medium difficulty5 min

Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary.

In one line: Cyclic opposite angles complete 180.

Practice This Concept

Understand it

If ABCD is cyclic, then angle A + angle C = 180 degrees and angle B + angle D = 180 degrees. The converse is also useful: if a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic.

Formula and rule

For cyclic quadrilateral ABCD: angle A + angle C = 180 degrees and angle B + angle D = 180 degrees.

Converse: if one opposite pair is supplementary, the quadrilateral is cyclic.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

If angle A = 80 degrees and angle C = x in cyclic quadrilateral ABCD, then x = 180 - 80 = 100 degrees.

Method you can reuse

1Check whether all four vertices lie on a circle.
2Identify opposite angles.
3Add them to 180 degrees or use the converse to prove cyclicity.

Sketch it in your answer

Draw quadrilateral ABCD inside a circle with all four vertices on the circumference and mark opposite angle pairs.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In cyclic quadrilateral ABCD, angle B = 115 degrees. What is angle D?

Think first, then show answer

Angle D is 65 degrees because opposite angles of a cyclic quadrilateral are supplementary, so angle B + angle D = 180 degrees.

Common wrong answer

If angle A = 70 degrees, writing adjacent angle B = 110 degrees without more information.

Why it loses marks: The rule applies to opposite angles.

Don't mix this up

Students sometimes add adjacent angles instead of opposite angles. The cyclic quadrilateral rule is for opposite angle pairs.

Exam tip

In a cyclic quadrilateral, mark opposite angle pairs with different symbols and write their sums separately.