Angles in Cyclic Quadrilaterals (Cambridge (CIE) IGCSE International Maths)

Revision Note

Written by: Amber

Reviewed by: Dan Finlay

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°

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

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!

The circle below has centre, O.

Find the value of $x$ .

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

Add to the diagram as you work through the problem

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°

$\begin{array}{rcl} 2 x + 4 + 20 + 18 & = & 180 \\ 2 x & = & 138 \end{array}$

$x = 69 °$

Last updated: 27 May 2024

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