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Cyclic quadrilateral: opposite angles

2026-09-30Plane GeometryMs Ho

A quadrilateral is cyclic if a circle passes through all four vertices; opposite angles sum to 180°.

Key formulas

∠A + ∠C = 180°
∠B + ∠D = 180°

Test

If opposite angles of a quadrilateral sum to 180°, the points are concyclic (converse).

Example

Angles 95°, 70°, 85°, x. Opposite 95 and 85 already sum to 180. Then 70+x=180 → x=110.

Exterior angle

Exterior angle of a cyclic quadrilateral equals the opposite interior angle — another quick tool.

Worked example

Worked example

Problem. In cyclic quadrilateral ABCD, ∠A = 95° and ∠B = 70°. Find ∠C and ∠D.

Solution. Opposite angles sum to 180°: ∠C = 180° − 95° = 85°, ∠D = 180° − 70° = 110°.

Not every kite

A kite is cyclic only in special cases; do not assume it.

Review quiz

Cyclic quadrilateral has ∠B=100°. Opposite ∠D is:

Practice questions

  1. Opposite angles 102° and y. Find y.
  2. Three angles of a cyclic quadrilateral are 80°, 95°, 100°. Find the fourth.

Study notes

Rectangle?

Every rectangle is cyclic because all angles are 90°.

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