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Maths Gcse Wjec-Eduqas Higher

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Exam code:C300

Cyclic quadrilaterals

Circle theorem: Opposite angles in a cyclic quadrilateral add up to 180°

  • A quadrilateral that is formed by four points on the circumference of a circle, (a cyclic quadrilateral), will have pairs of opposite angles that add up to 180°

A circle with points A, B, C and D on its circumference. The quadrilateral ABCD formed is a cyclic quadrilateral.

  • To spot this theorem in a diagram

    • look for quadrilaterals that have all four points on the circumference

  • When explaining this theorem in an exam you must use the keywords: 

    • Opposite angles in a cyclic quadrilateral add up to 180°

  • The theorem only works for cyclic quadrilaterals

    • The diagram below shows a common scenario that is not a cyclic quadrilateral

Not cyclic quad point at centre, IGCSE & GCSE Maths revision notes

Examiner Tips and Tricks

  • Cyclic quadrilaterals are often easy to spot in a busy diagram

    • Mark on their angles (even if you think you don’t need them) as they may help you later on!

Worked Example

The circle below has centre, O.

Find the value of x.

Q1 Circle Theorems 3, IGCSE & GCSE Maths revision notes

Identify both the cyclic quadrilateral and the radius perpendicular to the chord

Add to the diagram as you work through the problem

Q1 CT3 Working in red, IGCSE & GCSE Maths revision notes

The radius bisects the chord and so creates two congruent triangles

Use this to work out 72° (equal to the equivalent angle in the other triangle)
And 18° (angles in a triangle add up to 180°)

Then use that opposite angles in a cyclic quadrilateral add up to 180°

table row cell 2 x plus 4 plus 20 plus 18 end cell equals 180 row cell 2 x end cell equals 138 end table

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