Circle theorems and angle reasoning

Name the chord, arc or tangent that makes a circle theorem applicable.

Angles subtended by the same chord in the same segment are equal. The angle at the centre is twice the angle at the circumference when both stand on the same arc. Opposite angles of a cyclic quadrilateral sum to 180°. Check the positions of the points: the same chord alone does not put two vertices in the same segment.

A radius is perpendicular to the tangent at its point of contact. The perpendicular from the centre to a chord bisects the chord. The alternate segment theorem relates an angle between a tangent and chord to the angle made by that chord in the opposite segment. Write each angle deduction with its reason, and use triangle or straight-line angle sums where needed.

Worked example

ABCD is a cyclic quadrilateral. Angle DAB = 3x + 10° and angle BCD = 2x + 20°. Find both angles.

The two angles are opposite angles in a cyclic quadrilateral.
Their sum is 180°, so 5x + 30 = 180 and x = 30.
Substitute into each expression.

Answer: 100° and 80°

Practise circle theorems and angle reasoning

Course mapping

These specification references show where the topic occurs. The questions cover only some parts of each topic.

8365 · core · 3.7: Chord length from its distance to the centre
8365 · core · 6.1: An angle in a cyclic quadrilateral
7M20 · core · G4.1: An angle in a cyclic quadrilateral

Next practice: Writing a geometric proof, Circles and tangents.