Cyclic quadrilateral

Definition

In a cyclic quadrilateral opposite angles are supplementary and the exterior angle equals the interior opposite angle. Proving a quadrilateral is cyclic is a standard Euclidean geometry question.

Worked example

If ∠A = 105° in cyclic quadrilateral ABCD, then ∠C = 75°.

In cyclic quadrilateral ABCD with BC extended to E, if exterior ∠DCE = 80°, then ∠A = 80° (exterior angle of a cyclic quadrilateral equals the interior opposite angle).

Where it appears in the exam

Paper 2 Euclidean geometry asks you to prove that a quadrilateral is cyclic, or to use cyclic quadrilateral properties to find angles, with reasons.

Step-by-step method

1. Check that all four vertices lie on one circle.
2. Use opposite angles supplementary, or exterior angle = interior opposite angle.
3. Write every statement with its reason.
4. To prove a quadrilateral is cyclic, use the converse theorems.

Common mistake

Writing a statement without its reason, which earns no mark in Euclidean geometry.

Related euclidean geometry terms

• Similar triangles

Frequently asked questions

What does cyclic quadrilateral mean in Grade 12 Maths?

In a cyclic quadrilateral opposite angles are supplementary and the exterior angle equals the interior opposite angle. Proving a quadrilateral is cyclic is a standard Euclidean geometry question.

Which NSC paper tests cyclic quadrilateral?

Cyclic quadrilateral belongs to the Euclidean geometry section of CAPS Grade 12 Mathematics, which is examined in Paper 2.

How do I practise questions involving cyclic quadrilateral?

Snap a question that uses cyclic quadrilateral into the Snap&Learn AI solver, or work through the euclidean geometry topic guide and NSC past papers.