Constructing Angle Bisectors
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.
|1.||Construct an Angle Bisector With a Compass|
|2.||How to Prove an Angle Bisector?|
|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.
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.
Constructing Angle Bisectors Solved Examples
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°
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.
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.
Example 2: Construct an angle bisector for an ∠AOB = 60°.
Please follow the steps of construction shown below to construct the angle bisector.
Step 1: Draw a line AB of any length. WIth 'A' as the center, and by placing the center of the protractor on A, mark 60°, and label the point as 'C'. Please note that this step could be done with a compass as well. Click here to know more.
Step 2: Using the compass, with any width as radius, draw an arc such that it intersects at two points on the lines AC and AB and label them as 'D" and 'E' respectively.
Step 3: Now with 'D' and 'E' as centers, and without changing the radius draw two arcs such that they intersect at a point 'F'.
Step 4: Now, join the points 'A' and 'F'. The ray AF is the angle bisector, where ∠DAF and ∠FAE are 30° each.
The construction of the angle bisector is shown in the figure below.
FAQs on Constructing Angle Bisectors
How Do You Construct an Angle Bisector?
An angle bisector divides an angle into two congruent angles. To construct an angle bisector for an angle, follow the steps given below.
- Step 1: Draw an angle of any measure. Here, we can observe that one ray is horizontal just to ease out our constructions.
- Step 2: With one end of the horizontal ray which makes the angle as the center and measuring any width (less than the length of the ray drawn) in the compass, draw an arc that intersects the two rays of the angle at any two points.
- Step 3: With the intersection points as the center, mentioned in step 2, and without any change in the radius, draw two arcs such that they intersect each other and lie between the intersection points on the legs of the angle.
- Step 4: Join the point of intersection with the vertex of the angle. We will get our required angle bisector.
Can We Construct Angle Bisectors For Angles of Any Measure?
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 right-angle measures 90°. When an angle bisector is constructed, we get two congruent angles measuring 45° each.
How to Construct an Angle Bisector With a Protractor and a Compass?
A protractor is not always needed to construct an angle bisector. When there is a need to bisect an angle for which the measure is given, then we make an angle with the given measurements and use the compass to construct the angle bisector. The steps are as follows.
- Step 1: Draw an angle of the given measure using the protractor and label the point. Here, we can observe that one ray is horizontal.
- Step 2: With one end of the horizontal ray which makes the angle as the center, and measuring any width in the compass, draw two arcs that intersect at the two rays of the angle.
- Step 3: With the intersection points as the center, mentioned in step 2, and without any change in the radius, draw two arcs such that they intersect each other.
- Step 4: Join the point of intersection with the vertex of the angle.
Are There Any Triangles Formed by Constructing an Angle Bisector?
Yes, two congruent triangles are formed by constructing an angle bisector which is similar to the SSS congruence of triangles. The common side shared by these two triangles is the angle bisector line. All the sides of these two triangles are equal and also it makes two equal angles, which proves that the angle bisector line bisects the given angle.