GCSE Maths · Higher · Chapter 20: Properties of circles · Lesson 1

Circle theorems

Grades 5–8About 45 minutesNo calculator35 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Angles on a straight line are $3 x$ and $x$ . Find $x$ .
[1 mark]
Answer
A2
From chapter 16
Rationalise the denominator of $\frac{6}{3}$ .
[1 mark]
Answer
A3
From chapter 18
30 fish are tagged. Later 40 are caught and 2 are tagged. Estimate the population.
[1 mark]
Answer
A4
From chapter 18
A cumulative frequency graph shows 60 values. At what cumulative frequency do you read the median?
[1 mark]
Answer

Grades 5–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

$O$ is the centre of the circle. Angle $A O B = 12 4^{\circ}$ . Work out angle $A C B$ .

Not drawn accurately

Hint The angle at the centre is twice the angle at the circumference.

B1
$O$ is the centre of the circle. Angle $A C B = 4 7^{\circ}$ . Work out angle $A O B$ . Give a reason for your answer.
[2 marks]
Not drawn accurately
Answer

Example 2

$A B C D$ is a cyclic quadrilateral. Work out angle $D A B$ .

Not drawn accurately

Hint Opposite angles add up to

18 0^{\circ}

B2
$P QR S$ is a cyclic quadrilateral. Work out angle $S P Q$ .
[3 marks]
Not drawn accurately
Answer

Grades 5–7

C1
$O$ is the centre. Work out angle $A C B$ .
[3 marks]
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Answer
C2
Work out angle $A D B$ . Give a reason for your answer.
[2 marks]
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Answer

C3
$O$ is the centre. Work out the reflex angle $A O B$ .
[2 marks]
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Answer
C4
$O$ is the centre. Work out the value of $x$ .
[3 marks]
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Answer

Grades 6–8

D1
Ali draws a quadrilateral $A B C D$ with angle $D A B = 7 0^{\circ}$ and angle $B C D = 12 0^{\circ}$ . He says, "I can draw a circle that passes through $A$ , $B$ , $C$ and $D$ ."
Explain why Ali is wrong.
[2 marks]

D2
$A$ , $B$ and $C$ are points on a circle with centre $O$ . $A C$ is a diameter. Prove that angle $A B C = 9 0^{\circ}$ . You must give a reason for each stage of your working.
[4 marks]
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D3
$B D$ is a diameter of the circle with centre $O$ . $C B E$ is a straight line.
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(a)
Write down angle $B A D$ . Give a reason for your answer.
[2 marks]
Answer
(b)
Work out angle $A B C$ . You must show your working.
[3 marks]
Answer
(c)
Work out angle $A B E$ . Give a reason for your answer.
[2 marks]
Answer
Total for D3: 7 marks

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the angle at the centre and the angle in a semicircle.
I can use cyclic quadrilaterals and angles in the same segment.
I can prove a circle theorem, with reasons.

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