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In this worksheet, students will apply the cyclic quadrilateral theorem in relation to circles.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, AQA, Eduqas

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Circles

Difficulty level:

### QUESTION 1 of 10

Question

Why does nobody talk to circles?

Because there is no point.

Ah but within one particular circle theorem there are points within the circle in the shape of a quadrilateral.

One of the circle theorems is about angles in a cyclic quadrilateral.

Lets investigate

The facts

In the example 76 + 104 = 180° and 92 + 88 = 180°

Remember also that angles in a quadrilateral  also add up to 360°:

The quadrilateral has to fill the circle.  i.e all four points have to touch the circumference

As ever in these investigations you will be asked to recall other angle properties.

 76° 108° 70° 68° Angle x is Angle y is

The value of x is

90°

169°

94°

75°

True or False angle x = 68° and angle y = 91°

The value of x is 85°, 108°, 167° and the value of y is 95° 85° 108°

The value of x is 85°, 108°, 167° and the value of y is 95° 85° 108°

 116° 92° 64° 88° Angle x = Angle y =

Find the correct answers from the list below.

a= 89°

a= 91°

b= 89°

b= 91°

c= 89°

c= 91°

Find the value of x.

Find the value of x and also the value of y.

x = 180°

x = 60°

x = 90°

y = 36°

y = 72°

y = 56°

• Question 1

 76° 108° 70° 68° Angle x is Angle y is
EDDIE SAYS
Did you remember that the opposites add up to 180°? Angle x 180 - 72 = 108° Angle y 180 - 110 = 70° If you add all the angles up you will find they come to 360° Don't you think it amazing how many times 180° and 360° come into our maths investigations.
• Question 2

The value of x is

75°
EDDIE SAYS
I hope you picked the correct opposite angle. They are the ones diagonally opposite to each other. So in this example 180 - 105 = 75°
• Question 3

True or False angle x = 68° and angle y = 91°

False
EDDIE SAYS
The wrong pairs of angles have been identified here. x = 180 - 89 = 91° y = 180 - 112 = 68° It is important to remember that it is the diagonals that are required.
• Question 4

EDDIE SAYS
180 - 58 = 122° 180 - 121 = 59° Are you cycling around these now with ease?
• Question 5

The value of x is 85°, 108°, 167° and the value of y is 95° 85° 108°
EDDIE SAYS
Hopefully you are finding this a little easier now. A good way to check if you are correct is to see if all the angles add up to 360°. Its always good to check.
• Question 6

EDDIE SAYS
Ah yes we need to dig into what else we know about angles. Examiners like to sneak this sort of thing in. 180 - 101 = 79° Now use this to find x. 180 - 79 = 101° It has to be 101° as the rules angles on a straight line and opposite angles in a cyclic quadrilateral add up to 180° Angle y = 180 - 96 = 84°
• Question 7

 116° 92° 64° 88° Angle x = Angle y =
EDDIE SAYS
This is one of those questions that give you all the information you need. It just takes a little time to work it all out or maybe not. Here again we need to use angles on a straight line add up to 180° No bother for us is there. For angle x 180 - 92 = 88° on the straight line and 180 - 88 = 92°. You may have spotted the 180° rule for both properties and known that x was 92° without the need to do any working out. For angle y either 180 - 116 = 64 ° and then 180 - 64 = 116° or you have spotted the same rule applies therefore no working out required. Double check 116 + 64 + 92 + 88 = 360°
• Question 8

Find the correct answers from the list below.

a= 91°
b= 91°
c= 89°
EDDIE SAYS
Look at the quadrilateral, it is showing parallel sides. You should recall that interior angles within parallel lines add up to 180° First find the angle not identified. 180 - 89 = 91° Then work in any order. Using the rule that interior angles add up to 180° c = 180 - 91 = 89° a 180 - 89 = 91° (opposite angles = 180°) b is 180 - 91 = 89° (interior angles add up to 180°)
• Question 9

Find the value of x.

36
EDDIE SAYS
Oh the curved ball, except this is a quadrilateral. Oh well. Our skills of algebra are tested here. We know in total there are 10 lots of x and that the angles in a quadrilateral = 360° 10x = 360 so x = 36 Test it out. Substitute in 36+ 72 + 108 + 144 = 360° Job done.
• Question 10

Find the value of x and also the value of y.

x = 60°
y = 36°
EDDIE SAYS
All you need to remember here is that opposite angles in the cyclic quadrilateral = 180° 3x = 180° 180 ÷ 3 = 60° 5y = 180° 180 ÷ 5 = 36°: Substitute your answers in to check 3 x 60 = 180° 5 x 36 = 180°: Angles in a quadrilateral = 360°
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