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Cyclic Quadrilaterals (AQA GCSE Maths)
Revision Note
Cyclic Quadrilaterals
Circle theorem: Opposite angles in a cyclic quadrilateral add up to 180°
- This theorem states that if any quadrilateral is formed by four points that are on the circumference of a circle, then the angles opposite each other will add up to 180°
- A cyclic quadrilateral must have all four vertices on the circumference
- The theorem only works for cyclic quadrilaterals
- Do not be fooled by other quadrilaterals in a circle
- The diagram below shows a common scenario that is NOT a cyclic quadrilateral
- If giving the cyclic quadrilateral theorem as a reason in an exam, use the key phrase
- "Opposite angles in a cyclic quadrilateral add up to 180°"
- The word supplementary means angles that add up to 180° and could be used here as well you must reference a cyclic quadrilateral
Examiner Tip
- Identifying cyclic quadrilaterals quickly in a busy circle theorem question can help find angles and speed up answering these questions in an exam
- Just remember to check that all four vertices lie on the circumference
Worked example
The circle below has centre, O, find the value of x.
This is a busy diagram with a lot going on.
Identify both the cyclic quadrilateral and the radius that is 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 the cyclic quadrilateral theorem.
2x + 4 + 20 + 18 = 180
2x = 138
x = 69°
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