Circle theorems
Cambridge IGCSE Mathematics (0580) · Unit 4: Geometry · 9 flashcards
Circle theorems is topic 4.7 in the Cambridge IGCSE Mathematics (0580) syllabus , positioned in Unit 4 — Geometry , alongside Angles, Angles in polygons and Parallel lines. In one line: The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
This topic is examined across Paper 1 (Core) or Paper 2 (Extended) — non-calculator — and Paper 3 (Core) or Paper 4 (Extended) — calculator. It is a Supplement (Extended-tier) topic, so it appears only on the Extended-tier papers.
The deck below contains 9 flashcards — 4 definitions, 2 key concepts and 1 application card — covering the precise wording mark schemes reward. Use the 4 definition cards to lock down command-word answers (define, state), then move on to the concept and application cards to handle explain, describe and compare questions.
State the angle in a semicircle theorem
The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
Questions this Circle theorems deck will help you answer
- › Angles subtended by the same chord in the same segment of a circle are what?
- › A line is drawn from the centre of a circle to bisect a chord. What is the angle between the line and the chord?
- › A tangent touches a circle at point T. Chord AB is drawn from T. If the angle between the tangent and chord AB is 60°, what is the angle at the circumference in the alternate segment, opposite angle ATB?
The angle at the centre of a circle is 130°. What is the angle at the circumference subtended by the same arc?
The angle at the centre is twice the angle at the circumference when subtended by the same arc. Therefore, the angle at the circumference is 130°/2 = 65°.
State the angle in a semicircle theorem.
The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
What is a cyclic quadrilateral, and what is the relationship between its opposite angles?
A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180°.
A tangent to a circle meets the radius at what angle?
A tangent to a circle is perpendicular to the radius at the point of contact. This means they meet at an angle of 90°.
Describe the alternate segment theorem.
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. The alternate segment is the area of the circle cut off by the chord.
Angles subtended by the same chord in the same segment of a circle are what?
Angles subtended by the same chord, on the same side of it, are equal. Visualize drawing two different triangles from the chord to the circumference; the angles at the circumference will be equal.
A line is drawn from the centre of a circle to bisect a chord. What is the angle between the line and the chord?
A line drawn from the centre of a circle to bisect a chord is perpendicular to the chord. Therefore, the angle is 90°.
In a cyclic quadrilateral ABCD, angle A = 70°. What is the size of angle C?
In a cyclic quadrilateral, opposite angles are supplementary (add up to 180°). Therefore, angle C = 180° - 70° = 110°.
A tangent touches a circle at point T. Chord AB is drawn from T. If the angle between the tangent and chord AB is 60°, what is the angle at the circumference in the alternate segment, opposite angle ATB?
According to the Alternate Segment Theorem, the angle between the tangent and the chord equals the angle in the alternate segment. So, the angle in the alternate segment is also 60°.
Key Questions: Circle theorems
State the angle in a semicircle theorem.
The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
What is a cyclic quadrilateral, and what is the relationship between its opposite angles?
A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180°.
A tangent to a circle meets the radius at what angle?
A tangent to a circle is perpendicular to the radius at the point of contact. This means they meet at an angle of 90°.
Describe the alternate segment theorem.
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. The alternate segment is the area of the circle cut off by the chord.
More topics in Unit 4 — Geometry
Circle theorems sits alongside these Mathematics decks in the same syllabus unit. Each uses the same spaced-repetition system, so progress in one informs the next.
Cambridge syllabus keywords to use in your answers
These are the official Cambridge 0580 terms tagged to this section. Mark schemes credit responses that use the exact term — weave them into your answers verbatim rather than paraphrasing.
Key terms covered in this Circle theorems deck
Every term below is defined in the flashcards above. Use the list as a quick recall test before your exam — if you can't define one of these in your own words, flip back to that card.
Related Mathematics guides
Long-read articles that go beyond the deck — cover the whole subject's common mistakes, high-yield content and revision pacing.
How to study this Circle theorems deck
Start in Study Mode, attempt each card before flipping, then rate Hard, Okay or Easy. Cards you rate Hard come back within a day; cards you rate Easy push out to weeks. Your progress is saved in your browser, so come back daily for 5–10 minute reviews until every card reads Mastered.
Study Mode
Space to flip • ←→ to navigate • Esc to close
You're on a roll!
You've viewed 10 topics today
Create a free account to unlock unlimited access to all revision notes, flashcards, and study materials.
You're all set!
Enjoy unlimited access to all study materials.
Something went wrong. Please try again.
What you'll get:
- Unlimited revision notes & flashcards
- Track your study progress
- No spam, just study updates