The theorems that decide Paper 2 marks — centre vs circumference, semicircle, same segment, cyclic quadrilateral, tangent–radius and tangent–chord — each with the reason to quote, a worked example, a diagram and a worksheet.
Euclidean (circle) geometry rewards memory and careful reasoning more than any other topic. The theorems are fixed and full marks go to learners who state the correct reason for every step. This guide gives each key theorem with a worked example and the reason to quote.
1The core theorems (and their reasons)
- Angle at the centre = 2 × angle at the circumference (same arc). Reason: ∠ at centre = 2 × ∠ at circ.
- The angle in a semicircle is 90°. Reason: ∠ in semicircle.
- Angles in the same segment (same arc) are equal. Reason: ∠s in same segment.
- Opposite angles of a cyclic quadrilateral add to 180°. Reason: opp ∠s of cyclic quad.
- A tangent is perpendicular to the radius at the point of contact. Reason: tan ⊥ radius.
- Tangent–chord angle = angle in the alternate segment. Reason: tan–chord.
2Type 1: Centre and circumference
- 1Angle at circumference = 80° ÷ 2 = 40°.∠ at centre = 2 × ∠ at circumference (same arc).
- 2State the reason alongside your answer.The reason is worth a mark of its own.
3Type 2: Angle in a semicircle
- 1∠APB = 90°.Angle in a semicircle (AB is a diameter).
- 2∠ABP = 180° − 90° − 35° = 55°.Angles of triangle APB add to 180°.
4Type 3: Cyclic quadrilateral
- 1Opposite angle = 180° − 95° = 85°.Opposite ∠s of a cyclic quad add to 180°.
5Type 4: Same segment
- 1The other angle = 48°.Angles in the same segment (same arc) are equal.
6Type 5: Tangent–chord (alternate segment)
- 1Angle in the alternate segment = 62°.Tangent–chord angle = angle in the alternate segment.
Make flashcards with the theorem statement on one side and the accepted 'reason' wording on the other. In the exam you must quote the reason, not just the result.
Don't assume a line is a diameter or a quadrilateral is cyclic unless it's given or you can prove it. Using a theorem whose conditions aren't met loses marks.
These same theorems return in Grade 12, so the effort counts twice. Attempt every rider — correct statements with reasons pick up marks even when you can't finish the whole proof. Practise below.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Arc subtends 120° at the centre. Angle at the circumference? (centre/circ)
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120 ÷ 2 = 60° - 2Angle at the circumference is 35°. Angle at the centre (same arc)? (centre/circ)
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2 × 35 = 70° - 3AB is a diameter, P on the circle. What is ∠APB? (semicircle)
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90° (angle in a semicircle) - 4In a semicircle, ∠PAB = 40°. Find ∠ABP. (semicircle)
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180 − 90 − 40 = 50° - 5Cyclic quad angle is 110°. Opposite angle? (cyclic quad)
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180 − 110 = 70° - 6Cyclic quad angle is 75°. Opposite angle? (cyclic quad)
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180 − 75 = 105° - 7Two angles stand on the same arc; one is 52°. The other? (same segment)
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52° (equal) - 8Tangent–chord angle is 70°. Angle in the alternate segment? (tan–chord)
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70° - 9A tangent meets a radius at the point of contact. What is the angle? (tan–radius)
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90° (tan ⊥ radius) - 10Angle at centre = 2x and equals 100°. Find x, the angle at the circumference. (solve)
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2x = 100 → x = 50°
Now practise it
Download Grade 11 past papers and worksheets on this topic.