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Angle rules Types of anglesAngles on a straight line

This topic is relevant for:

Here we will learn about **complementary angles** including how to find missing angles by applying knowledge of complementary angles to a context.

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

**Complementary angles** are two angles that add up to

E.g.

When we add together complementary angles we get a right angle.

Before we start looking at specific examples it is important we are familiar with some **key words**, **terminology, rules **and **symbols **required** **for this topic:

**Angle:**defined as the amount of turn around a common vertex.**Vertex**: the point created by two line segments (*plural is vertices)*.

**How to label an angle**:

We normally label angles in two main ways:

- By giving the angle a ‘name’ which is normally a lowercase letter/symbol such as
a ,x ory or the greek letter (theta).

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 the diagram for the angle we call

**Angles on a straight line equal 180°**:

Angles on one part of a straight line **always add up to **

However in the diagram below **not **equal

Angles around a point will always equal 360°. See the diagram for an example where 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. See below for an example:

Here the two angles labelled

The same applies to angles labelled as

**Note:** Sometimes these are called vertical angles.

In order to solve problems involving complementary angles:

**Identify which angles are complementary.***If appropriate write this down using angle notation e.g.*AOC + BOC = 90° **Clearly identity which of the unknown angles the question is asking you to find the value of.****Solve the problem and give reasons where applicable.****Clearly state the answer using angle terminology.**

Get your free complementary and supplementary angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free complementary and supplementary angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREETwo angles ‘

**Identify which angles are complementary**.

The two angles are complementary and therefore equal

\[x+y=90\]

2**Clearly identity which of the unknown angles the question is asking you to find the value of**.

Find the angle that is not

3**Solve the problem and give reasons where applicable**.

\[\begin{aligned}
x+y&=90 \\
x+17&=90 \\
x&=73
\end{aligned}\]

4**Clearly state the answer using angle terminology**.

The size of the other angle is

Two angles are complementary. One is double the size of the other. What is the size of the smaller angle?

**Identify which angles are complementary**.

The two non-identified angles are complementary and therefore equal

**Clearly identity which of the unknown angles the question is asking you to find the value of**.

You are being asked to find the smaller angle.

**Solve the problem and give reasons where applicable**.

If you call the first angle ‘

We can now solve this equation:

\[\begin{aligned}
a+2a&=90 \\
3 a&=90 \hspace{3cm} \text{simplify the equation} \\
a&=30 \hspace{3cm} \text{divide each side of the equation by 3}
\end{aligned}\]

The two angles are therefore of size

**Clearly state the answer using angle terminology. **

The smaller angle is

**Identify which angles are complementary**.

In this question you are not told which angles are complementary.

**Clearly identity which of the unknown angles the question is asking you to find the value of**.

You are trying to find the angles that are complementary. Therefore we are looking for two angles that when added together equal

**Solve the problem and give reasons where applicable**.

We know the sum of interior angles for a triangle is

**Clearly state the answer using angle terminology**.

Angles

** **Angles

\[A = 2x – 8\]

\[B = 5x – 7\]

Find the size of angle

**Identify which angles are complementary**.

The two angles given as

Therefore

**Clearly identity which of the unknown angles the question is asking you to find the value of**.

Find the size of angle

**Solve the problem and give reasons where applicable**.

We can create an equation from the information given:

\[\begin{aligned}
A+B&=90 \\
2 x-8+5 x-7&=90 \\
7 x-15&=90 \\
7 x&=105 \\
x&=15
\end{aligned}\]

Remember you need to find the value of angle A so we substitute the value

\[\begin{aligned}
A=2x-8\\
A=2(15)-8\\
A=30-8\\
A=22
\end{aligned}\]

**Clearly state the answer using angle terminology**.

Angle

**Identify which angles are complementary**.

In this question you are not told which angles are complementary.

**Clearly identity which of the unknown angles the question is asking you to find the value of**.

You are trying to find the angles that are complementary and therefore you are looking for two angles that when added together equal

**Solve the problem and give reasons where applicable**.

Below is the diagram (given in the question) where the two sets of angles have been labelled separately. This will help you spot which are complementary

The two angles labelled are both right angles as they are created by adjacent line segments of a rectangle. Therefore when added together they make

The diagram shows the two angles make a right angle and are therefore complementary

**Clearly state the answer using angle terminology**.

Angle

Can a parallelogram contain a pair of complementary angles? If so state their size.

**Identify which angles are complementary**.

In this question you are not told which angles are complementary.

**Clearly identity which of the unknown angles the question is asking you to find the value of**.

You are being asked to find whether a parallelogram contains a pair of angles that when added together equal to

**Solve the problem and give reasons where applicable**.

The interior angles at vertex

The interior angles at vertex

**Clearly state the answer using angle terminology**.

A parallelogram can contain complementary angles.

The two acute angles in the parallelogram would be

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

Complementary angles is part of our series of lessons to support revision on angle rules. You may find it helpful to start with the main angle rules 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:

1. Two angles ‘ x and y ‘ are complementary and one of them is 47^{\circ} . What is the size of the other angle?

x=33

x=43

x=133

x=313

If two angles are complementary, they add up to 90 , 43+47=90 .

2. Two angles ‘ x and y ‘ are complementary and one of them is 3^{\circ} . What is the size of the other angle?

x=3

x=87

x=93

x=177

If two angles are complementary, they add up to 90 , 3+87=90

3. Two angles are complementary. One is four times the size of the other. What is the size of the smaller angle?

20^{\circ}

36^{\circ}

18^{\circ}

72^{\circ}

The angles make up a 90 degree angle. There are four parts in one angle and one part in the other, so five parts in total. If we divide 90 by 5 we get 18 , so this is the size of the smaller angle.

4. Two angles are complementary. One is x – 16 and the other is 2x – 29 . Find the value of x .

x=16

x=29

x=45

x=90

The sum of the two angles must equal 90 , so the equation we must solve is 3x-45=90 . Using the standard methods for solving a linear equation gives the solution x=45 .

5. Two angles A and B are complementary. Find the difference between the two angles if: A=2x-33 and B= 5x-31

33^{\circ}

22^{\circ}

68^{\circ}

79^{\circ}

The sum of the two angles must equal 90 , so the equation we must solve is 7x-64=90 . Using the standard methods for solving a linear equation gives the solution x=22 so the two angles are 11^{\circ} and 79^{\circ}

6. Does a rectangle have any pairs of complementary angles?

Yes

No

Since each corner of a rectangle is a right angle, it is not possible for any two angles to add up to 90 .

1. Find the size of the angle marked a :

**(2 marks)**

Show answer

90-61

**(1)**

**(1)**

2. Are angles x and y complementary? Give reasons for your answer.

**(3 marks)**

Show answer

135 + 135 = 270

**(1)**

360-270 = 90 - angles in a quadrilateral add to 360^{\circ}

**(1)**

`Yes they are complementary as they add up to 90^{\circ} `

**(1)**

3. Find the size of the larger angle:

**(4 marks)**

Show answer

3x + 5x + 2 = 90

**(1)**

**(1)**

**(1)**

**(1)**

You have now learned how to:

- Use conventional terms and notation for angles
- Define angles that are complementary
- Apply the properties of complementary angles
- Apply angle facts and properties to solve problems

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