Overview
What you’ll learn
- State and apply the angle in a semicircle and the tangent–radius right angle (Core).
- Convert between angles at the centre and angles at the circumference on the same arc (Extended).
- Collect equal angles from the same segment and complete cyclic quadrilaterals to 180° (Extended).
- Apply the alternate segment theorem to move an angle onto the tangent–chord pair (Extended).
- Use chord bisection, equal tangents, and the isosceles triangles made by two radii to finish chases (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 4.7Use the angle in a semicircle (90°) and the perpendicularity of a tangent and the radius at the point of contact.
Extended — Extended candidates only
- 4.7Use the angle at the centre (twice the angle at the circumference on the same arc) and angles in the same segment. (Extended)
- 4.7Use opposite angles of a cyclic quadrilateral summing to 180° and the alternate segment theorem. (Extended)
- 4.8Use properties of chords: the perpendicular from the centre bisects a chord, and equal chords are equidistant from the centre. (Extended)
- 4.8Use equal tangents from an external point and the isosceles triangles formed by radii. (Extended)
In 30 seconds
- 1The angle in a semicircle is 90°: any triangle on a diameter has a right angle at the rim.
- 2A tangent meets the radius at 90° — draw that right angle the moment you see a tangent.
- 3The angle at the centre is twice the angle at the circumference standing on the same arc (Extended).
- 4Angles in the same segment (from the same chord, same side) are equal (Extended).
- 5Cyclic quadrilateral: four points on a circle → opposite angles sum to 180° (Extended).
- 6Alternate segment: the angle between tangent and chord equals the angle in the alternate segment (Extended).
- 7A perpendicular from the centre bisects a chord; tangents from one external point are equal — and every “two radii + chord” triangle is isosceles (Extended).
Notes · 1 of 4
The two Core theorems
A diameter’s rim angle and a tangent’s right angle.
If AB is a diameter and C is any other point on the circle, the angle ACB = 90° — the angle in a semicircle is a right angle. This is the quickest fact in the topic: see the diameter, stamp the 90°. Combined with the angle sum of a triangle, one given angle on a diameter-side triangle finishes the other two.
A tangent touches the circle at exactly one point, and the radius to that point is perpendicular to it: tangent and radius meet at 90°. Every tangent question starts by drawing the radius to the point of contact and marking the right angle. The triangle that then appears (centre, contact point, external point) contains two radii — so it is isosceles, which supplies equal base angles for free.
Notes · 2 of 4
Centre, circumference and the same segment
Double at the middle; equal along the same arc. (Extended)
Extended. Stand one angle at the centre and one on the rim, both leaning on the same chord. The centre angle is exactly double the rim angle: centre 110° fixes every rim angle on that arc at 55°. The proof sketch is the two radii making an isosceles triangle — the exterior-angle rule does the doubling.
Angles in the same segment — from the same chord, standing on the same side — are equal: rim angles of 44° and its unseen cousins are all 44°. Together these two rules turn one labelled angle into a whole diagram’s worth of answers. Watch which arc an angle leans on: the other side of the chord belongs to the opposite segment and behaves differently.
Notes · 3 of 4
Cyclic quadrilaterals
Four points on a circle: opposite angles make a straight line’s worth. (Extended)
Extended. A cyclic quadrilateral has all four corners on a circle — any quadrilateral inscribed like this. Its opposite angles sum to 180°: one angle of 95° forces the opposite to 85°, while the other pair handles its own 180°. The full total is 360° as for any quadrilateral, but the paired structure is what earns marks.
The same fact reads backwards: if a quadrilateral’s opposite angles sum to 180°, it is cyclic — useful for proving four points lie on one circle. Combined with the angle-in-a-semicircle (a diameter makes the right angle) the cyclic family covers most “four points on a circle” exam diagrams.
Notes · 4 of 4
Alternate segment and tangent properties
Bounce angles off the tangent line; equal tangents, chord bisection. (Extended)
Extended. The alternate segment theorem says the angle between a tangent and a chord equals the angle in the alternate segment — the triangle corner on the far side of the chord. A tangent–chord angle of 52° means every triangle vertex in that alternate segment reads 52°. It is the theorem that moves angles across the circle without any drawing of new lines.
Two more tangent facts: tangents from one external point are equal (PX = PY), and with the two radii they form a kite whose angle at the centre is 180° minus the angle at the point — a quadrilateral of two right angles, the external angle, and the centre angle. Chords behave symmetrically: the perpendicular from the centre bisects any chord, which turns “chord length + distance from centre” questions into right triangles: chord 16 cm sitting 6 cm from the centre means a radius of √(8² + 6²) = 10 cm.
Revise
Key terms
The words examiners expect you to use precisely.
- Angle in a semicircle
- The angle on the circumference facing a diameter is 90°.Diameter AB with C on the rim → angle ACB = 90°.
- Tangent–radius theorem
- A tangent and the radius to its contact point meet at 90°.Angle OTP = 90° for tangent TP.
