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.

Start practising Read the lesson

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
$P$ , $Q$ and $R$ are points on a circle, centre $O$ .

$SPT$ is the tangent to the circle at $P$ .Angle $TPQ = 67°$

Angle $RPS = 49°$

Work out the size of angle $OQR$ .

Write down any circle theorems that you use.

(4 marks)
Level 5 · 4 marksTry it
2
$A$ , $B$ and $C$ are points on a circle with centre $O$ .

Angle $ACB = 58°$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ (angle $OAB$ ). You must give reasons for each stage of your working.
Level 5 · 4 marksTry it
3
$A$ , $B$ , $C$ and $D$ are points on a circle. The chords $AC$ and $BD$ cross at the point $E$ .

Angle $ADB = 40°$ and angle $DBC = 33°$ .Diagram NOT accurately drawn

(a) Write down the size of angle $ACB$ . Give a reason for your answer.

(b) Work out the size of angle $AEB$ , marked $x$ .
Level 5 · 4 marksTry it