GCSE Maths Geometry and Measure Angle Rules

Angles on a Straight Line

# 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 angles in a quadrilateral:

2. Subtract the angle sum from 180°.
3. Form and solve the equation.

### How to find missing angles on a straight line ## Angles on a straight line examples

### Example 1: obtuse angles

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

$78^{\circ}$

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.

$154^{\circ}$

$180-154=26^{\circ}$

$x=26^{\circ}$

### Example 3: right angles

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

$90+47= 137$

$180-137=43^{\circ}$

$x=43^{\circ}$

### Example 4: vertically opposite angles

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

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

$112$

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

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

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.

$60+3x+x+10+2x+20=6x+90$

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.

$90+45=135$

$180-135=6x$

### 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.

### Practice angles on a straight line questions

1. AOB is a straight line. Calculate the size of angle x . 95^{\circ} 25^{\circ} 115^{\circ} 295^{\circ} 180-65=115^{\circ}

2. AOB is a straight line. Calculate the size of angle x . 238^{\circ} 58^{\circ} 32^{\circ} 122^{\circ} 180-122=58^{\circ}

3. Calculate the size of the angle 2x . Hence find the value of x . 26^{\circ} 52^{\circ} 76^{\circ} 116^{\circ} 2x+90+38=180

2x+128=180 4. AB and CD are straight lines. Calculate the size of angle BOD. Hence find the value of x . 8^{\circ} 40^{\circ} 220^{\circ} 12^{\circ} 5x+140=180 5. AOB is a straight line. By finding the value for x , calculate the size of each angle in the diagram below. 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}  The circle with centre C has a tangent at point O. Calculate the value of x correct to 2 decimal places. 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)

360\div5=72^{\circ}

(1)

180-72=108^{\circ}

(1)

2.

(a) 3 straight lines intersect to form a triangle. Calculate the size of angle x .
(b) What type of triangle is ABC?

(4 marks)

a)

180-131=49

(1)

180-87=93

(1)

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

(1)

b)

A scalene triangle.

(1)

3. AB and CD are parallel lines. OE transects both lines. Calculate the value of x .

(3 marks)

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)

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

## Still stuck?

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