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Apply the Angles in the Same Segment Rule

In this worksheet, students will focus on one specific theory relating to circles (angles in the same segment are always equal) and use this to solve related geometric problems.

'Apply the Angles in the Same Segment Rule' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Circles

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Circles are all around us in everyday life.

 

How many examples can you think of?

 

Can you see any circular shapes in objects around you at this moment?

 

 

Mathematicians have investigated circles for centuries because of their love of pure mathematics.

The ability to learn, understand and apply circle theorem is an essential, but challenging, geometric skill. 

For this reason, we usually only see circle theorem questions on the Higher GCSE exam paper. 

It is also common for circles and angles to be linked in questions, so we may need to apply our knowledge of angle properties too. 

 

 

In this activity, we will focus on one specific theory relating to circles:

Angles in the same segment are always equal. 

 

 

Let's look at how this theory works now and why it is always true. 

 

1.Circle2.Circle with  triangle in

3.4.

 

 

Let's put this theory into practice now in some geometric problems. 

Review the diagram below:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

47°

43°

133°

180°

Investigate this new diagram now:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

Can't tell

66°

48°

114°

Explore the circle and sectors below:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

Can't tell

66°

48°

114°

Take a careful look at the pair of triangles shown within the circle here:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

Can't tell

66°

48°

114°

Check out this next circle challenge now:

 

Diagram of a circle with two interior triangles

 

What are the values of angles a and b?

 64°18°116°46°
a =
b =

Review the diagram below:
 

Diagram of a circle with two interior triangles

 

What are the values of each of the unknown angles shown?

 40°86°94°47°68°
a =
b =
c =

Investigate this new diagram now:

 

Diagram of a circle with three interior triangles

 

What are the values of each of the unknown angles shown?

 

Type only numbers into the gaps below to express your answers. 

 40°86°94°47°68°
a =
b =
c =

Explore the circle and sectors below:

 

Diagram of a circle with three interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

 40°86°94°47°68°
a =
b =
c =

Take a careful look at the trio of triangles shown within the circle below:

 

Diagram of a circle with three interior triangles

 

Angle a and b have the same value

 

What is the value of angle a or b?

88°

44°

92°

46°

Check out this final circle challenge to finish:

 

Diagram of a circle with two interior triangles

 

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

88°

44°

92°

46°

  • Question 1

Review the diagram below:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

CORRECT ANSWER
47°
EDDIE SAYS
Can you draw a chord line which connects angle a and other known angles within the same segment? A chord is a line which passes between two sides of the circle without passing through the centre. If we draw a line from the point by angle a to the point by 47°, we can connect them within the same segment. We know that angles in the same segment are equal, so angle a will have the same value as the angle present, which is 47°.
  • Question 2

Investigate this new diagram now:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

CORRECT ANSWER
48°
EDDIE SAYS
These problems can be a little confusing, as there is also an angle present below 48°. Another way to think of this rule is that: "The angles at the circumference created by the same arc are equal." We can see that angle a and our 48° have both been created by the same arc, so they are within the same segment. Therefore, they must be equal, so angle a has a value of 48°.
  • Question 3

Explore the circle and sectors below:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

CORRECT ANSWER
EDDIE SAYS
Can you identify the important chord here? The left, vertical line is the one to notice, as both angles are positioned above this chord. This means that both angles are already in the same segment so they are equal and both have a value of 36°. Are you getting the hang of these now? Let's take the difficulty up a notch then...
  • Question 4

Take a careful look at the pair of triangles shown within the circle here:

 

Diagram of a circle with two interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

CORRECT ANSWER
EDDIE SAYS
This time, there are two angles to choose from. We have two options to find our answer here: 1) Draw a chord in horizontally at the top or bottom; 2) Work out which angles at the circumference have been created by the same arc. Whichever option we choose, we can see that angle a is in the same segment as 22°, so these angles are equal.
  • Question 5

Check out this next circle challenge now:

 

Diagram of a circle with two interior triangles

 

What are the values of angles a and b?

