Angles on a straight line

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In order to access this I need to be confident with:

Types of angles Collecting like terms Solving equations

This topic is relevant for:

GCSE Maths Geometry and Measure Angles Angle Rules

Angles On A Straight Line

Here we will learn about angles on a straight line, including the sum of angles on a straight line, how to find missing angles, and using these angle facts to generate equations and solve problems.

There are also angles on a straight line 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 angles on a straight line?

Angles on a straight line relate to the sum of angles that can be arranged together so that they form a straight line.

Angles on a straight line add to 180°.

Let us look at this visually:

Let’s take the three angles of a, b, and c.

a is a right angle as it measures 90°, b = 38°, and c = 52°

If we move these three angles so that each vertex meets, we get an arrangement that looks like this:

These three angles create a straight line.

By adding together a = 90°, b = 38°, and c = 5° we can see the sum of the angles on a straight line is 180°.

We can also look at this in reverse by considering how many degrees it takes to do a full turn.

If you stand facing North, and turn to face East, you have turned 90° clockwise. A further turn of 90° clockwise will mean you would now be facing South. Repeating this turn twice more, you would first face West, and then back to North (the full turn). Half a turn is therefore 2 lots of 90° which is 180°.

We can therefore state that the sum of angles on a straight line is equal to 180°. If we split any straight line into smaller angles, all of these angles would add to make 180°, the same as with a triangle.

Step by step guide: Angles in a triangle

Angles in a straight line are a problem solving tool for many geometric problems. These include: properties of shapes, circle theorems, angles in parallel lines, calculating angles in shapes (interior and exterior angles), trigonometry and more.

What are angles on a straight line?

How to find missing angles on a straight line

In order to find missing angle on a straight line:

Add all known angles.
Subtract the angle sum from 180°.
Form and solve the equation.

How to find missing angles on a straight line

Angles on a straight line worksheet

Get your free angles on a straight line worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Angles on a straight line worksheet

Get your free angles on a straight line worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Angles on a straight line examples

Example 1: obtuse angles

AB is a straight line through O. Calculate the missing angle x.

Add all known angles.

\[78^{\circ}\]

2Subtract the angle sum from 180°.

\[180-78=102^{\circ}\]

\[x=102^{\circ}\]

Example 2: acute angles

AB is a straight line through O. Calculate the missing angle x.

Add all known angles.

\[154^{\circ}\]

Subtract the angle sum from 180°.

\[180-154=26^{\circ}\]

\[x=26^{\circ}\]

Example 3: right angles

AB is a straight line through O. Calculate the missing angle x.

Add all known angles.

\[90+47= 137\]

Subtract the angle sum from 180°.

\[180-137=43^{\circ}\]

\[x=43^{\circ}\]

Example 4: vertically opposite angles

AB and CD are straight lines. Calculate the value x.

Add all known angles.

As CD is a straight line, we are going to find angle BOD, then angle AOD.

\[112\]

Subtract the angle sum from 180°.

\[180-112=68^{\circ}\]

\[BOD=68^{\circ}\]

\[AOD = 180 – BOD = 180-68 = 112^{\circ}\]

\[AOD=112^{\circ}\]

Form and solve the equation.

As AOD = 112° and AOD = 4x, we can equate them to find the value of x:

This example shows that vertically opposite angles are equal.

Example 5: forming and solving equations

AB is a straight line through O. Calculate the size of all the angles that make up the line AB.

Add all known angles.

\[60+3x+x+10+2x+20=6x+90\]

Form and solve the equation.

Angles on a straight line add up to 180° so we have the equation 6x+90=180.

As x = 15, substitute this into each angle to find their values:

\[3x=3\times15=45^{\circ}\]

\[x+10=15+10=25^{\circ}\]

\[2x+20=2\times15+20=50^{\circ}\]

We can check the solution by adding up the angles:

\[60+45+25+50=180^{\circ}\]

Example 6: circles and tangents

AB is a tangent to the circle with centre C. The tangent intersects the circle at the point O on the circumference. Use this information to calculate the value of x.

Add all known angles.

\[90+45=135\]

Form and solve the equation.

\[180-135=6x\]

Common misconceptions

The sum of angles on a straight line is equal to 360°

The angle sum is remembered incorrectly as 360°, rather than 180°. The sum of angles on a straight line is half of a full turn, which is 180°.

Solving equations

There are many misconceptions around forming and solving equations. See the lesson on Solving Equations for further information.

Step by step guide: Solving equations

Using a protractor

When you are asked to calculate a missing angle, a common error is to use a protractor to measure the angle. When using angle facts to determine angles, diagrams are deliberately not drawn to scale unless the angle is 90° or 180° as these are important angles to recognise. You should not use a protractor for this style of question.

Angles on a straight line 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 angles on a straight line questions

95^{\circ}

25^{\circ}

115^{\circ}

295^{\circ}

$180-65=115^{\circ}$

238^{\circ}

58^{\circ}

32^{\circ}

122^{\circ}

$180-122=58^{\circ}$

26^{\circ}

52^{\circ}

76^{\circ}

116^{\circ}

$2x+90+38=180$

$2x+128=180$

8^{\circ}

40^{\circ}

220^{\circ}

12^{\circ}

$5x+140=180$

x=35^{\circ}, COD=175^{\circ}, DOE=15^{\circ}, EOB=165^{\circ}

x=12.5^{\circ}, COD=72.5^{\circ}, DOE=7.5^{\circ}, EOB=75^{\circ}

x=11.1^{\circ}, COD=55.5^{\circ}, DOE=8.9^{\circ}, EOB=69.4^{\circ}

x=2.05^{\circ}, COD=10.2^{\circ}, DOE=17.95^{\circ}, EOB=33.18^{\circ}

19.11^{\circ}

10.59^{\circ}

9.11^{\circ}

5.29^{\circ}

Angles on a straight line GCSE questions

1. Given that the sum of exterior angles of any regular polygon is equal to $360^{\circ}$ , calculate the interior angle of a regular pentagon.

(2 marks)

Show answer

$360\div5=72^{\circ}$

(1)

$180-72=108^{\circ}$

(1)

(a) $3$ straight lines intersect to form a triangle.

Calculate the size of angle $x$ .
(b) What type of triangle is $ABC$ ?

(4 marks)

Show answer

$180-131=49$

(1)

$180-87=93$

(1)

$x=180-(93+49)=38^{\circ}$

(1)

A scalene triangle.

(1)

3. $AB$ and $CD$ are parallel lines. $OE$ transects both lines.

Calculate the value of $x$ .

(3 marks)

Show answer

$4x+22.4=180$

(1)

$4x=157.6$

(1)

$x=39.4^{\circ}$

(1)

4. $AB$ is a straight line that is a tangent to the circle at point $O$ . Show that $OCD$ is a right angle triangle.

(3 marks)

Show answer

$ODC=180-112=68^{\circ}$

(1)

$180-(68+22)=90^{\circ}$

(1)

One angle in triangle $OCD$ is equal to $90^{\circ}$ so it is a right angle triangle.

(1)

Learning checklist

You have now learned how to:

Find unknown angles on a straight line

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