Angle Bisector

Here we will learn about angle bisectors, including how to construct an angle bisector using a pencil, a ruler and a pair of compasses.

There are also constructions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is an angle bisector?

An angle bisector is the name given to an accurate drawing where an angle is cut in half by a straight line. Bisector means to cut in half; in two equal pieces.

To do this we need to use a pencil, a ruler (a straight-edge) and compasses.

E.g.

Here is an angle bisector of angle ABC .

Arcs are drawn with compasses at the vertex B . This arc creates two new points and two more arcs are drawn. The final straight line is drawn from B to the intersection of the last two arcs. The final straight line bisects the original angle into two equal angles. The two new angles are congruent.

What is an angle bisector?

Angle bisector theorem

The angle bisector theorem involves a triangle ABC. 

It involves the relative lengths of the two segments that a side of a triangle is divided into when one of the angles of a triangle is bisected to create a new point D .

It states that  \frac{\text { Length of } AB}{\text { Length of } A C}=\frac{\text { Length of } BD}{\text { Length of } D C}

How to construct an angle bisector

In order to construct the bisector of an angle:

1. Use compasses to draw an arc.
2. Use compasses to draw two more arcs.
3. Join the vertex with the point where the arcs intersect.

How to construct an angle bisector

Related lessons on constructions

Angle bisector is part of our series of lessons to support revision on construction and loci and construction. You may find it helpful to start with the main loci and construction lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Loci and construction
• Constructions
• Loci
• Bearings
• Perpendicular bisector
• Constructing triangles
• How to draw a hexagon
• How to construct a 30, 60, 45, 90 degree angle
• Constructions between points and lines

Angle bisector examples

Example 1: Angle bisector

Construct an angle bisector of angle ABC

Use compasses to draw an arc.

Set your compasses to a length that is less than the shortest arm. Putting the point of the compasses on B , draw one arc going through both AB and BC .

Use compasses to draw two more arcs.

Put the point of the compasses on the point where the first arc crossed AB and draw an arc. Keep the compass on the same setting. Repeat by putting the point of the compasses on the point where the first arc crossed BC and draw an arc. These two arcs need to intersect.

Join the vertex with the point where the arcs intersect.

Using a straight-edge – a ruler, join up the point where the arcs intersect each other with the vertex B .

The new straight line is the angle bisector of the original angle ABC and splits it into two equal parts. 

You can check that the new straight line bisects the angle ABC by using a protractor.

Example 2: Angle bisector

Construct an angle bisector of angle PQR

Use compasses to draw an arc.

Set your compasses to a length that is less than the shortest arm. Putting the point of the compasses on B , draw one arc going through both PQ and QR .

Use compasses to draw two more arcs.

Put the point of the compasses on the point where the first arc crossed PQ and draw an arc. Keep the compass on the same setting. Repeat by putting the point of the compasses on the point where the first arc crossed BC and draw an arc. These two arcs need to intersect.

Join the vertex with the point where the arcs intersect.

Using a straight-edge – a ruler, join up the point where the arcs intersect each other with the vertex Q .

The new straight line is the angle bisector of the original angle PQR .

You can check that the new straight line bisects the angle PQR by using a protractor to measure. The angle ABC should have been cut into two equal angles. The two new angles are congruent.

Common misconceptions

• The construction arcs must not be removed

The arcs drawn should be drawn lightly so can be adjusted if needed but they must be visible in your final answer. This is to show that you have used the correct method to draw the perpendicular bisector accurately.

• The pencil should be sharp

A sharp pencil helps your diagram to be accurate. Using a small pencil in compasses can also be helpful.

The next lessons are

• Vectors
• Transformations
• Congruence and similarity

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