CORRECT ANSWER
 64°18°116°46°
a =
b =
EDDIE SAYS
Again, we can opt to draw chords in or work out which angles at the circumference have been created by the same arc. Whichever option we choose, we can see that angle a is in the same segment as 46°, so these angles are equal. Also, we can observe that angle b is in the same segment as 18°, so these angles are also equal.
  • Question 6

Review the diagram below:
 

Diagram of a circle with two interior triangles

 

What are the values of each of the unknown angles shown?

CORRECT ANSWER
 40°86°94°47°68°
a =
b =
c =
EDDIE SAYS
This is becoming trickier, isn't it? We need to put another angle rule we know into practice here too: "There are 180° total in a triangle". Applying this, we know that: a + b + 86° = 180° We also know that angles a and b are equal as they are both within the same segment, (this chord has already been drawn for us to use). 180 - 86 = 94 94 ÷ 2 = 47° Angle c is in the same segment as 40°, as they are both positioned above the chord already present, so they are also equal.
  • Question 7

Investigate this new diagram now:

 

Diagram of a circle with three interior triangles

 

What are the values of each of the unknown angles shown?

 

Type only numbers into the gaps below to express your answers. 

CORRECT ANSWER
EDDIE SAYS
Angle b is in the same segment as 34° so must be equal, as we can use the almost horizontal chord line already present. If we draw a horizontal chord along the bottom, we can see that angle c is the same section as 78°, so these angles are also equal. Angle a is the trickiest one to work out. It is not in the same segment as 28°, but it is within the same triangle. We need to work out the missing angle in the centre, so that we can subtract both angles from 180° to work out the value of a. We can find the missing angle in the left-hand triangle: 180 - 78 - 34 = 68° There are also 180° in a straight line, so we can subtract this to find our missing angle in the top triangle: 180 - 68 = 112° Now we have all the information we need to find the value of a, as we know the two other angles (alongside a) in the upper triangle are 112° and 28°: 180 - 112 - 28 = 40° Phew, what a lot of steps there - well done if you got this correct on your own!
  • Question 8

Explore the circle and sectors below:

 

Diagram of a circle with three interior triangles

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

CORRECT ANSWER
EDDIE SAYS
This time the provided angle spans two triangles, so we will need to split them up. Firstly, let's find the missing angles in the left-hand triangle which has 94° positioned at its peak. This triangle is an isosceles triangle, so the two other angles will be equal: 180 - 94 = 86° 86 ÷ 2 = 43° Angle a is in the same segment as one of these angles, as we can draw a chord between them, almost horizontally, at the top. So angle a will also have a value of 43°.
  • Question 9

Take a careful look at the trio of triangles shown within the circle below:

 

Diagram of a circle with three interior triangles

 

Angle a and b have the same value

 

What is the value of angle a or b?

CORRECT ANSWER
46°
EDDIE SAYS
All of the diagonal lines present here originate from the centre of the circle - did you spot the 'O' label? This means that they are all radii (the plural of radius). Therefore, all the triangles created with the addition of chords, will be isosceles triangles. So, if we find the value of the central angle at the tip of one of these triangles, we can divide the remainder by 2 to find the other angles. Angles on a straight line add up to 180° so: 180 - 88 = 92° 92 ÷ 2 = 46°
  • Question 10

Check out this final circle challenge to finish:

 

Diagram of a circle with two interior triangles

 

 

What is the value of angle a?

 

Type only numbers into the gap below to express your answer. 

CORRECT ANSWER
EDDIE SAYS
We need to find the angles in the smaller triangle first, as one of the missing angles is in the same segment as a. Vertically opposite angles are equal, therefore, one of the missing angles is 92°. Angles in a triangle add up to 180°, so we can work out the final angle in the small triangle by subtracting the other two angles: 180 - 92 - 60 = 28° This angle is in the same segment as angle a, if we draw a chord between them, so they both have a value of 28°. Well done! You can now use the theory that angles in the same segment are always equal to solve related geometric problems, often involving other angle theories too. Why not try another circle theorem activity now, if you are up to the challenge?
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