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Mathematics 05804.2

Circle theorems

The angle rules in circles, and how to justify them.

Review these first

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 4.2.1Use circle theorems to calculate angles and give reasons.Supplement

6 minute read

Circle theorems

Circle theorems are pure recall plus careful diagram-reading. Each theorem has an exact phrase that earns the reason mark.

The theorems

  1. Angle at the centre is twice the angle at the circumference, when both stand on the same arc.
  2. Angle in a semicircle is 90° — a triangle drawn on a diameter always has a right angle at the circumference.
  3. Angles in the same segment are equal — angles subtended by the same arc, on the same side, are equal.
  4. Opposite angles in a cyclic quadrilateral add to 180° (a quadrilateral with all four vertices on the circle).
  5. A tangent is perpendicular to the radius at the point of contact.
  6. Tangents from an external point are equal in length, which creates an isosceles triangle.
  7. Alternate segment theorem — the angle between a tangent and a chord equals the angle in the alternate segment.

How to approach a problem

  • Mark every angle you can find, working outward from what is given.
  • Look first for a diameter (theorem 2) or a tangent meeting a radius (theorem 5) — these give instant right angles.
  • Watch for isosceles triangles formed by two radii; the base angles are equal, and this is often the hidden step.
  • Write the theorem name beside each step as you go.

Where marks are lost

Nearly always by giving the correct number with no reason, or by applying a theorem when its conditions do not hold — for example using "angles in the same segment" when the two angles stand on different arcs.

Think of it like this

Circle theorems are keys on a ring: each fits exactly one lock. The skill is not forcing a key but scanning the diagram for the feature — a diameter, a tangent, a cyclic quadrilateral — that tells you which key to reach for.

Worked examples

Method, step by step

A, B and C lie on a circle with centre O. Angle AOC at the centre is 130°. Find angle ABC at the circumference.

  1. 1Angles AOC and ABC both stand on the same arc AC.
  2. 2The angle at the centre is twice the angle at the circumference.
  3. 3So angle ABC = 130 ÷ 2 = 65°.

65° (angle at centre is twice the angle at the circumference)

Common misconceptions

  • Applying "angles in the same segment" to angles standing on different arcs. The arc must be the same.
  • Assuming any four-sided shape inside a circle is cyclic. All four vertices must lie **on** the circumference.
  • Halving instead of doubling in theorem 1. The angle at the **centre** is the larger one.

In the exam

  • Name the theorem for every step. A page of correct arithmetic with no reasons typically scores about half marks.
  • Look for two radii forming an isosceles triangle — it is the most commonly missed intermediate step in multi-stage problems.