GCSE Maths: Angles in Triangles & Quadrilaterals
Introduction
When it comes to angles in triangles and quadrilaterals there are a few essential facts that you need to know:
A triangle is a three sided shape. There are two types of triangles that you should be aware of:
- Equilateral – all lengths of the triangle are the same and all angles are also equal.
- Isosceles – two lengths are the same and the bottom two angles are equal.
In all cases, the sum of angles in a triangle adds up to 180°.
A quadrilateral is any four sided shape and the sum of angles is 360°.
In order to find missing angles in a triangle or a quadrilateral you will also need to use the fact that the sum of angles on a straight line is equal to 180°.
Angles in Triangles and Quadrilaterals - An Example
Take a look at the following angles in triangles and quadrilaterals question where you want to find the missing angles:
It is important that you understand what a diagram is telling you. The two dashes on the edge of the triangle are telling you that the lengths are the same. This means that you are dealing with an isosceles triangle.
The question does not provide any additional information so it is important to understand the information in the diagram.
Clearly you can see a triangle but what type? The dashes on the edge of the triangle are telling you that the lengths are the same. Because if two lengths are the same then the type of triangle is isosceles. This then means that the angles at the bottom are also the same.
In order to find the value of x consider the following part of the diagram only:
This is where you need to use the fact the same of angles on a straight line is equal to 180°. In this case 126° + x°=180° ∴x=54°.
Now concentrating on the triangle only:
Because the angles at the base are equal then the sum at the bottom of the angles is 108°. This is based on the fact that angles on a straight line = 180°.
With these types of questions you are relying on your knowledge of angles, facts about triangles and also facts about quadrilaterals or indeed knowledge of polygons. These topics all link together. If you are finding this topic a little tricky then you can use an online maths tutor for GCSE. Additional guidance can be given on how best to answer these types of questions and also provide the necessary techniques in the run up to the final examinations.
Angles in Triangles and Quadrilaterals - Another Example
Take a look at the following angles in triangles and quadrilaterals question where you want to find the missing angles:
You will not always be able to instantly find any unknown angles straight away. Be prepared to calculate any other angles first before you can actually answer the question.
This shows a quadrilateral and you need to remember that the sum of angles in any four sided shape is 360°.
The missing angle insider the quadrilateral can be found as follows:
360-78-119-105=58°In order to find the angle that is being asked for, x°, you need to use the fact that the sum of angles on a straight line adds up to 180°.
∴58+x=180 →x=122°
Question Practice
Try the following angles in triangles and quadrilaterals question on your own before looking at the solution.
Question Practice Solution
So how did you get on? Hopefully you found that the answer to be y= 48°, x= 30°
Looking at the triangle, in order to find the value of x you need to ask what type of triangle you are actually dealing with. Because there are two dashes on just two of the edges this means that the triangle is isosceles so the angles at the bottom must be the same i.e. they must both have an angle of x.
This means that 120+x+x=180 ∴120+2x=180
2x = 60, x = 30Next you need to deal with the quadrilateral and just as in a previous example, just because an angle is not asked for does not mean that you do not need to work out its value:
The question has not specifically asked you, but you first need to find the missing angle within the quadrilateral before you can proceed further with the question.
Let the missing angle be z so 30+z=180 ∴z=150°
Three angles of the quadrilateral are now known so the angle y can be found as follows:
108+54+150+y=360
312+y=360
∴y=48°
From the examples that you have seen so far, the diagrams, for some, can be a little off putting but you need to remember that in most cases you are dealing with a triangle. It is important to use the information within the diagram and as seen you are more than likely going to need to find additional angles before you can find the actual angle that you want.
As a part of your GCSE Maths Revision it is important that you are doing questions that link various topics. This will help to reinforce other topic areas and help you develop your own understanding of how to answer questions effectively and correctly.
Keep trying the examples again if you have any uncertainties and check your answer with our solutions.