Edexcel International GCSE Mathematics A · 4MA1
Circle theorems
Circle theorems connect angles, chords, radii and tangents. A useful solution identifies the relevant theorem before calculating. Be precise about which arc or chord an angle subtends; a diagram’s appearance alone is not a reason.
Key method
The angle at the centre is twice the angle at the circumference standing on the same arc. Opposite angles of a cyclic quadrilateral total 180°. A tangent is perpendicular to the radius at contact.
Worked example
A, B, C and D lie in order on a circle. Angle ABC is 112°. Find angle ADC.
- ABCD is a cyclic quadrilateral because all four vertices lie on the circle.
- ABC and ADC are opposite angles.
- Use their sum of 180°: angle ADC = 180° − 112°.
Answer and interpretation
68°, because opposite angles in a cyclic quadrilateral are supplementary.
Common mistake
Assuming any pair of angles on the circumference is equal ignores whether they stand on the same chord and segment.
Practise independently
A radius meets a tangent at T. What angle do they form?
Check your answer
90°, by the tangent–radius theorem.
Syllabus connection
This lesson develops circle theorems within the geometry area. Start with the worked method, attempt the independent question without looking at its answer, then explain any correction in your own words. Check the official specification for your tier’s full requirements.
Related resources
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