GCSE · Circles, arcs & sectors

Circle theoremsExam-style questions

The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
$A$ , $B$ and $C$ are points on a circle with centre $O$ .

Angle $ABC = 36°$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ . Give a reason for your answer.
Level 2 · 3 marksTry it
2
$A$ , $B$ , $C$ and $D$ are points on a circle.Diagram NOT accurately drawn

Angle $DAB = 3x°$ and angle $BCD = (x + 20)°$ .

Work out the value of $x$ . Give a reason for your answer. (3 marks)
Level 3 · 3 marksTry it
3
$A$ , $B$ and $C$ are points on a circle with centre $O$ . Angle $AOB = 140°$ .Diagram NOT accurately drawn

(a) Work out the size of angle $ACB$ . Give a reason for your answer. (2 marks)

(b) Work out the size of angle $OAB$ . Give a reason for your answer. (2 marks)
Level 3 · 4 marksTry it
4
$P$ , $Q$ and $R$ are points on a circle with centre $O$ . $PQ$ is a diameter.

Angle $QPR = 27°$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ . You must give reasons for each stage of your working.
Level 3 · 4 marksTry it
5
$ABCD$ is a cyclic quadrilateral.

Angle $DAB = 104°$ and angle $ABC = 88°$ .Diagram NOT accurately drawn

(a) Work out the size of the angle marked $x$ . Give a reason for your answer.

(b) Work out the size of the angle marked $y$ .
Level 3 · 4 marksTry it

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