Circle theorems

Circle theorems are facts about angles in circles. Each one helps you find missing angles, and in proofs you must name the theorem as your reason.

Key facts

  • The angle at the centre is twice the angle at the circumference.
  • The angle in a semicircle is 90°.
  • Angles in the same segment are equal.
  • Opposite angles in a cyclic quadrilateral add to 180°.
  • A tangent meets a radius at 90°; tangents from a point are equal.
  • Alternate segment theorem: the angle between a tangent and chord equals the angle in the alternate segment.

Worked examples

Example 1

O is the centre. The angle at the circumference is 35°. Find the angle at the centre on the same arc.

  1. Angle at the centre = 2 × angle at the circumference.
  2. 2 × 35 = 70°

Answer: 70°

Example 2

ABCD is a cyclic quadrilateral with angle A = 104°. Find angle C.

  1. Opposite angles in a cyclic quadrilateral add to 180°.
  2. 180 − 104 = 76°

Answer: 76°