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Use Angle Properties in a Triangle

In this worksheet, students will use the key fact, that angles in a triangle always add to 180°, to find the value of unknown angles in triangles and identify triangles accurately.

'Use Angle Properties in a Triangle ' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Eduqas, AQA, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Illustration of the pyramids in EgyptPhoto of a musical triangleImage of the fire triangle

 

Oh yes, the humble triangle, we take it for granted.

Its shape has been used by Egyptian kings, musicians and in warning signs, to name just a few of its many uses!

 

There are lots of properties (facts) about triangles that people have learned over the centuries, and now it is your turn.

 

 

 Special Triangles

 

Diagram of special types of triangles

 

 Diagram of positions of right angles in triangles

 

We need to look out for the markings on these triangles, as these quite often tell us which type of triangle we are looking at.

e.g. A triangle marked |||, || and | on the sides show that each side and angle is different, so therefore it must be scalene.

e.g. The two red angles in the isosceles triangle, highlight the base angles which are the same.

 

 

A Key Triangle Fact:

Angles in a triangle always add up to 180°.

 

 

Using this knowledge, we can find missing angles in a triangle.

Let's look at how we can do this now. 

 

 

 

e.g. What is the value of angle f in the first triangle below? And the value of angles a and b in the second triangle?

 

Diagram of finding a missing angle in an isosceles triangle

 

From the markings on these triangles, we can see these are both isosceles triangles.

As a result, we know that the base angles must be the same.

 

 

 

e.g. What is the value of angle b in the diagram below? 

 

Diagram of finding an external angle of a triangle

 

Here, we have been asked to find an angle that is external to the triangle.

 

We need to put two rules we know together here - if we extend the base of the triangle we have a straight line.

What facts do we know about straight lines?

 

The angles on a straight line will also add up to 180°, so we can use this fact to find the missing exterior angle. 

 

 

 

Right then, let's put what we know into action now. 

 

In this activity, we will use the key fact, that angles in a triangle always add to 180°, to find the value of unknown angles in triangles, identify triangles accurately and solve problems involving triangles. 

A blue triangle

 

This triangle has two equal sides and two equal angles on the base.

 

What type of triangle is this?

 

Before you answer, try to explain why you think this is the case in your head or on a piece of paper. 

Scalene

Equilateral

Right-angle

Isosceles

Use your knowledge of different types of triangles to match the triangle terms on the left to their correct definitions on the right. 

Column A

Column B

Angles in a triangle...
show that it is an equilateral triangle
A scalene triangle...
has two sides and two angles which are the same
An isosceles triangle...
add up to 180°
An equilateral triangle...
has angles which are all the same size and sides w...
The same markings on each side of a triangle...
has angles that are all different sizes and length...

Use your knowledge of different types of triangles to match the groups of angles on the left to the scenario they describe on the right. 

Column A

Column B

38°, 38°, 104°
Isosceles triangle
60°, 60°, 60°
Equilateral triangle
90°, 57°, 33°
Scalene triangle
40°, 140°
External angle of a triangle

Review the diagram below:

Diagram of a triangle with an angle missing

Calculate the value of angle x in this triangle.

Column A

Column B

38°, 38°, 104°
Isosceles triangle
60°, 60°, 60°
Equilateral triangle
90°, 57°, 33°
Scalene triangle
40°, 140°
External angle of a triangle

Consider this new diagram:

Diagram of a triangle with two angles missing

Calculate the value of angles a and b in this triangle.

Column A

Column B

38°, 38°, 104°
Isosceles triangle
60°, 60°, 60°
Equilateral triangle
90°, 57°, 33°
Scalene triangle
40°, 140°
External angle of a triangle

Consider this triangle and its associated markings:

Picture of a green triangle with triangle notation

What type of triangle is this?

 

Before you answer, try to explain why you think this is the case in your head or on a piece of paper. 

 EquilateralIsoscelesScalene
This triangle is...

Review the equilateral triangle below and its associate markings:

 

Diagram of an equilateral triangle with notation

What is the size of the angles in this equilateral triangle?

 

Before you type your answer, try to explain why you think this is the value of the angles in your head or on a piece of paper. 

 EquilateralIsoscelesScalene
This triangle is...

This triangle is a scalene triangle:

 

Image of a yellow scalene triangle

 

What are the correct properties of this triangle shown in the list below?

Angles add up to 180°

Two sides and angles are the same

All angles and sides are different

All sides and angles are equal

Explore the triangle below:

 

Image of a triangle with a missing angle

 

Calculate the value of angle d in this triangle.

Angles add up to 180°

Two sides and angles are the same

All angles and sides are different

All sides and angles are equal

Which of the collections of angles below, accurately describe a valid triangle

 

a)  38°  57°  86°
b)  115°  25°  40°
c)  65°  55°  60°
d)  96°  48°  38°
a)

b)

c)

d)

  • Question 1

A blue triangle

 

This triangle has two equal sides and two equal angles on the base.

 

What type of triangle is this?

 

Before you answer, try to explain why you think this is the case in your head or on a piece of paper. 

CORRECT ANSWER
Isosceles
EDDIE SAYS
Oh no, there are no markings on this triangle... not to worry! If we apply our triangle facts, we know that a triangle with two equal sides and two equal angles is an isosceles triangle. In text books and exam papers, triangles will either be marked with their specific symbols or explicitly named. Either way, you will need to know the significance of these names or symbols so that you can use these facts to help you solve the problem you are being asked.
  • Question 2

Use your knowledge of different types of triangles to match the triangle terms on the left to their correct definitions on the right. 

