GCSE Maths: Angles in Parallel Lines
Introduction
With GCSE Maths you need to recall a number of facts with regards to angles in parallel lines. One of the most common types of questions is dealing with questions that involve parallel lines.
It is important that you are able to recall the exact name of these types of angles and are able to remember their properties such as when angles are equal or when angles sum to 180°.
There are a number of important factors that you need to remember when it comes to calculating angles.
GCSE Maths Angles In Parallel Lines Facts
Opposite angles are always equal.
The above shows two parallel lines and what you can see is something that looks like the letter Z. This is referred to as the Z shape angle or the correct mathematical term is alternate angles. The angles in an alternate angle are equal.
In this diagram we again have a set of parallel lines that if you look carefully you will see that the lines resemble an upside down letter F. This F shape angle is referred to as a corresponding angle and corresponding angles are equal.
From the diagram shown above you what resembles a letter C shape. This letter C shape is known as a co-interior angle and the sum of these angles is 180°.
You need to know the correct mathematical names for these shapes and these shapes can appear as you would see them in the alphabet, upside down as well as backwards. The challenge in a gcse maths exam is to be able to identify them correctly.
You should also be aware of the fact that angles on a straight line add up to 180°.
Example
Take a look at the following question:
It is always important to use the information in the diagram as effectively as possible and when it comes to questions like this there is usually more than one way to go about them.
As labelled on the diagram, the angle can be found next to the angle of 47°. Using the fact that the angles on line add up to 180° this angle would have a value of 180°- 47°=133°.
As drawn on the diagram you can see a “F” shape so these are corresponding angles. So angle x is 133°.
Example
Take a look at the following question:
There is very little information given in this question so the diagram needs to be maximised. You can see two lines that are parallel to each other and also there is a triangle with two lines on two of the edges. This is telling you that the triangle is isosceles.
From the diagram diagram you can see a “Z” shape (from B to A to E to D) which is an alternative angle. So the angle at E inside the triangle will have a value of 38°.
Now you know that the triangle is isosceles so the angles at the bottom must be the same. So 180°-38°=142°. This is the sum of the two angles at the bottom and because they are equal, each one will be 71°.
In order to find the value of x you need to use the fact for angles on a straight line. x+71=180 ∴x=109°.
Question Practice
Try the following question on your own before looking at the solution.
Question Practice Solution
You should have hopefully found that the missing angle has a value of 25°.
The diagram is not the easiest to look at but you can clearly see two parallel lines. What you need to do is to look at one of the angles that are given and from there see if you can spot a “F”, “Z” or “C” shape.
From the diagram below a “F” shape has been found when going from C to B to F to H.
This means that a corresponding angle exists, so the angle at F is 53°. Again at F, using angles on a straight line = 180°, this angle would be 127°.
At E, you have opposite angles, so the angle is 28°.
You can now focus on the triangle EFH and use the fact the sum of angles in a triangle add up to 180° ∴x=25°
Calculating angles in parallel lines can appear in both the foundation and higher gcse maths papers. There is never one set method to find a missing angle, but what is important is that you are labelling any diagrams correctly and you are using the correct mathematical terms when providing any reasoning to your answer.
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