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Types of anglesThis topic is relevant for:

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.

**Angle rules **enable us to calculate unknown angles:

**Angles on a straight line equal 180****º**

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^{º }and 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**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 ‘

The same applies to angles labelled as ‘

**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.

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

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

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**.

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

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

DOWNLOAD FREEFind angles

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

You need to find angles labelled

**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

**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

Angle

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**.

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

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°.

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°.

Find angle

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

Find the angle at vertex

Angle ^{o }are 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**.

**Clearly state the answer using angle terminology**

Angle

In the diagram below:

- Angle AOB is a right angle.

AOE andEOD are complementary angles.

- Angle
AOE is50 degrees.

Find angle

**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

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

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

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

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

**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**

1. Find angle x

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}

2. Can angles 40^{\circ}, 100^{\circ}, 115^{\circ}, 105^{\circ} lie around a single point?

Yes

No

The sum of these angles is 360 .

3. Find angle a:

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}

4. Are angles 60^{\circ} \text{ and } 90^{\circ} supplementary angles?

Yes

No

The sum of these angles is not 180 .

5. Are angles 75^{\circ} \text{ and } 15^{\circ} complementary angles?

Yes

No

The sum of these angles is 90 .

6. Find angle z

z=115^{\circ}

z=60^{\circ}

z=65^{\circ}

z=55^{\circ}

Vertically opposite angles are equal

1. Work out the size of angle z .

**(2 marks)**

Show answer

360-169-83

**(1)**

**(1)**

2.

(a) Find the size of angle a .

(b) Find the size of angle b .

**(3 marks)**

Show answer

a)

49^{\circ}**(1)**

b)

180-49**(1)**

**(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)**

**(1)**

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

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