The core circle theorems
Angle at the centre angle at the circumference (same arc); angle in a semicircle ; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to ; a tangent meets a radius at .
Always give the proper reason
In "give a reason" parts you must name the theorem precisely, e.g. "the angle at the centre is twice the angle at the circumference".
Drawn from real examiner reports.
Giving an inadequate reason
Vague statements such as "it is double 54" or naming the wrong theorem score nothing.
Reasons such as "it is double 54" or "kite theorem" did not explain the centre/circumference relationship on Nov 2024 Paper 2H (Q16); the precise wording must be learnt.
Assuming a right angle that is not there
Treating an angle as 90° (or assuming a diameter) when the diagram does not justify it.
On Nov 2024 Paper 2H (Q16) some students wrongly assumed 90° angles were present.
Misapplying the cyclic-quadrilateral theorem
Adding the two given angles and subtracting from 180 instead of using opposite angles.
Some added both marked angles and took them from 180, showing a lack of understanding of the cyclic-quadrilateral theorem (Nov 2024 Paper 2H, Q16).
Halving instead of doubling
For the centre/circumference theorem, dividing the circumference angle by instead of multiplying — or doubling the centre angle by mistake. Centre circumference on the same arc.
(general exam technique)
Missing the tangent–radius right angle
Not marking the where a tangent meets a radius, so the right-angled triangle needed for the next step is missed. A tangent always meets a radius at .
(general exam technique)
Name the theorem you used
Marking the angle and quoting the exact theorem secures the reasoning mark.
Learn the exact wording
For "give a reason", the phrase must be exact, e.g. "the angle at the centre is twice the angle at the circumference". Vague wording such as "double 54" scores nothing.
Look for a diameter or tangent
Scan the diagram for a diameter (angle in a semicircle ) or a tangent (meets the radius at ) before choosing a theorem — these unlock most parts.
Angle in a semicircle?
Points , and lie on a circle with centre . The angle at the circumference, , standing on arc is . Find the angle at the centre, , and give a reason.
