Angle Bisector Theorem: Definition, Formula, Proof, Examples

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What Is Angle Bisector Theorem?

Angle bisector theorem states that an angle bisector divides the opposite side into two line segments that are proportional to the other two sides.

Angle bisector theorem

Here, in $\Delta ABC$, the line AD is the angle bisector of $\angle A$.

AD bisects the side BC in two parts, c and d.

a and b are the lengths of the other two sides.

By the angle bisector theorem, we have

$\frac{a}{b} = \frac{c}{d}$

Consider an example:

Example on application of angle bisector theorem

Angle Bisector Theorem Statement

The bisector of an interior angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle.

Interior Angle Bisector Theorem

The interior angle bisector theorem is simply referred to as angle bisector theorem. Hence, an interior angle bisector of a triangle divides the opposite side in two parts such that the ratio of the two line segments is proportional to the ratio of the other two sides.

interior angle bisector theorem

Let us consider $\Delta ABC$ as shown above. Here, AD is the angle bisector (interior angle bisector) of $\angle BAC$, which meets the opposite side BC at point D. Hence, as per this theorem, the ratio of the BD to DC is equal to the ratio of the lengths of other sides, AB to AC.

$\frac{BD}{DC} = \frac{AB}{AC}$.

External Angle Bisector Theorem

A triangle’s external angle bisector divides one side externally in proportion to the ratio of the sides containing the angle. Most of the time, non-equilateral triangles have this condition.

Exterior angle bisector theorem

Let us consider $\Delta ABC$ as shown above.  

The side AB is extended. $\angle CAP$ is an external angle.

Here, AD is the angle bisector (exterior angle bisector) of $\angle CAP$. 

CE is drawn parallel to DA, such that it meets the line AB of a triangle at E. 

Hence, according to the external angle bisector theorem:

 $\frac{BD}{DC} = \frac{AB}{AC}$

Angle Bisector Theorem Proof

Statement: The angle bisector of a triangle divides the opposite side in such a way that the ratio of the two line segments is proportional to the ratio of the other two sides.

Let’s prove this statement.

Given: Let us consider $\Delta ABC$, in which AD is the angle bisector of $\angle BAC$, which meets BC at point D.

To prove: $\frac{BD}{DC} = \frac{AB}{AC}$.

Construction: Draw a ray CE parallel to DA. Extend BA such that it intersects CE at E.

Angle bisector theorem proof

Since $CE || DA$, AC acts as the transversal. 

$\angle DAC = \angle ACE$ (alternate interior angles) …(1)

$\angle BAD = \angle AEC$ (corresponding angles) …(2)

As we know that AD is the angle bisector of $\angle BAC$,

$\angle BAD = \angle DAC$ …(3)

From (1), (2) and (3), we have

$\angle ACE = \angle AEC$ this implies that $\Delta ACE$ is an isosceles triangle. 

Therefore, opposite sides of equal angles are equal.

Thus for $\Delta ACE$, we can write,

$AE = AC$ …(4)

Now, in $\Delta BCE$ we have, $CE || DA$.

By triangle proportionality theorem, a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.

$\frac{BD}{DC} = \frac{AB}{AE}$ …(5)

From equation (4) we have $AE = AC$, so on replacing AE by AC in equation (5), we get

$\frac{BD}{DC} = \frac{AB}{AC}$

Hence, the angle bisector theorem is proved.

Converse of Angle Bisector Theorem

The converse of the angle bisector theorem states that if the interior point of an angle of a triangle is equidistant from the two sides (arms) of a triangle, then that point lies on the angle bisector of the angle.

Converse of angle bisector theorem

D is a point in the interior of $\angle BAC$. 

If the perpendicular distances BD and DC are equal then, the line AD is the angle bisector of angle $\angle BAC$.

If $BD = CD$, then $\angle BAD = \angle CAD$.

Converse of angle bisector theorem

Facts about Angle Bisector Theorem

  • The points on an angle bisector are all equidistant from the arms of the angle.
  • In real life, angle bisectors have applications in games of billiards, rugby, football, architecture, etc.
  • Angle bisector theorem helps to find unknown lengths of sides of a triangle.
  • The point of concurrency of the angle bisectors of a triangle is known as the incenter of a triangle.

Conclusion

In this article, we learned about angle bisector theorem, its proof, interior angle bisector and exterior angle bisector theorem, the converse of angle bisector theorem, and application of these theorems in problem-solving as well in real life.

Solved Examples on Angle Bisector Theorem

  1. In the figure given below, find the ratio AD:DC.
Angle bisector theorem example

Solution:

$AB = 6,\; BC = 3$.

