Angles of a Cyclic Quadrilateral
Simple Explanation
A cyclic quadrilateral is a four-sided figure whose vertices all lie on a single circle. In any cyclic quadrilateral, each pair of opposite angles adds up to exactly 180° (they are supplementary).
Why Do We Need It?
This lets you find a missing angle in a cyclic quadrilateral instantly from its opposite angle, without any further measurement.
See It
A four-sided figure ABCD inscribed in a circle, with opposite angles at B and D marked
Formula
Opposite Angles of a Cyclic Quadrilateral
∠A + ∠C = 180° and ∠B + ∠D = 180°
In any quadrilateral whose four vertices all lie on a circle (a cyclic quadrilateral), each pair of opposite angles adds up to exactly 180°.
- ∠A, ∠B, ∠C, ∠D
- — the four interior angles of the cyclic quadrilateral, in order around the circle
When to use it: Whenever a quadrilateral is inscribed in a circle and you need to find a missing angle from its opposite angle.
Worked Example
Find a missing angle in a cyclic quadrilateral
In cyclic quadrilateral ABCD, ∠A = 105°. Find ∠C.
Why Does This Work?
Stated simply: draw the two diagonals from the centre O to each vertex (or use the inscribed angle theorem directly). Angle B is an inscribed angle subtending the arc ADC (the arc NOT containing B), and angle D is an inscribed angle subtending the arc ABC (the arc not containing D) — together, these two arcs make up the entire circle, 360°. Since each inscribed angle is half its arc, ∠B + ∠D = (1/2)(arc ADC + arc ABC) = (1/2)(360°) = 180°.
Real-Life Example
Designing four-sided structures around a circular plaza
An architect designs a four-cornered walkway with all four corner posts placed on the boundary of a circular plaza.
Knowing the cyclic quadrilateral angle rule, the architect can verify or design the corner angles so opposite corners always sum to exactly 180°.
Practice
In cyclic quadrilateral PQRS, ∠Q = 98°. Find ∠S.
HardCommon mistake
Adding adjacent angles instead of opposite angles — the 180° rule applies specifically to opposite pairs (A&C, B&D), not to angles that share a side.
Quick Review
- Opposite angles of a cyclic quadrilateral sum to 180°: ∠A+∠C=180°, ∠B+∠D=180°.
- Proof idea: opposite angles are inscribed angles subtending arcs that together make the whole circle (360°).
- Only applies when all four vertices lie on the same circle.