- Angle at the centre (Extended)
- Twice any circumference angle on the same arc.Centre 110° → rim 55°.
- Same segment (Extended)
- Rim angles from the same chord, same side, are equal.44° and 44° from one chord.
- Cyclic quadrilateral (Extended)
- A quadrilateral whose vertices all lie on a circle.Opposite angles sum to 180°.
- Alternate segment theorem (Extended)
- The tangent–chord angle equals the angle in the alternate segment.Tangent–chord 52° → rim angle 52°.
Show 2 more key termsShow fewer key terms
- Equal tangents (Extended)
- Tangents drawn from one external point have equal lengths.PX = PY.
- Chord bisection (Extended)
- The perpendicular from the centre cuts a chord in half.Chord 16 cm, 6 cm from centre → radius 10 cm.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Doubling the circumference angle instead of halving the centre angle.
The centre holds the larger of the pair.
Instead: Centre 110° → circumference 55°, never 220°.
Mistake: Equating angles from opposite segments.
Same-segment equality holds only on the same side of the chord.
Instead: Opposite-segment angles are supplementary: 58° and 122°.
Show 11 more mistakesShow fewer mistakes
Mistake: Summing adjacent angles of a cyclic quadrilateral to 180°.
It is the opposite pair that is supplementary.
Instead: A with C, B with D: 82 + 98 = 180.
Mistake: Forgetting the 90° where tangent meets radius.
The right angle is the entry point of every tangent problem.
Instead: Draw the radius to the contact point and stamp 90° first.
Mistake: Missing the isosceles triangle made by two radii.
Radii are equal, so the base angles match.
Instead: Triangle OAB with OA = OB: angle OAB = angle OBA.
Mistake: Assuming different tangent lengths from one point.
Equal tangents is a theorem, not a coincidence of the diagram.
Instead: PX = PY always, for tangents from the same external point.
Mistake: Using the full chord with the centre distance in Pythagoras.
The perpendicular bisects the chord — use half its length.
Instead: Chord 16, distance 6 → radius √(8² + 6²) = 10.
Mistake: Quoting the alternate segment for the wrong side.
The theorem pairs the tangent–chord angle with the segment on the far side of the chord.
Instead: Trace the chord to the triangle vertex across the circle.
Mistake: Leaving out reasons in a circle-theorem chase.
Most marks are awarded for the theorem, not the arithmetic.
Instead: Name it: “angle at centre = 2 × angle at circumference”.
Mistake: Doubling an angle that already sits in the same segment.
Same-segment angles are equal; only the centre angle is double.
Instead: PRQ = PSQ whenever R and S stand on the same arc from chord PQ.
Mistake: Treating angles from opposite arcs as equal.
Points on opposite arcs see supplementary angles, summing to 180°.
Instead: 70° on the major arc pairs with 110° on the minor arc for the same chord.
Mistake: Halving the angle between tangents to get the centre angle.
The kite OXTY links them by subtraction, not by halving.
Instead: angle XOY = 180° − angle XPY: 64° at P gives 116° at O.
Mistake: Adding tangent lengths from the same point as if they differed.
The two tangents from one external point are equal in length.
Instead: A perimeter with a tangent pair uses each length twice or doubled.
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Apply the angle in a semicircle
AB is a diameter and C lies on the circle. Angle ACB is
Ready for more practice on Circle theorems?
Generate a fresh 5-question adaptive drill grounded in this topic’s 0580 key points. Free with your study account.
Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 13 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 13: the angle in a semicircle, tangent–radius, angle at the centre, same-segment angles, cyclic quadrilaterals, the alternate segment theorem, equal tangents, and chord bisection.
Mark scheme
PROTopic 13 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Which circle theorem should I use first in a chase?
Scan for free gifts in this order: a diameter (stamp 90°), a tangent (stamp 90° at the contact point), two radii (mark the isosceles base angles). Once one angle is known, the centre–circumference, same-segment and cyclic rules cascade quickly.
How do I tell which segment an angle is in?
Trace the chord, then see which side of it the angle’s vertex sits on. Same side = same segment = equal angles. Opposite side = the supplementary partner (they complete 180° as a cyclic quadrilateral).
Is the alternate segment angle the same as the tangent–chord angle or supplementary?
Equal — that is the whole theorem. The tangent–chord angle of 52° appears again as the triangle angle in the alternate segment. It is the opposite-segment angle (132°) that is supplementary, via the cyclic quadrilateral.
Do Core candidates meet cyclic quadrilaterals?
No — at Core the circle work is the angle in a semicircle and the tangent–radius right angle. Centre–circumference, same segment, cyclic quadrilaterals, the alternate segment and the chord/tangent properties are all Extended statements.
Is QuizQuip connected to Cambridge?
No. Cambridge IGCSE and Cambridge are trademarks of Cambridge University Press & Assessment. QuizQuip is independent — not affiliated with, endorsed by, or connected to Cambridge. All our notes and questions are original material written against the published 0580 syllabus; we do not host Cambridge past papers.
Up next
Topic 14: Transformations and vectors