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Use Angle Properties Around a Point

In this worksheet, students will learn about angle properties around a point. They will also find missing angles using number and algebra.

'Use Angle Properties Around a Point' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Nighttime image of a fairground

 

Have you ever wondered about the maths that makes a fairground work?

When you are on a fast ride, trying not to scream, you may think to yourself, "I really hope the designer got the angle measurements correct here!"

 

Angles are all around us, everywhere.

We may not often think about them, but rather just trust that engineers and designers have got this element right. 

 

In this activity, we are going to investigate angles around a point.

 

The key fact to remember is that angles positioned around a point always add up to 360°.

If they don't, then sack the designer!

 

A girl looking through a magnifying glass

 

Let's see this fact in action in an example now. 

 

 

So how do we use this information to help us find the value of an unknown angle? 

 

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

Firstly, we need to add up the value of the angles provided:

105° + 110° + 48° = 263°

 

Then we can subtract this amount from 360° to find the value of the unknown angle:

 360° - 263° = 97°

 

It's as simple as that!

 

 

 

Now let's take our newfound skills for a spin...

In this activity, we will use the key fact, that angles around a point always add to 360°, to find the value of unknown angles using numbers and algebra and solve problems involving collections of angles. 

What is the value of angle b in the diagram below?

Angles around a point diagram

What is the value of angle C in the diagram below?

Angles around a point diagram

283°

177°

77°

83°

What is the value of the unknown angle (?) in the diagram below?

 

Angles around a point diagram

283°

177°

77°

83°

What is the value of angles d and e in the diagram below?

Angles around a point diagram

 193°167°83.5°105.5°
Angle d
Angle e

Review the diagram below and answer the associated question by underlining the correct word:

Angles around a point diagram

There are one / two / three acute angles in this diagram.

This ball is divided into 6 equal segments:

 

A brightly-coloured beach ball

 

If we consider this ball as a 2D circle, what does each segment from the centre of this ball measure?

30°

40°

50°

60°

Angles around a point diagram

30°

40°

50°

60°

Now consider the unknown angle in the diagram below which has been represented algebraically

Angles around a point diagram

Find the value of x in this diagram.

 

Round your answer to one decimal place.

30°

40°

50°

60°

Next think about the angles represented algebraically in this diagram: 

Angles around a point diagram

Then use what we know about angles around a point, to select the correct values in the table below. 

 130°120°94°24°32°72°98°
x is equal to...
3x is equal to...
5x is equal to...

This clock face is divided into 5 minute segments:

 

Blank clock face with Roman numerals

 

What is the angle of each segment from the point at the centre of the clock?

 130°120°94°24°32°72°98°
x is equal to...
3x is equal to...
5x is equal to...
  • Question 1

What is the value of angle b in the diagram below?

Angles around a point diagram

CORRECT ANSWER
EDDIE SAYS
Was your head in a spin here, or were you spinning with success? We just need to follow our two steps to find the unknown angle. 1) Add the values provided: 117 + 48 + 79 = 244° 2) Subtract this amount from 360 to find the unknown angle: 360 - 244 = 116° We're off the mark!
  • Question 2

What is the value of angle C in the diagram below?

Angles around a point diagram

CORRECT ANSWER
77°
EDDIE SAYS
Let's apply our same process again. 1) Add the values provided: 132 + 63 + 88 = 283° 2) Subtract this amount from 360 to find the unknown angle: 360 - 283 = 77° Did you know that snowboarders call a full turn on their board 'a 360'? We know why now, don't we?!
  • Question 3

What is the value of the unknown angle (?) in the diagram below?

 

Angles around a point diagram

CORRECT ANSWER
EDDIE SAYS
Firstly, we need to add: 90 + 25 + 105 = 220° Then we need to subtract this value from 360: 360 - 220 = 140° Are you getting the hang of these now?
  • Question 4

What is the value of angles d and e in the diagram below?

Angles around a point diagram

CORRECT ANSWER
 193°167°83.5°105.5°
Angle d
Angle e
EDDIE SAYS
This time, we have two angles to find. Where this occurs with no additional information, we know that these angles must be the same, otherwise it would be impossible to find their value without more information. So let's treat angles d and e as one angle initially. Let's add together the values we have: 120 + 48 + 25 = 193° Now let's subtract this value from 360: 360 - 193 = 167° As both angles must be the same size, we can divide this amount in two in order to find the value of angles d and e: 167 ÷ 2 = 83.5°
  • Question 5

Review the diagram below and answer the associated question by underlining the correct word:

Angles around a point diagram

CORRECT ANSWER
There are one / two / three acute angles in this diagram.
EDDIE SAYS
Did you recall the fact that acute angles are always less than 90°? We can see that there is one acute angle present in the diagram already (53°). So we just have to work out the values of the missing angle to see if this is acute of not. 128 + 53 + 91 = 272° 360 - 272 = 88° So there are two acute angles present in total. Here's a joke for you now... Why was the obtuse angle upset? Because he wasn't a-cute!
  • Question 6

This ball is divided into 6 equal segments:

 

A brightly-coloured beach ball

 

If we consider this ball as a 2D circle, what does each segment from the centre of this ball measure?

CORRECT ANSWER
60°
EDDIE SAYS
We have all the information we need, although this question may be phrased differently. There are six equal angles at the centre of the ball, so we need to take 360 and divide this by six: 360 ÷ 6 = 60°
  • Question 7

Angles around a point diagram

CORRECT ANSWER
EDDIE SAYS
This one may look tricky, but let's just apply what we know and follow our usual steps. The ratio totals add up to 12. Now we need to divide 360 by 12: 360 ÷ 12 = 30° The largest section of the ratio is 5 parts: 30 × 5 = 150° Now, when your maths teacher asks if you understand angles, you can say "yes to a degree!" Great work! You have shown that you can use the key fact, that angles around a point always add to 360°, to find the value of unknown angles using numbers and algebra.
  • Question 8

Now consider the unknown angle in the diagram below which has been represented algebraically

Angles around a point diagram

Find the value of x in this diagram.

 

Round your answer to one decimal place.

CORRECT ANSWER
EDDIE SAYS
Good old algebra, it just can't keep its nose out of anything! Let's follow our usual process to initially find the value of 5x: 105 + 91 = 196° 360 - 196 = 164° A useful way to think about this is that 164 is the value of 5 lots of x. To find x, we can divide this amount by 5: 164 ÷ 5 = 32.8°
  • Question 9

Next think about the angles represented algebraically in this diagram: 

Angles around a point diagram

Then use what we know about angles around a point, to select the correct values in the table below. 

CORRECT ANSWER
 130°120°94°24°32°72°98°
x is equal to...
3x is equal to...
5x is equal to...
EDDIE SAYS
As long as we remember our 360 rule, then this question shouldn't take us around the houses too much. If we add all the x's present together, we find that there are 15 lots of x. So to find the value of x, we need to divide 360 by 15: 360 ÷ 15 = 24° Now we can multiply this amount to find the values of 3x and 5x: 3 × 24 = 72° 5 × 24 = 120°
  • Question 10

This clock face is divided into 5 minute segments:

 

Blank clock face with Roman numerals

 

What is the angle of each segment from the point at the centre of the clock?

CORRECT ANSWER
EDDIE SAYS
Let's take our time to think about this one... Firstly, let's count the number of segments which the clock has been divided into. There are 12. We know there are 360° in any circle, so we can divide this total by 12 to find the value of each segment: 360 ÷ 12 = 30° We can check we have the correct answer by working backwards... If we add 12 segments with a value of 30° together, do we reach 360°?
---- OR ----

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