# Complementary Angles

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.

## What are complementary angles?

Complementary angles are two angles that add up to 90 degrees. They do not have to be adjacent or share a vertex.

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:

1. By giving the angle a ‘name’ which is normally a lowercase letter/symbol such as a, x or y 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 ABC:

• Angles on a straight line equal 180°:

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

However in the diagram below a and b do not equal 180° because they are not on one single part of a straight line, i.e. they do not share a vertex and are not adjacent to one another:

Angles around a point will always equal 360°. See the diagram for an example where angles a, b and c are equal to 360°:

• 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 ‘a’ are equal to one another because they are ‘vertically opposite’ at the same vertex.

The same applies to angles labelled as ‘b’.

Note: Sometimes these are called vertical angles.

## How to solve problems involving complementary angles

In order to solve problems involving complementary angles:

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

## Complementary Angles examples

### Example 1: finding an angle which is complementary to another

Two angles ‘x and y‘ are complementary and one of them is 17°. What is the size of the other angle?

1. Identify which angles are complementary.

The two angles are complementary and therefore equal 90°

$x+y=90$

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

Find the angle that is not 17°.

3Solve the problem and give reasons where applicable.

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

4Clearly state the answer using angle terminology.

The size of the other angle is 73°.

### Example 2: finding an angle which is complementary to another

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.

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.

### Example 3: finding complementary angles from a diagram

ABC is a right-angled triangle. Which of the following pair of angles is complementary?

Identify which angles are complementary.

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.

### Example 4: finding a given angle using complementary angles

Angles A and B are complementary to one another.

$A = 2x – 8$
$B = 5x – 7$

Find the size of angle A.

Identify which angles are complementary.

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.

### Example 5: identifying complementary angles within a polygon

ABCD is a rectangle. Which of these is a pair of complementary angles: DAB and DCB or DAC and CAB?

Identify which angles are complementary.

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.

### Example 6: complementary angles within a polygon

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

Identify which angles are complementary.

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.

### Common misconceptions

• 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:

### Practice complementary angles questions

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 .

### Complementary Angles GCSEquestions

1. Find the size of the angle marked a :

(2 marks)

90-61

(1)

29^{\circ}

(1)

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

(3 marks)

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)

3x + 5x + 2 = 90

(1)

8x + 2 = 90

(1)

\begin{aligned} 8x&=88\\ x&=11 \end{aligned}

(1)

5 \times 11 + 2 = 57

(1)

## Learning checklist

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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