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.
Assuming a right angle that is not there
Treating an angle as 90° (or assuming a diameter) when the diagram does not justify it.
In examiner reports 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 (reported by examiners).
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 in examiner reports; the precise wording must be learnt.
Doubling when both angles are at the edge
The rule applies only from the centre to the circumference. Two angles both standing on the same arc at the circumference are equal (same segment), not double.
Missing the tangent–radius right angle
A tangent meets a radius at at the point of contact. Overlooking this right angle is common, yet it is usually the key that unlocks the rest of the diagram.
Name the theorem you used
Marking the angle and quoting the exact theorem secures the reasoning mark.
Chase every angle you can find
Mark the diagram and fill in each angle you can justify from a theorem, one step at a time. A part-way angle often reveals the isosceles triangle or arc you need for the answer.
Spot the tangent–chord angle
When a tangent and a chord meet, the angle between them equals the angle in the alternate segment. Recognising this pattern is often the fastest route to the answer.
Angle between a tangent and a radius?
is a cyclic quadrilateral. Angle . Find angle .
