Bisecting an Angle

Bisecting an angle means dividing the angle into two equal halves.

For Example
Observe the below angle AOB which is 60 degree in measurement.
If we divide the angles into two equal halves (i.e. 30 degree) it means we have bisected the angle

Observe the above image.

In the first image simple angle AOB of 60 degree is shown.

In the second image, the line OC bisected the angle AOB into two halves.
Now we have two angles of 30 degree each.
Line OC is also known as Angle Bisector of angle AOB

How to bisect an angle

Here we will learn how to bisect an angles using two methods
(a) Angle Bisection using Compass
(b) Angle bisection using Protractor

(a) Angle Bisection using Compass

Step 01
Take an angle of your choice
Given below is the angle AOB of 70 degree

Step 02
Take a compass with certain width and place it at vertex O
From the vertex, mark points on the line OA & OB

From O, marking at point OA

Without changing the compass measurement, mark point Q on line OB

Step 03
After marking points P & Q, you can adjust the compass measurement for the next step

Place the compass at point P and mark an arc between the angle AOB as shown in below image

Without changing the measurement, place the compass at point Q and cut the arc drawn previously.
Mark the point M at the intersection of two arcs

Step 04
Remove the compass and join line OM

Line OM is the angle bisector which divides angle AOB into two equal halves.
\angle A\ and \ \angle B is equal to 75 degree

(B) Angle Bisection using Protractor

Step 01
Take an angle of your choice
Given below is the angle AOB of 70 degree

Step 02
Take a Protractor and place it over line OA

Step 03
Here angle AOB = 70 degree
Half of angle AOB = 35 degree

So draw 35 degree angle using Protractor

Hence line OM is the bisector of angle AOB

Proof of Angle Bisection

We have already seen how to bisect an angle using compass.
Here we will prove theoretically if the above method is really bisecting the angle or not

Below is the angle bisector diagram which have been used above.
We have joined line QM & PM to form triangle OQM & OPM

To Prove
Angle AOB is bisected into two halves
\angle A\ = \ \angle B

Proof
Compare triangles OQM & OPM

Side OQ = OP {As same compass width is used}
Side QM = PM {Again same compass width is used to draw the arc}
Side OM = OM {Common side of the triangles}

Hence by SSS congruency, triangle OQM & OPM are congruent.

Since both triangles are congruent, it means:
Angle AOM = QOM
i.e \angle A\ = \ \angle B

This means that lime OM bisected angle AOB into two halves

Properties of Angle Bisector

(01) Perpendicular line drawn from angle bisector to the angle line are equal in length

In the above figure, OM is the angle bisector of angle AOB
Two perpendicular lines are drawn from line OM to line OA & OB.

According to the property of angle bisector:
length of X = length of Y

Try to remember the property as it will help us to solve different higher grade questions

(02) The angle bisector divide the opposite side in the same ratio as the adjacent sides

In the above figure:
ABC is a triangle with angle bisector AD bisecting angle A

The bisector AD divides the line BC in the same ratio as adjacent sides AB & AC
i.e. \frac{BD}{DC} \ =\frac{AB}{AC} \

This property is also called Angle Bisector Theorem

Questions on Angle Bisector

(01) Angle AOB = 56 degree
Line OC is the angle bisector of angle AOB.
Find value of angle AOC

Read Solution

(02) M & N are parallel lines intersected by transversal.
\angle A = 100 degree
Find the value of \angle B , if line K bisect \angle DOE

Read Solution