What you need to know

Two straight lines are parallel if they are always the same distance away from each other, no matter how long the lines are extended. In other words, they’re going the exact same direction and will never meet. Parallel lines are marked with matching arrows (see below).

If you draw a straight line that passes right through two parallel lines, a whole bunch of angles are formed. 8, to be specific, but the fact is that all of those angles are related to all the other angles in some way, and that’s what this topic is about.

Before that however, we must know two important facts: the angle around a point is always 360\degree, and the angle on a straight line is always 180\degree.

Furthermore, if there are several angles which together go all the way around a point, they will add up to make 360\degree, and similarly if there are several angles which together form a straight line, they will add to make 180\degree.

 

Here are all of the different pairings of angles, their names, and their relationships.

 

Note: vertically opposite angles don’t require the presence of the parallel lines – they occur when any two straight lines cross.

 

Some of these facts might look obvious, you might think the angles *look* the same, but if a question asks you to find a missing angle using these facts, it will almost certainly ask you to quote the fact you used, so make sure you know them all.

 

It’s important to understand that A and B, marked on the first diagram, are not the only pair of vertically opposite angles in picture. The same goes for all the other types of pairs and their diagrams, too.

Example: BD and EG are parallel lines. Find the angle marked x in the picture below. State which angle fact you used at each step.

 

These two angles do not go together in any of the pairs seen above, so to find x we will need to use more than one of our angle facts (each of which will be bolded, just to be clear).

 

There are multiple correct ways to do this, we’ll go through one.

 

Firstly, we will use the fact that angles on a straight line add to 180. Specifically, angle EFC and angle GFC add to make 180, so if we subtract EFC from 180, then we get the size of angle CFG.

So, we get

\text{Angle CFG } = 180 - 32 = 148\degree.

 

Now, looking at the diagram to the right (and it helps a lot to draw on your diagrams when doing these questions on paper), we can see that angle CFG and the missing angle x are corresponding angles.

 

Therefore, we get

 

x = 148\degree.

 

As mentioned, there are multiple ways to do this question. How else might you do it?

Example Questions

1) BD and EG are parallel lines. Find the angle marked x in the picture below. State which angle fact you use at each step.

 

Answer

Firstly, because angle HFG and angle EFC are vertically opposite, we get

 

\text{angle EFC } = 48\degree

 

Secondly, because angle EFC and angle BCA (angle x) are corresponding angles, we get

 

\text{angle BCA } = x = 48\degree

 

 

There are other possible methods for doing this question. As long as you’ve correctly applied angle facts, explained each step, and got the answer to be 48\degree, your answer is correct.

2) CE and FH are parallel lines. Find the angle marked x in the picture below. State which angle fact you use at each step.

 

Answer

Firstly, using the fact that angles FGJ and CDG are corresponding angles, we get

 

\text{angle CDG } = 121\degree.

 

Secondly, because angles on a straight line add to 180, and angles CDG and CDA are on a straight line, we get

 

\text{angle CDA } = 180 - 121 = 59\degree.

 

Thirdly, again using the fact that angles on a straight line add to 180, and angles CDA, BDE, and ADB (otherwise known as angle x) are on a straight line, we get

 

x + 50 + 59 = 180,\text{ so } x = 180 - 109 = 71\degree.

 

 

 

3) CF and GJ are parallel lines. Find the angle marked x in the picture below. State which angle fact you use at each step.

 

Answer

Firstly, because angles BEF and EHJ are corresponding angles, we get

\text{angle EHJ } = 39\degree.

 

Next, because angles EDH and DHG are alternate angles, we get

 

\text{angle DHG } = 76\degree.

 

Then, because angles DHG, DHE, and EHJ are angles on a straight line and angles on a straight line add to 180, we get

 

\text{angle DHE } = 180 - 76 - 39 = 65\degree

 

Finally, because angle DHE and angle x are vertically opposite angles, we get

 

x = 65\degree.

 

 

There are other possible methods for doing this question. As long as you’ve correctly applied angle fact, explained each step, and got the answer to be 71\degree, your answer is correct.

Whether you are a GCSE Maths tutor in Harrogate looking for AQA GCSE Maths geometry questions or a teacher in London looking for geometry homework sheets, the foundation resources on this page should be of great use. Take a look at our geometry worksheets and revision materials and see what you would incorporate into your teaching programme.