Constructing angles bisectors for an angle divides the given angle exactly into two halves. The term “bisect” refers to dividing into two equal parts. Constructing angle bisectors makes a line that gives two congruent angles for a given angle. For example, when an angle bisector is constructed for an angle of 70°, it divides the angle into two equal angles of 35° each. Angle bisectors can be constructed for an acute angle, obtuse angle, or a right angle too.
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| 1. | Construct an Angle Bisector With a Compass |
| 2. | How to Prove an Angle Bisector? |
| 3. | Solved Examples |
| 4. | Practice Questions |
| 5. | FAQs on Constructing Angle Bisectors |
Construct an Angle Bisector With a Compass
An angle bisector is a line that bisects or divides an angle into two equal halves. 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. Let us consider the angle AOB shown below.
Note that the measure of the angle is not mentioned here. So, we do not need a protractor in constructing the angle bisector. This point is important to understand. When no angle measurements have been asked for, we must avoid using a protractor, and use only a ruler and a compass. This challenge is a fundamental idea behind geometrical constructions.
Follow the sequence of steps mentioned below to construct an angle bisector.
Step 1: Span any width of radius in a compass and with O as the center, draw two arcs such that it cut the rays OA and OB at points C and D respectively.
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Note that OC = OD, since these are radii of the same circle.
Step 2: Without changing the distance between the legs of the compass, draw two arcs with C and D as centers, such that these two arcs intersect at a point named E (in the image).
Note that CE = DE, since the two arcs were drawn in this step was of the same radius.
Step 3: Join the ray OE. This is the required angle bisector of angle AOB.
The proof of constructing an angle bisector is given below.
How to Prove an Angle Bisector?
From the above figure, we see that the angle bisector is constructed for the ∠AOB. The constructed angle bisector has created two similar triangles. Let us see how equal angles are made using the angle bisector with proof.
Compare ΔOCE and ΔODE:
1. OC = OD (radii of the same circular arc)
2. CE = DE (arcs of equal radii)
3. OE = OE (common)
By the SSS criterion, the two triangles are congruent, which means that ∠COE = ∠DOE. Thus, ray OE is the angle bisector of ∠COD or ∠AOB. It is to be noted that no angle measurements were required for this construction. If in other cases we know the measurement of the angle on which angle bisector is to be constructed, then we can simply use a protractor to construct an angle with half of the measurement of the given angle.
Topics Related to Constructing Angle Bisectors
Check out some interesting topics related to constructing angle bisectors.
Example 1: By constructing an angle bisector for each of the following angles, find out the angles that will be formed after the construction. a) 50° b) 74° c) 105°
Solution:
When an angle bisector is constructed for an angle of the measure say ∠X, then, it is divided into two equal halves, which means, two angles of measure ∠X/2 are formed.
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a) Let ∠X = 50°. The angle bisector makes two angles of measure ∠X/2, which is 25° each.b) Let ∠X = 74°. The angle bisector makes two angles of measure ∠X/2, which is 37° each.c) Let ∠X = 105°. The angle bisector makes two angles of measure ∠X/2, which is 52.5° each.
