Angle rules

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Types of angles

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GCSE Maths Geometry and Measure Angles

Angle Rules

Here we will learn about angle rules including how to solve problems involving angles on a straight line, angles around a point, vertically opposite angles, complementary angles and supplementary angles.

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

What are angle rules?

Angle rules enable us to calculate unknown angles:

Angles on a straight line equal 180º
Angles on a straight line always add up to 180^º.

E.g.

Here the two angles are labelled 30º and 150^º. When added together they equal 180^ºand therefore lie on a straight lie. These angles share a vertex.

However, below we can see an example of where two angles do not equal 180º:

This is because they do not share a vertex and therefore do not lie on the same line segment.

Note: you can try out the above rule by drawing out the above diagrams and measuring the angles using a protractor.

Step-by-step guide: Angles on a straight line

Angles around a point equal 360^o
Angles around a point will always equal 360º.

E.g.

The three angles above share a vertex and, when added together equal 360^o.

Step-by-step guide: Angles around a point

Supplementary angles
Two angles are supplementary when they add up to 180º , they do not have to be next to each other.

E.g.

These two angles are supplementary because when added together they equal 180º.

Step-by-step guide: Supplementary angles

Complementary angles
Two angles are complementary when they add up to 90º , they do not have to be next to each other.

E.g.

These two angles are supplementary because when added together they equal 90º.

Step-by-step guide: Complementary angles

Vertically opposite angles
Vertically opposite angles refer to angles that are opposite one another at a specific vertex and are created by two lines crossing.

E.g.

Here the two angles labelled ‘a’ are equal to one another because they are ‘vertically opposite’ at the same vertex.

The same applies to angles labelled as ‘b’.

Step-by-step guide: Vertically opposite angles

When solving problems involving angles sometimes we use more than the above rules. Below you will see a range of problems involving angles with links to lessons that will go into more detail with more complex questions.

What are angle rules?

Keywords

It is important we are familiar with some key words, terminology and symbols for this topic:

Angle: defined as the amount of turn round a common vertex.

Vertex: the point created by two line segments (plural is vertices).

We normally label angles in two main ways:

1By giving the angle a ‘name’ which is normally a lower case letter such as a, x or y or the greek letter θ (theta). See below for an example:

This image has an empty alt attribute; its file name is Angle-Rules-image-1.svg

2By referring to the angle as the three letters that define the angle. The middle letter refers to the vertex at which the angle is e.g. see below for the angle we call ABC:

This image has an empty alt attribute; its file name is Angle-Rules-image-2.svg

How to use angle rules

In order to solve problems involving angles you should follow these steps:

Identify which angle you need to find.
Identity which angle rule/s apply to the context and write them down.
Solve the problem using the above angle rule/s. Give reasons where applicable.
Clearly state the answer using angle terminology.

How to use angle rules

Angle rules worksheet

Get your free Angle Rules worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Angle rules worksheet

Get your free Angle Rules worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Angle rules examples

Example 1: angles on a straight line

Find angles a and b.

Identify which angle you need to find (label it if you need to).

You need to find angles labelled a and b.

2Identify which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

Angles on a straight line at the same vertex always add up to 180^o.

Notice how angles a and b do not share a vertex.

3Solve the problem using the above angle rule/s. Give reasons where applicable.

\[\begin{aligned} a+110&=180 \\ a=&70 \end{aligned}\]

\[\begin{aligned} b+55&=180 \\ b&=125 \end{aligned}\]

4Clearly state the answer using angle terminology.

Angle a = 70°

Angle b = 125°

Example 2: angles around a point

Find the size of θ:

Identify which angle you need to find (label it if you need to).

You need to find the angle labelled θ (theta).

Identity which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

All the angles are around a single vertex and we know that angles around a point equal 360^o.

Solve the problem using the above angle rule/s. Give reasons where applicable.

\[\begin{aligned} \theta+120+95+30&=360 \\ \theta+245&=360 \\ \theta&=115 \end{aligned}\]

Clearly state the answer using angle terminology.

\[\text{Angle } \theta = 115^{\circ}\]

Example 3: using supplementary angles

Two angles are supplementary and one of them is 127°. What is the size of the other angle?

Identify which angle you need to find (label it if you need to).

You need to find the other angle in a pair of supplementary angles where one is 127°. We will call this angle ‘a’.

Identity which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

Supplementary angles add up to 180°.

Solve the problem using the above angle rule/s. Give reasons where applicable.

\[\begin{aligned} 127 +a &= 180\\ a&=53\\ \end{aligned}\]

Clearly state the answer using angle terminology.

