Angle Bisector of a Triangle – Definition, Properties & Examples

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Triangles are one of the most studied 2D shapes in mathematics. A triangle has some important properties like medians and altitudes. Angle bisectors of a triangle are one such important concept.

Let’s understand the angle bisector of a triangle and its properties with examples.

Angle Bisector

The angle bisector is a line that divides a given angle into two congruent (or equal) angles. For example, an angle bisector of a $60^{\circ}$ angle will divide it into two angles of $30^{\circ}$ each. 

In the above figure, $\text{OC}$ is the angle bisector of $\angle \text{AOB}$, such that $\angle \text{AOC} = \angle \text{COB} = \frac{1}{2} \angle \text{AOB}$.

What is an Angle Bisector of a Triangle?

In a triangle, the angle bisector of an angle is a straight line that divides the angle into two congruent(or equal) angles. In any triangle, there can be three angle bisectors, one for each vertex. 

The point where these three angle bisectors meet in a triangle is known as its incenter. The incenter is equidistant from all three vertices of a triangle, i.e., the distance between the incenter to all the vertices of a triangle is the same. 

In the figure below $\text{AD}$, $\text{BE}$, and $\text{CF}$ are the angle bisectors of $\angle \text{CAB}$, $\angle \text{ABC}$, and $\angle \text{BCA}$ respectively. $\text{G}$ is the point of intersection of all three bisectors which is known as the incenter.

In the above figure, you can see that the distance (perpendicular distance) of incenter $\text{G}$ from each of the vertices $\text{A}$, $\text{B}$, and $\text{C}$ is equal.

Properties of an Angle Bisector of a Triangle

The following are the important properties of an angle bisector of a triangle.

• An angle bisector divides an angle into two equal parts.
• Any point on the bisector of an angle is equidistant from the sides or arms of the angle.
• In a triangle, it divides the opposite side into the ratio of the measure of the other two sides.

Interior Angle Bisector Theorem

The interior angle bisector theorem states that an angle bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In the above figure, in $\triangle \text{ABC}$, $\text{AD}$ is the angle bisector of $\angle \text{A}$, then by angle bisector theorem, $\frac{\text{CD}}{\text{DB}} = \frac{\text{AC}}{\text{AB}}$.

Let’s prove the theorem.

Draw $\text{BE} || \text{AD}$

Extend $\text{CA}$ to meet $\text{BE}$ at $\text{E}$

By the Basic Proportionality theorem,

$\frac{\text{CD}}{\text{DB}} = \frac{\text{CA}}{\text{AE}}$ ——————— (1)

$\angle 4 = \angle 1$ ($\angle 4$ and $\angle 1$ are corresponding angles)

Since, $\text{AD}$ is an angle bisector of the angle $\angle \text{CAB}$, therefore,  $\angle 1 = \angle 2$

Also, by the Alternate Interior Angle theorem, $\angle 2 = \angle 3$

Therefore, by transitive property $\angle 4 = \angle 3$

Since, $\angle 3$ and $\angle 4$ are equal, therefore, $\triangle {ABE}$ is an isosceles triangle, where $\text{AE} = \text{AB}$

Replacing $\text{AE}$ by $\text{AB}$ in (1), we get 

$\frac{\text{CD}}{\text{DB}} = \frac{\text{AC}}{\text{AB}}$

Practice Problems

1. What is an angle bisector?
2. What is an angle bisector of a triangle?
3. How many angle bisectors a triangle can have?
4. The point of intersection of all the angle bisectors of a triangle is called its ____________.
5. In $\triangle \text{ABC}$, $\text{AD}$ is the bisector of $\angle \text{A}$ meeting side $\text{BC}$ at $\text{D}$, if $\text{AB} = 10 \text{cm}$, $\text{AC} = 14 \text{cm}$ and $\text{BC} = 6 \text{cm}$, find $\text{BD}$ and $\text{DC}$.

FAQs

Conclusion

In a triangle, the angle bisector of an angle is a straight line that divides the angle into two congruent(or equal) angles. In any triangle, there can be three angle bisectors, one for each vertex. The point of intersection of all the three angle bisectors in a triangle is called its incenter.

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