What are the steps in constructing an angle bisector with a compass and straightedge?

In this section, you will learn how to construct angle bisector. 

To construct an angle bisector, you must need the following instruments.

1. Ruler

2. Compass

Steps to Construct Angle Bisector

Step 1 :

Construct an angle of given measure at O using protractor.

Step 2 :

With ‘O’ as center draw an arc of any radius to cut the rays of the angle at A and B.

Step 3 :

With ‘A’ as center draw an arc of radius more than half of AB, in the interior of the given angle.

Step 4 :

With ‘B’ as center draw an arc of same radius to cut the previous arc at ‘C’.

Step 5 :

Join OC.

OC is the angle bisector of the given angle.

This construction clearly shows how to draw the angle bisector of a given angle with compass and straightedge or ruler. The angle bisector divides the given angle into two equal parts. 

For example, if we draw angle bisector for the angle 60°, the angle bisector will divide 60° in to two equal parts and each part will measure 30°.

Key Concept - Angle Bisector

The angle bisector of a given angle is the line of symmetry which divides the angle into two equal parts.

Practice Question

Construct ∠AOB  =  80° and draw its angle bisector.

Step 1 :

Construct ∠AOB = 80° angle at the point ‘O’ using protractor

Step 2 :

With ‘O’ as center draw an arc of any radius to cut the rays OA and OB at the points X and Y respectively.

Step 3 :

With ‘X’ as center draw an arc of radius more than half of XY in the interior of the angle.

Step 4 :

With ‘Y’ as center draw an arc of the same radius to cut the previous arc at C. Join OC.

Using the steps explained above, in the above figure, angle bisector is drawn at O.

OC is the angle bisector of the given angle 80°.

Hence, OC divides the angle 80° in to two equal parts and each part measures 40°.

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You can bisect an angle just as you can bisect a line. To bisect means to divide something into two equal parts. There are two methods for bisecting an angle. You can use the first method if you have a protractor, and if you need to find the degree measurement of the bisector. You can use the second method if you have a compass and straightedge, and only need to draw the bisector, not measure it.

1. 1

   Measure the angle. Place the origin hole of the compass on the angle’s vertex, lining up the base line with one of the angle’s rays. Look at the degree mark where the other ray falls. This will tell you the degree measurement of the angle.

   • For example, you might measure an angle that is 160 degrees wide.
   • Remember that protractors have two sets of numbers. To know which set of numbers to look at, think about the size of the angle. Obtuse angles are greater than 90 degrees, and acute angles are less than 90 degrees.

2. 2

   Divide the number of degrees in half. An angle bisector divides an angle into two equal parts.[1] X Research source Go to source So, to find where the angle bisector lays, divide the number of degrees in the angle by 2.[2] X Research source Go to source

   • For example, if the angle is 160 degrees, you would calculate
     
     . So, the angle bisector is at the 80-degree mark of the angle.

3. 3

   Draw a point marking the measurement of the bisector. Again, line up the origin hole with the angle’s vertex, and line up the base line with one of the rays. Find the halfway point using the protractor. Mark this point in the angle’s interior.

   • For example, if the bisector of a 160-degree angle lies at the 80-degree point, you would find the 80-degree mark on the protractor and draw a point at this location in the angle’s interior.

4. 4

   Draw a line from the vertex to the point. Use the straightedge of the protractor to connect the vertex to the halfway point of the angle. The line you draw is the angle’s bisector.[3] X Research source Go to source

1. 1

   Draw an arc across both rays. Open the compass to any width, and place the point of the compass at the angle’s vertex. Swing the compass so that the pencil draws an arc that crosses both rays of the angle.[4] X Research source Go to source

   • For example, you might have angle BAC. Place the compass tip on point A. Swing the compass so that it draws an arc intersecting ray AB at point D, and ray AC at point E.

2. 2

   Draw an interior arc. Move the compass so that the point sits on the location where the first arc intersects the first ray. Swing the compass, drawing an arc inside of the angle.[5] X Research source Go to source

   • For example, place the compass tip on point D and draw an arc inside the angle.

3. 3

   Draw a second interior arc intersecting the first interior arc. Without changing the width of the compass, move the point to the location where the first arc intersects the second ray. Swing the compass, drawing an interior arc that intersects the first interior arc you drew.[6] X Research source Go to source

   • For example, place the compass tip on point E and draw an arc intersecting the first interior arc. Label the point of their intersection point F.

4. 4

   Draw a line from the vertex to the point where the arcs intersect. Use a straightedge to ensure that the line is accurate. This line bisects the angle.[7] X Research source Go to source

   • For example, use a straightedge to draw a line connecting points F and A.

• Question

  What is the number of bisectors that can be drawn of a given angle?

  One.

• Question

  How do I prove that the ray drawn bisects the angle?

  Check the angles between either of the original rays and the bisector line. If they are equal, then the line is the exact bisector.

• Question

  How can I draw more than one angle bisector?

  Any given angle has exactly one bisector.

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Updated: October 5, 2021

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Categories: Geometry

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