What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided shape whose four vertices all lie on one circle. Its angles follow two neat rules that make many circle problems quick.
Key idea
Opposite angles of a cyclic quadrilateral add up to 180° (they are supplementary). The exterior angle equals the interior opposite angle.
Using the opposite-angles rule
Label the quadrilateral ABCD. Then angle A + angle C = 180° and angle B + angle D = 180°. If angle A = 85°, its opposite angle C = 180 − 85 = 95°. If angle B = 115°, then angle D = 180 − 115 = 65°.
Worked example
In cyclic quadrilateral PQRS, angle P = 100° and angle Q = 70°. Opposite pairs give angle R = 180 − 100 = 80° and angle S = 180 − 70 = 110°. Check: 100 + 70 + 80 + 110 = 360°, as any quadrilateral should.
The exterior angle
Extend one side. The exterior angle at a vertex equals the interior angle at the opposite vertex. If that opposite interior angle is 75°, the exterior angle is also 75°. This follows from the opposite-angles rule, because the exterior angle and its own interior angle already add to 180°.
These two rules together let you find every angle of a cyclic quadrilateral once you know just two of them, so look first for a pair of opposite angles or an extended side whenever a four-sided shape sits inside a circle.
Remember
- Only works when all 4 vertices are on the circle.
- Opposite angles sum to 180°, not adjacent ones.
- All four interior angles still add to 360°.