Angle Bisector Theorem
Angle bisector theorem states that an angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. An angle bisector is a ray that divides a given angle into two angles with equal measures. In this short lesson, we will be focusing more on the most important theorem on the angle bisector. To be more precise, we will be learning about the angle bisector theorem proof, angle bisector theorem examples, triangle angle bisector theorem, how to construct angle bisector, and other interesting properties and facts around angle bisectors.
1.  What is Angle Bisector Theorem? 
2.  How to Construct an Angle Bisector? 
3.  Proof of Angle Bisector Theorem 
4.  Angle Bisector Theorem Formula 
5.  FAQs on Angle Bisector Theorem 
What is Angle Bisector Theorem?
Triangle angle bisector theorem states that in a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle. Consider the figure below.
Here, AD is the bisector of ∠A. According to the angle bisector theorem, \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\).
Definition of Angle Bisector
An angle bisector is a line or ray that divides a triangle and an angle into two equal measures. The main aspects of an angle bisector are that any point on the bisector of an angle is equidistant from the sides of the angle and the angle bisector divides the opposite side of a triangle in the ratio of the adjacent sides.
How to Construct an Angle Bisector?
To geometrically construct an angle bisector, we would need a ruler, a pencil, and a compass, and a protractor if the measure of the angle is given. Any angle can be bisected using an angle bisector. Follow the sequence of steps mentioned below to construct an angle bisector.
 Step 1: Draw any angle, say ∠ABC.
 Step 2: Taking B as the center and any appropriate radius, draw an arc to intersect the rays BA and BC at, say, E and D respectively.
 Step 3: Now, taking D and E as centers and with a radius more than half of DE, draw an arc to intersect each other at F.
 Step 4: Draw ray BF. This ray BF is the required angle bisector of angle ABC.
Proof of Angle Bisector Theorem
The angle bisector theorem statement: In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle. Let us see the proof of this.
Draw a ray CX parallel to AD, and extend BA to intersect this ray at E.
By the Basic Proportionality Theorem, we have that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
In \(\Delta CBE\), DA is parallel to CE.
\(\dfrac{BD}{DC}=\dfrac{BA}{AE}\;\;\;\;\;\; \cdots (1)\)
Now, we are left with proving that AE = AC.
Let's mark the angles in the above figure.
Since DA is parallel to CE, we have
 ∠1= ∠2 (corresponding angles)
 ∠3= ∠4 (alternate interior angles)
Since AD is the bisector of ∠BAC, we have ∠1= ∠3.
So, we can say that ∠2= ∠4.
Since sides opposite to equal angles are equal, AC = AE.
Substitute AC for AE in Equation (1).
\(\dfrac{BD}{DC}=\dfrac{BA}{AC}\)
Hence proved.
Angle Bisector Theorem Formula
Triangle angle bisector theorem states that "In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle". i.e., the angle bisector formula is:
\(\dfrac{\text{BD}}{\text{DC}} = \dfrac{\text{AB}}{\text{AC}}\)
The easy method to remember the angle bisector theorem of a triangle is
Related Topics
Listed below are a few interesting topics related to the angle bisector theorem, take a look.
Examples on Angle Bisector Theorem

Example 1: Amy drew a triangle on the board. Where, AB = 4in, AC = 6in, BD = 1.6in, and DC = 2.4in. She asks if \(\dfrac{AB}{AC}=\dfrac{BD}{DC}\). Can you answer this?
Solution:
Let's find the ratio \(\dfrac{AB}{AC}\).
\[\begin{align}\dfrac{AB}{AC}=\dfrac{4}{6}=\dfrac{2}{3}\end{align}\]
Let's find the ratio \(\dfrac{BD}{DC}\).
\[\begin{align}\dfrac{BD}{DC}=\dfrac{1.6}{2.4}=\dfrac{2}{3}\end{align}\]
Therefore, both the ratios are equal.

Example 2: In ΔXYZ, XE is the bisector of ∠X. Let XY = 4 units, YE = 2 units, and EZ = 3 units. Can you find the length of XZ?
Solution:
Given that, XE is the bisector of ∠X.
So, we can use the angle bisector theorem.
According to the question,
\[\begin{align}\dfrac{YE}{EZ}&=\dfrac{XY}{XZ}\\\dfrac{2}{3}&=\dfrac{4}{XZ}\\XZ&=\dfrac{4}{2}\times 3\\XY&=6\end{align}\]
Therefore, the length of XZ = 6 units.

Example 3: Look at ΔABC shown below.
If AD bisects ∠A, can you find the value of x?
Solution:
Given that, AD is the bisector of ∠A.
So, we can use the angle bisector theorem.
According to the question,
\[\begin{align}\dfrac{AB}{AC}&=\dfrac{BD}{DC}\\\dfrac{x}{x2}&=\dfrac{x+2}{x+1}\\x(x1)&=(x2)(x+2)\\x^2x&=x^24\\x&=4\\x&=4\end{align}\]
Therefore, the value of x is 4.
FAQs on Angle Bisector Theorem
What is the Formula for Angle Bisector?
Let AD be the bisector of ∠A in ΔABC. According to the angle bisector theorem, \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\).
How are the SideSplitter Theorem and the Angle Bisector Theorem Similar?
The only similarity between the sidesplitter theorem and the angle bisector theorem is that both the theorems related the proportions of side lengths of the triangle.
What is the Converse of the Angle Bisector Theorem?
If AD is drawn in the Δ ABC such that \(\dfrac{BD}{DC}=\dfrac{AB}{AC}\), then AD bisects the ∠A.
How Do You Find the Angle Bisector of a Triangle?
Follow the steps mentioned below to bisect ∠PQR.
 Let Q be the center and with any radius, draw an arc intersecting the ray \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\), say at the points E and D respectively.
 Now, taking D and E as centers and the same radius, draw arcs intersecting each other say at F. <<DE is not a line. Pls check. Also, I guess the radius should be same as the step 1>>
 Draw the ray \(\overrightarrow{QF}\).
Here, \(\overrightarrow{QP}\) is the angle bisector of ∠PQR.
How Do You Solve an Angle Bisector Problem?
We solve an angle bisector problem by using the triangle angle bisector theorem.
When Can You Use the Angle Bisector Theorem?
The angle bisector theorem is used when we know the angle bisector and length of sides of the triangle.
When an Angle Bisector is Constructed For a Right Angle, What is the Measure of the Two Angles Formed?
An angle bisector line divides or makes two congruent angles for any given angle. The same concept applies to a right angle too. A rightangle measures 90°. When an angle bisector is constructed, we get two congruent angles measuring 45° each.
visual curriculum