT || PS and extend PQ to meet T
Since, PS || RT
∠ QPS = ∠ PTR (corresponding angles are equal) -------(3)
∠SPR = ∠ ∠PRT (alternate interior angles are equal) -------(4)
Consider triangle QRT
By Thales' theorem
QP / PT = QS / SR ------(2)
By equation (1) and (2)
PQ / PR = PQ / PT
PR = PT
Therefore, triangle PTR is an isosceles triangle
∠ PTR = ∠ PRT
From equation (3) and (4)
∠QPS = ∠SPR
Therefore, PS is angle bisector of angle P
Hence, prove d.
Related Articles
Examples of Angle Bisector Theorem
Example 1: In triangle PQR the lengths of sides PQ and PR are 10 units and 12 units, respectively. The PS is the angle bisector bisecting the angle QPR and meets the third side QR at S. If QS = 8 units, then find the length of SR.
Solution:
Given
- PQ = 10 units
- PR = 12 units
- QS = 8 units
Using Angle Bisector Theorem
PQ / PR = QS / SR
10 / 12 = 8 / SR
SR = (8 × 12) / 10
SR = 96 / 10
SR = 9.6 units
Example 2: In triangle ABC the lengths of sides AB and AC are 6 units and 8 units, respectively. The AD is the angle bisector bisecting the angle BAC and meets the third side BC at D. If CD = 4 units, then find the length of BD.
Solution:
Given
- AB = 6 units
- AC = 8 units
- CD = 4 units
Using Angle Bisector Theorem
BD / CD = AB / AC
BD = (AB × CD) / AC
BD = (6 × 4) / 8
BD = 3 units
Example 3: In triangle XYZ the lengths of sides XY and XZ are 5 units and 9 units, respectively. The XW is the angle bisector bisecting the angle YXZ and meets the third side YZ at W. If YW = 3 units, then find the length of YZ.
Solution:
Given
- XY = 5 units
- XZ = 9 units
- YW = 3 units
Using Angle Bisector Theorem
YW / WZ = XY / XZ
WZ = (YW × XZ) / XY
WZ = (3 × 9) / 5
WZ = 27 / 5
WZ = 5.4 units
YZ = YW + WZ
YZ = 3 + 5.4
YZ = 8.4 units
Example 4: In triangle XYZ the lengths of sides XY and XZ are a units and a - 1 units, respectively. The XW is the angle bisector bisecting the angle YXZ and meets the third side YZ at W. If YW = a + 3 units and WZ = a + 1, then find the length of all the sides.
Solution:
Given
- XY = a
- XZ = a - 1
- YW = a + 3
- WZ = a + 1
Using Angle Bisector Theorem
YW / WZ = XY / XZ
(a+ 3) / (a + 1) = a / (a - 1)
(a + 3) (a - 1) = a (a + 1)
a2 + 2a - 3 = a2 + a
a = 3
So, the sides are XY = 3 units, XZ = 2 units, YW = 6 units, WZ = 4 units and YZ = YW + WZ = 10 units
Example 5: In a triangle PQR, a line PS is drawn on QR with side lengths PQ = 10 units, PR = 5 units, QS = 8 units, and SR = 4 units. Prove that PS is the angle bisector of angle P.
Solution:
Given
- PQ = 10 units
- PR = 5 units
- QS = 8 units
- SR = 4 units
To Prove: PS is angle bisector of angle P.
To prove this, we will use converse angle bisector theorem
If PQ / PR = QS / SR then, PS is the angle bisector of angle P
First determine PQ / PR and QS / SR
PQ / PR = 10 / 5 = 2 units
QS / SR = 8 / 4 = 2 units
Since, PQ / PR = QS / SR
The line PS is angle bisector of the angle P.
Practice Questions on Angle Bisector Theorem
Q1. In triangle PQR the lengths of sides PQ and PR are 25 units and 32 units, respectively. The PS is the angle bisector bisecting the angle QPR and meets the third side QR at S. If QS = 14 units, then find the length of SR.
Q2. In triangle ABC the lengths of sides AB and AC are 12 units and 14 units, respectively. The AD is the angle bisector bisecting the angle BAC and meets the third side BC at D. If CD = 7 units, then find the length of BD.
Q3. In triangle XYZ the lengths of sides XY and XZ are 7 units and 12 units, respectively. The XW is the angle bisector bisecting the angle YXZ and meets the third side YZ at W. If YW = 4 units, then find the length of YZ.
Q4. In triangle XYZ the lengths of sides XY and XZ are p units and p - 3 units, respectively. The XW is the angle bisector bisecting the angle YXZ and meets the third side YZ at W. If YW = 2p units and WZ = p + 1, then find the length of all the sides.
Q5. In a triangle ABC a line AD is drawn on BC with side lengths AB = 15 units, AC = 3 units, BD = 10 units, and SR = 2 units. Prove that AD is the angle bisector of angle A.