Also, BD is the angle bisector.

According to the angle bisector theorem, BD divides AC in the ratio proportional to the ratio of the other two sides. 

Thus, the ratio of AD to DC is the same as the ratio of AB to BC.

$\frac{AB}{BC} = \frac{AD}{DC}$

$\Rightarrow \frac{6}{3} = \frac{AD}{DC}$

$\Rightarrow AD:DC = 6:3$

$\Rightarrow AD:DC = 2:1$

  1.  In the figure given below, if PS is the angle bisector, find the length of segments RS and QS.
Converse of angle bisector theorem example

Solution:

RS $= 3x \;-\; 8,\; SQ = 2x + 2$

From the figure, point S lies on the bisector of angle QPR.

Thus, the point S is equidistant from the sides PR and PQ. 

RS $= SQ$

$\Rightarrow 3x\;-\;8 = 2x + 2$

$\Rightarrow 3x\;-\;2x = 2 + 8$

$x = 10$

RS $= 3x\;-\;8 = 3(10)\;-\;8 = 22$

SQ $= 2x + 2 = 2(10) + 2 = 22$

$\therefore$ RS $= SQ = 22$ units.

  1. In $\Delta PQR$ as shown below, PS is the angle bisector of $\angle P$. Find the lengths of sides of ΔPQR.
Angle bisector theorem example

Solution:

According to the angle bisector theorem, $\frac{PQ}{PR} = \frac{QS}{SR}$

$\Rightarrow \frac{x}{x \;-\; 2} = \frac{x + 2}{x \;-\; 1}$

$\Rightarrow x(x\;-\;1) = (x\;-\;2)(x + 2)$

$\Rightarrow x^2\;-\;x = x^2\;-\;4$

$\Rightarrow \;-\;x = \;-\;4$

$\Rightarrow x = 4$

Hence, length of the sides are 

PQ $= x = 4$, 

PR $= x\;-\;2 = 4\;-\;2 = 2$, 

QR $= (x + 2) + (x\;-\;1) = 2x + 1 = 2(4) + 1 = 9$.

Practice Problems on Angle Bisector Theorem

Angle Bisector Theorem: Definition, Formula, Proof, Examples

Attend this quiz & Test your knowledge.

1

If the time on a clock is exactly 04:00, then where should the bisector of the hour hand and minute-hand point?

At 12
At 1
At 2
At 6
CorrectIncorrect
Correct answer is: At 12
If the time on a clock is exactly 04:00, then the angle bisector and it points to 2.
Angle Bisector Theorem: Definition, Formula, Proof, Examples
2

Which of the following options represent the statement of angle bisector theorem for the given triangle?

Angle Bisector Theorem: Definition, Formula, Proof, Examples
$\frac{a}{x} = \frac{y}{b}$
$\frac{a}{x} = \frac{2b}{y}$
$\frac{a}{b} = \frac{x}{y}$
None of the above
CorrectIncorrect
Correct answer is: $\frac{a}{b} = \frac{x}{y}$
According to the angle bisector theorem, $\frac{a}{b} = \frac{x}{y}$.
3

The __________ of a triangle divides the opposite sides in the ratio equal to the ratio of the remaining two sides.

angle bisector
median
Perpendicular bisector
height
CorrectIncorrect
Correct answer is: angle bisector
An angle bisector divides the opposite side into two line segments that are proportional to the remianing two sides.
4

The bisectors of the angles of a triangle are _________.

concurrent
congruent
perpendicular
None of the above
CorrectIncorrect
Correct answer is: concurrent
The bisectors of the angles of a triangle pass through a common point. Hence, bisectors of the angles of a triangle are concurrent.

Frequently Asked Questions on Angle Bisector Theorem

The perpendicular drawn from the center of a circle to a chord bisects the chord.

Yes, the angle bisector theorem is also known as the triangle angle bisector theorem.

According to the exterior angle bisector theorem, a triangle’s external angle bisector divides one side externally in proportion to the ratio of the sides containing the angle.

The converse of the angle bisector theorem states that the point on the opposite side of an angle is assumed to be on its angle bisector if the sides of a triangle satisfy the following condition: “If a line drawn from a vertex of a triangle divides the opposite side into two parts such that they are proportional to the other two sides of the triangle.”

The missing lengths of a triangle’s sides can be determined by applying the triangle angle bisector theorem. Let $\angle A$ be divided by the bisector AD. Hence, according to the triangle angle bisector theorem $\frac{BD}{DC} = \frac{AB}{AC}$.