Circle theorems

Higher only

The idea

The angle at the centre is twice the angle at the circumference. Angles in the same segment are equal. Opposite angles in a cyclic quadrilateral add to $180^\circ$ . The angle in a semicircle is $90^\circ$ . A tangent meets a radius at $90^\circ$ .

Look for radii (isosceles triangles), diameters and tangents.
Find one angle at a time.
Give the theorem's full wording as the reason.

Watch out Check the theorem's condition holds, for example both angles stand on the same arc.

See it worked

Worked exampleNon-calculator2 marks

$A$ , $B$ and $C$ are points on a circle with centre $O$ . Angle $AOB = 130^\circ$ and $C$ is on the major arc $AB$ .

Work out the size of angle $ACB$ . Give a reason for your answer.

The angle at the centre is twice the angle at the circumference standing on the same arc $AB$ .
$\angle ACB = 130 \div 2 = 65$

Answer $65^\circ$

Your turn

Question 1Non-calculator3 marks

$ABCD$ is a cyclic quadrilateral.

Work out the size of angle $BAD$ .

Question 2Non-calculator3 marks

$A$ , $B$ , $C$ and $D$ are points on a circle. $AB = AD$ .

Work out the size of angle $BAD$ .

You must show your working.

Question 3Non-calculator4 marks

$A$ , $B$ and $C$ are points on a circle. $TAS$ is the tangent at $A$ . $AB = AC$ . Angle $TAB = 58^\circ$ , and $C$ and $T$ are on opposite sides of the line $AB$ .

Work out the size of angle $BAC$ . Give reasons for each stage of your working.

In past papers

48 official questions in 47 papers since 2017. The latest:

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