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Segment of a circle Circles, sectors and arcs Parts of a circle Angle bisector Perpendicular bisector

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Here we will learn about how to construct a 30, 60, 45 and 90 degree angle 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.

Constructing a 30, 60, 45, 90 degree angle is constructing these angles accurately without using a protractor.

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

E.g.

A 60 degree angle can be constructed by drawing an equilateral triangle.

Then an angle bisector will construct a 30 degree angle.

E.g.

A 90 degree angle can be constructed with a perpendicular bisector.

Then an angle bisector will construct a 45 degree angle.

In order to construct a 60 degree angle:

Get your free Constructions worksheet of 20+ questions and answers including constructing 30º, 60º, 90º angles without a protractor. Reasoning and applied questions.

COMING SOON

Get your free Constructions worksheet of 20+ questions and answers including constructing 30º, 60º, 90º angles without a protractor. Reasoning and applied questions.

COMING SOON

How to construct a 30, 60, 45, 90 degree angle 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:

Construct a 60° angle

With a pencil and a straight-edge draw a straight line

2From one end of the line draw an arc.

Use compasses centered on one end of the line, draw an arc.

3From where the arc crosses the line, draw another arc.

Keep the compasses at the same setting. The new arc should cross the first arc.

4Join the end point of the line to where the two arcs intersect.

Use a straight-edge to join the end point used in step 2 to the intersection of the two arcs.

In order to construct a 30 degree angle:

Construct a 30° angle

Construct a 60 degree angle.

Follow the steps in Example 1 to construct a 60 degree angle

Angle ABC is 60°

Construct an angle bisector of the 60 degree angle.

Place the point of your compasses at A and draw an arc. Keep the compasses at the same setting and repeat placing the point at C and drawing another arc. Using a straight-edge, join up the point where the arcs intersect each other with the vertex B .

The 60 degree angle has been bisected into two equal angles, both 30 degrees.

In order to construct a 90 degree angle:

Construct a 90° angle

Draw a line.

Construct a perpendicular bisector.

Set your compasses to about three-quarters of the length of your line. Place the point of your compasses on one endpoint of the line and draw an arc. Keeping the compasses set at the same setting, draw another arc from the other endpoint. Finally use a straight-edge to join the two intersections of the arcs.

The new line is the perpendicular bisector of the original line segment.

In order to construct a 45 degree angle:

Construct a 45° angle

Construct a 90 degree angle.

Follow the steps in Example 3 to construct a 90 degree angle

Construct an angle bisector of the 90 degree angle.

Place the point of your compasses at the centre A and draw an arc. Place the point of your compasses at B and draw an arc. Keep the compasses at the same setting place the compasses at C and draw another arc, crossing the one made at B . Using a straight-edge, join up the point where the arcs intersect each other with the centre A .

The 90 degree angle has been bisected into two equal angles, both 45 degrees.

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

1. Construct a 60 degree angle

The construction must be made with compasses kept to the same setting for both arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

2. Construct a 30 degree angle

The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

3. Construct a 90 degree angle

The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

4. Construct a 45 degree angle

The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

1. In the space below, using compasses and a ruler, construct a 90^{\circ} angle.

(2 marks)

For drawing the construction arcs

(1)

For completing the perpendicular bisector

(1)

2. In the space below, using compasses and a ruler, construct a 30^{\circ} angle.

(3 marks)

For drawing the construction arcs to make a 60 degree angle

(1)

For drawing the arcs to construct an angle bisector

(1)

For completing the construction of the 30 degree angle

(1)

You have now learned how to:

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