CORRECT ANSWER

Column A

Column B

Angles in a triangle...
add up to 180°
A scalene triangle...
has angles that are all different...
An isosceles triangle...
has two sides and two angles whic...
An equilateral triangle...
has angles which are all the same...
The same markings on each side of...
show that it is an equilateral tr...
EDDIE SAYS
Have you got the key facts straight in your head now? It can get a little confusing at first, but once we have them memorised and understood, our work on triangles becomes so much easier! The prefix 'tri-' means three, so with triangles we can ask ourselves three questions to help us decide which type we are observing: 1. Angles - Do they add up to 180°? 2. Sides - Are any of them the same length? 3. Angles - Are any of them equal? Depending on the answers to these key questions, we should be able to identify the correct triangle every time - let's put this to the test in the rest of this activity now!
  • Question 3

Use your knowledge of different types of triangles to match the groups of angles on the left to the scenario they describe on the right. 

CORRECT ANSWER

Column A

Column B

38°, 38°, 104°
Isosceles triangle
60°, 60°, 60°
Equilateral triangle
90°, 57°, 33°
Scalene triangle
40°, 140°
External angle of a triangle
EDDIE SAYS
Are you getting the rules clear in your mind now? We have three groups of angles containing three angles, which will relate to triangles and will all add up to 180°. Let's start by identifying the equilateral triangle, as all the angles will be the same. An isosceles triangle has two angles which are the same. A scalene triangle has three angles which are all different. The pair of angles must relate to an external angle of a triangle, as we would see only two angles present in this scenario.
  • Question 4

Review the diagram below:

Diagram of a triangle with an angle missing

Calculate the value of angle x in this triangle.

CORRECT ANSWER
EDDIE SAYS
The total of angles in a triangle must add up to 180°. Firstly, we can use this fact to find our missing interior angle: 48° + 74° = 122° 180° - 122° = 58° Now we have the missing angle, we can find the external angle, labelled x. Remember, that the total of angles in a straight line also add up to 180°, so: x = 180° - 58° = 122° Did you notice that the external angle is also the same as the sum of the two opposite angles inside the triangle? Another fact which you may want to add to your list...
  • Question 5

Consider this new diagram:

Diagram of a triangle with two angles missing

Calculate the value of angles a and b in this triangle.

CORRECT ANSWER
EDDIE SAYS
We can see that the two base angles are different, which makes this a scalene triangle. Angles in a triangle add up to 180°, so: Angle a = 180° - 65° - 48° = 67° Angles on a straight line add up to 180°, so: Angle b = 180° - 48° = 132°
  • Question 6

Consider this triangle and its associated markings:

Picture of a green triangle with triangle notation

What type of triangle is this?

 

Before you answer, try to explain why you think this is the case in your head or on a piece of paper. 

CORRECT ANSWER
 EquilateralIsoscelesScalene
This triangle is...
EDDIE SAYS
We need to look carefully at the triangle's markings to correctly identify its type. The single marks on each side of the triangle indicate that two sides are the same length, whilst the arcs in the two bottom corners indicate that two angles have the same value. If we put these two facts together, we know that we are looking at an isosceles triangle.
  • Question 7

Review the equilateral triangle below and its associate markings:

 

Diagram of an equilateral triangle with notation

What is the size of the angles in this equilateral triangle?

 

Before you type your answer, try to explain why you think this is the value of the angles in your head or on a piece of paper. 

CORRECT ANSWER
EDDIE SAYS
Knowing our triangle facts should make this easy for us! We know the angles in a triangle add up to 180° and we know that we are looking at an equilateral triangle. The angles in an equilateral triangle are the same, so: 180 ÷ 3 = 60° Were you able to explain those combinations of facts to yourself in your head?
  • Question 8

This triangle is a scalene triangle:

 

Image of a yellow scalene triangle

 

What are the correct properties of this triangle shown in the list below?

CORRECT ANSWER
Angles add up to 180°
All angles and sides are different
EDDIE SAYS
Sometimes there are no markings on a triangle. When this happens, we will usually be told its name. Consider the lengths of the sides and the size of the angles in this triangle - what can you observe? You would be right to say that they all look different, as this is the key defining feature of a scalene triangle. Remember that there is one fact that is always true for all triangles, which is that the interior angles will add up to 180°. Did you spot both of those facts in the list?
  • Question 9

Explore the triangle below:

 

Image of a triangle with a missing angle

 

Calculate the value of angle d in this triangle.

CORRECT ANSWER
EDDIE SAYS
There is always a sneaky question, isn't there? Don't worry though, we have all the knowledge to put together and solve this one! We know that external angles of a triangle can be added to the interior angle next to it to create a straight line. The angles in a straight line will add up to 180°, so the base angles of the triangle are: 180° - 125° = 55° 180° - 140° = 40° Now we have our two interior base angles, we can use these to work out the value of d, as all angles in a triangle add to 180° too: 55° + 40° = 95° 180° - 95° = 85° Did you follow all those steps?
  • Question 10

Which of the collections of angles below, accurately describe a valid triangle

 

a)  38°  57°  86°
b)  115°  25°  40°
c)  65°  55°  60°
d)  96°  48°  38°
CORRECT ANSWER
b)
c)
EDDIE SAYS
Sometimes we may be asked to prove that measurements provided are from a triangle. No problem, as we have a rule we can apply: All angles in a triangle add up to 180°. So in this question, we need to identify which of the groups provided do add up to 180°: a) 38° + 57° + 86° = 181° b) 115° + 25° + 40° = 180° c) 65° + 55° + 60° = 180° d) 96° + 48° + 38° = 182° So which of these are correct then? Amazing work! You can now use the key fact, that angles in a triangle always add to 180°, to find the value of unknown angles in triangles, identify triangles accurately and solve problems involving triangles.
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