GCSE Maths Geometry and Measure

Angles

Supplementary Angles

# Supplementary Angles

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

There are also angles 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 supplementary angles?

Supplementary angles are two angles that add up to 180 degrees. They do not have to be adjacent or share a vertex.

E.g.

When we add together supplementary angles we get a straight line. This is because a straight line is 180 degrees; the same as the total of supplementary angles.

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:

1By giving the angle a ‘name’ which is normally a lowercase letter/symbol such as a, x or y or the greek letter ϴ (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 see the next diagram for an example of where a and b do not equal 180° because they do not meet at one single point on the straight line, i.e. they do not share a vertex and are not adjacent to one another:

• Angles around a point equal 360°:

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 supplementary angles

In order to solve problems involving supplementary angles:

1. Identify which angles are supplementary.
If appropriate write this down using angle notation e.g. AOB + BDE = 180
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.

## Supplementary Angles examples

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

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

1. Identify which angles are supplementary.

The two angles are supplementary and therefore equal 180°:

$x+y=180$

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&=180 \\ x+17&=180 \hspace{3cm} \text{Replace y with the value 17} \\ x&=163 \hspace{3cm} \text{Subtract 17 from both sides} \end{aligned}

4Clearly state the answer using angle terminology.

The size of the other angle is 163°.

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

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

The two non-identified angles are supplementary and therefore equal 180°.

You are being asked to find the smaller angle.

If you call the first angle a then the other angle must be 2a as ‘one is double the size of the other.’ Therefore a + 2a = 180. We can now solve this equation

\begin{aligned} a+2 a&=180 \\ 3 a&=180 \hspace{3cm} \text{simplify the equation} \\ a&=60 \hspace{3cm} \text{divide each side of the equation by 3} \end{aligned}

The two angles are therefore of size 60° and 120°.

The smaller angle is 60°.

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

ABC is a right-angled triangle. Which of the following pairs of angles are supplementary?

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

You are trying to find the angles that are supplementary. Therefore we are looking for two angles that when added together equal 180°.

As angles that are supplementary equal 180° you know that two angles that lie on a straight line are therefore supplementary. BDC is a straight line and therefore the two angles either side of D on that lie are supplementary.

Angles BDA and CDA are supplementary.

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

Angles A and B are supplementary to one another.

$A = 3x-29$
$B = 3x +17$

Find the size of angle B.

The two angles given as A and B are supplementary and therefore equal 180°.

Therefore A + B = 180°.

Find the size of angle B.

We can create an equation from the information given:

\begin{aligned} 3 x-29+3 x+17&=180 \\ 6 x-12&=180 \\ 6 x&=192 \\ x&=32 \end{aligned}

Remember you need to find the value of angle B so we substitute the value x = 32 into the expression for angle B:

\begin{aligned} B&=3x+17\\ B&=3(32)+17\\ B&=96+17\\ B&=113 \end{aligned}

Angle B = 113°.

### Example 5: identifying supplementary angles

AB and CD are parallel. Which pair of angles are supplementary,

HGD and GFB’ or ‘HGD and HGC ’?

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

You are trying to find the angles that are supplementary. Therefore you are looking for two angles that when added together equal 180°. Remember you are given a choice of ‘HGD and GFB’ or ‘HGD and HGC’.

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 supplementary.

HGD and GFB.

The angles are both obtuse angles therefore cannot add together to be 180°.

They are also congruent angles and known as corresponding angles because of their relationship within the two parallel lines.

HGD and HGC.

The two angles here lie on a straight line and are therefore equal to 180° and are supplementary.

HGD and HGC are supplementary angles.

### Example 6: identifying supplementary angles

Which angles in the below trapezium ABCD are supplementary? You must give your answers using correct angle notation.

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

You are trying to find the angles that are supplementary. Therefore we are looking for two angles that when added together equal 180°.

A trapezium has one set of parallel lines. From your prior knowledge of properties of a trapezium you know that, when added together, adjacent angles are equal to 180° and are therefore supplementary.

They are also known as co-interior angles because of their relationship with the two parallel lines.

There are two sets of supplementary angles

• Angles ABC and BCD

### Common misconceptions

These are some of the common misconceptions around the above angle rules

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

Supplementary 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 supplementary angles questions

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

x=43

x=133

x=47

x=313

The two angles are supplementary, so they must have a sum of 180 and 47+133=180 .

2. Two angles ‘ x and y ‘ are supplementary and one of them is 123^{\circ} . What is the size of the other angle?

x=53

x=123

x=57

x=237

The two angles are supplementary, so they must have a sum of 180 and 57+123=180 .

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

20^{\circ}

45^{\circ}

18^{\circ}

72^{\circ}

The angles make a 180 degree angle. There are three parts in one angle and one part in the other, so four parts in total. If we divide 180 by 4 we get 45 , so this is the size of the smaller angle.

4. Two angles are supplementary. One is 4x – 40 and the other is 5x – 50 . Find the value of x .

x=30

x=36

x=45

x=90

The sum of the two angles must equal 180 , so the equation we must solve is 9x-90=180 . Using the standard methods for solving a linear equation gives the solution x=30 .

5. Which angles are supplementary in the diagram below:

ACD and DBC

ABD and DBC

DAC and ACD

DAB and DBC

Since ABD and DBC meet at a point on a straight line, they must be supplementary as angles on a straight line add up to 180 .

6. How many pairs of supplementary angles does a parallelogram have?

 0

 1

 2

 4

By considering angle rules, we know there are four pairs of co-interior angles in a parallelogram. Since co-interior angles add up to 180 they are supplementary.

### Supplementary angles GCSE questions

1. Find the size of the angle marked x .

(2 marks)

 180 - 112

(1)

 68^{\circ}

(1)

2.

(a) Which two pairs of angles are supplementary in this trapezium?

(3 marks)

(1)

b)  180 - 58

(1)

 122^{\circ}

(1)

3. Work out the size of the smaller angle.

(4 marks)

  7x - 4 + x + 16 = 180

(1)

  8x + 12 = 180

(1)

 \begin{aligned}
8x&=168\\
x&=21
\end{aligned} 

(1)

 21 + 16 = 37

(1)

## Learning checklist

You have now learned how to:

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

## Still stuck?

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