Circle theorems
Eight standard theorems used to deduce angles and lengths in circles. Higher tier.
Higher-tier requires knowing and applying eight theorems: angle at centre = 2× angle at circumference; angles in same segment equal; angle in semicircle = 90°; opposite angles in cyclic quadrilateral sum to 180°; tangent perpendicular to radius; alternate segment theorem; tangents from external point equal length; perpendicular from centre bisects chord.
Worked examples
Angle at centre = 2 × angle at circumference (subtending the same arc).
Angle in a semicircle = 90°. If a triangle has a side as diameter, the opposite angle is 90°.
Cyclic quadrilateral: opposite angles sum to 180°.
Frequently asked questions
How are these examined?
Find missing angles using one or more theorems. Justify with theorem name (this scores marks).
Alternate segment theorem?
Tangent-chord angle equals the angle in the alternate segment. Often the trickiest.