Angle Bisector - Definition, Construction, Properties, Examples

Angle Bisector

An angle bisector is defined as a ray, segment, or line that divides a given angle into two angles of equal measures. The word bisector or bisection means dividing one thing into two equal parts. In geometry, we divide an angle by a line or ray which is considered as an angle bisector.

1.	What is Angle Bisector?
2.	Angle Bisector of a Triangle
3.	Properties of an Angle Bisector
4.	Angle Bisector Construction
5.	Angle Bisector Theorem
6.	FAQs on Angle Bisector

What is Angle Bisector?

The angle bisector in geometry is the ray, line, or segment which divides a given angle into two equal parts. For example, an angle bisector of a 60-degree angle will divide it into two angles of 30 degrees each. In other words, it divides an angle into two smaller congruent angles. Given below is an image of an angle bisector of ∠AOB.

Angle Bisector of a Triangle

In a triangle, the angle bisector of an angle is a straight line that divides the angle into two equal or congruent angles. There can be three angle bisectors in every triangle, one for each vertex. The point where these three angle bisectors meet in a triangle is known as its incenter. The distance between the incenter to all the edges of a triangle is the same. Look at the image below showing the angle bisector of a triangle. Here, AG, CE, and BD are the angle bisectors of ∠BAC, ∠ACB, and ∠ABC respectively. F is the point of intersection of all three bisectors which is known as incenter and it is at an equal distance from each of the vertex.

Properties of an Angle Bisector

Till now you must be clear about the meaning of angle bisector in geometry. Now, let us learn some of the angle bisector properties listed below:

• 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.

Construction of Angle Bisector

Let's try constructing the angle bisector for an angle. In this section, we will see the steps to be followed for angle bisector construction.

Steps 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. (Refer to the figure below)

Step 3: Now, taking D and E as centers and with the same radius as taken in the previous step, draw two arcs to intersect each other at F.

Step 4: Join B to F and extend it as a ray. This ray BF is the required angle bisector of angle ABC.

Angle Bisector Theorem

Let's now understand in detail an important property of the angle bisector of a triangle as stated in the previous section. This property is known as the angle bisector theorem of a triangle. According to the angle bisector theorem, in a triangle, the angle bisector drawn from one vertex divides the side on which it falls in the same ratio as the ratio of the other two sides of the triangle.

Statement: An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In the above image, PS is the angle bisector of ∠P in ΔPQR. Therefore, by applying the angle bisector theorem we can say that PQ/PR = QS/SR or a/b = x/y.

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• Constructing An Angle of 60 Degrees