The other angle is 53°.

Example 4: using complementary angles

If two angles are complementary and one of them is 34°, what is the size of the other angle?

Identify which angle you need to find (label it if you need to).

You need to find the other angle in a pair of supplementary angles where one is 34°. We will call this angle ‘b’.

Identity which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

Complementary angles add up to 90°.

Solve the problem using the above angle rule/s. Give reasons where applicable.

\[\begin{aligned} 34 + b &=90\\ b &= 56\\ \end{aligned}\]

Clearly state the answer using angle terminology.

The other angle is 56°.

Example 5: vertically opposite angles

Find angle BCD.

Identify which angle you need to find (label it if you need to).

Find the angle at vertex C made from lines CD and CB. The diagram shows it labelled as x

Identity which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

Angle x and 80^oare vertically opposite one another at a vertex which has been created by two lines crossing.

Vertically opposite angles are equal to one another

Solve the problem using the above angle rule/s. Give reasons where applicable.

x = 80 because they are vertically opposite

Clearly state the answer using angle terminology

Angle BCD = 80°

Example 6: Applying multiple rules to solve a problem

In the diagram below:

Angle AOB is a right angle.

AOE and EOD are complementary angles.

Angle AOE is 50 degrees.

Find angle COD.

Identify which angle you need to find (label it if you need to).

Let’s start by labelling the diagram. We have labelled the angle we are trying to find x.

Identity which angle rule/s apply to the context and write them down (remember multiple rules may be needed).

Two angles are complementary when they add up to 90^o.

Angles around a point will always equal 360^o.

Angles on one part of a straight line always add up to 180^o.

Vertically opposite angles are equal.

Solve the problem using the above angle rule/s. Give reasons where applicable.

Note: there are multiple ways we can solve this problem. Below is just one method.

As AOE and EOD are complementary angles then they must equal 90 degrees. Therefore:

\[\begin{aligned} 50+E O D&=90 \\ E O D&=40 \end{aligned}\]

EOD and BOC are vertically opposite. Therefore:

\[\begin{aligned} B O C&=E O D \\ B O C&=40 \end{aligned}\]

Angles around a point are equal to 360 degrees. Therefore:

\[\begin{aligned} A O B+A O E+E O D+B O C+x=360 \\ 90+50+40+40+x=360 \\ 220+x=360 \\ x=140 \end{aligned}\]

Clearly state the answer using angle terminology.

Angle COD = 140°

Common misconceptions

Incorrectly labelling angles
Misuse of the ‘straight line’ rule where angles do not share a vertex
Mixing up the rules for supplementary and complementary angles
Finding the incorrect angle due to misunderstanding the terminology

Practice angle rules questions

x=30^{\circ}

x=60^{\circ}

x=90^{\circ}

x=180^{\circ}

Using angle on a straight line we have $180 – (90 + 30) = 60^{\circ}$

Yes

The sum of these angles is $360$ .

a=30^{\circ}

a=60^{\circ}

a=90^{\circ}

a=50^{\circ}

Using angles around a point, we have $360 – (125 + 125 + 50) = 360 – 300 = 60^{\circ}$

Yes

The sum of these angles is not $180$ .

Yes

The sum of these angles is $90$ .

z=115^{\circ}

z=60^{\circ}

z=65^{\circ}

z=55^{\circ}

Vertically opposite angles are equal

Angle rules GCSE questions

1. Work out the size of angle $z$ .

(2 marks)

Show answer

360-169-83

(1)

108^{\circ}

(1)

(a) Find the size of angle $a$ .
(b) Find the size of angle $b$ .

(3 marks)

Show answer

49^{\circ}

(1)

180-49

(1)

131^{\circ}

(1)

3. Work out the size of angle $x$ . Give reasons for your answer.

(3 marks)

Show answer

180-90-57=33

(1)

Angles on a straight line add up to $180^{\circ}.$

(1)

x=33^{\circ}

(1)

Learning checklist

You have now learned how to:

Use conventional terms and notation for angles

Apply the properties of angles on a straight line, around a point and on vertically opposite angles

Apply angle facts and properties (e.g. supplementary and complementary angles) to solve problems

The next lessons are

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Introduction

What are angle rules?

Keywords

How to use angle rules

Angle rules worksheets

Angle rules examples

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Example 1: angles on a straight Line Example 2: angles around a point Example 3: using supplementary angles Example 4: using complementary angles Example 5: vertically opposite angles Example 6: applying multiple rules to solve a problem

Common misconceptions

Practice angle rules questions

Angle rules GCSE questions

Learning checklist

Next lessons

Still stuck?