C
CraftExam
high importancemedium8 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.

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    If ABCD is cyclic and angle A = 72 degrees, then angle C = 108 degrees because opposite angles add to 180 degrees.

  4. 4

    Avoid this incomplete answer

    If angle A = 70 degrees, writing adjacent angle B = 110 degrees without more information is wrong because the rule applies to opposite angles.

Definition

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

Example

If ABCD is cyclic and angle A = 72 degrees, then angle C = 108 degrees because opposite angles add to 180 degrees.

Rule to remember

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.

Memory hook

Cyclic opposite angles complete 180.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

Problems may ask for a missing angle, proof that a quadrilateral is cyclic, or use the converse with supplementary opposite angles.

Avoid common mistakes

Common confusion

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

Common wrong answer

If angle A = 70 degrees, writing adjacent angle B = 110 degrees without more information is wrong because the rule applies to opposite angles.

Exam tip

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

Quick check

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

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

Study the cyclic quadrilateral diagram carefully

Use the labelled diagram to keep cyclic quadrilateral clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of cyclic quadrilateral in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on cyclic quadrilateral.

Revision cue

Revise cyclic quadrilateral through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary. 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. If ABCD is cyclic and angle A = 72 degrees, then angle C = 108 degrees because opposite angles add to 180 degrees.

3-mark answer

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. 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. If angle A = 80 degrees and angle C = x in cyclic quadrilateral ABCD, then x = 180 - 80 = 100 degrees. Problems may ask for a missing angle, proof that a quadrilateral is cyclic, or use the converse with supplementary opposite angles. If angle A = 70 degrees, writing adjacent angle B = 110 degrees without more information is wrong because the rule applies to opposite angles.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept