The circle theorems

3) Cyclic quadrilaterals

A cyclic quadrilateral is a quadrilateral drawn inside a circle. Every vertex of the quadrilateral must touch the circumference of the circle.

Cyclic and non-cyclic quadrilateral

The second shape is not a cyclic quadrilateral. One vertex does not touch the circumference. The opposite angles in a cyclic quadrilateral add up to 180°.

Cyclic quadrilateral with angles a, b, c and d

α + c = 180° and b + d = 180°


Example

Calculate the angles α and b.

Cyclic quadrilateral with angles a, b, 60 degrees and 140degrees

The opposite angles in a cyclic quadrilateral add up to 180°.

b = 180° – 140° = 40° and α = 180° – 60° = 120°


Proof

Let angle CDE = x and angle EFC = y.

Cyclic quadrilateral (angles x and y at the circumference)

The angle at the centre is twice the angle at the circumference.

Angle COE = 2and the reflex angle COE = 2x.

Cyclic quadrilateral with angles x and y at the circumference and 2x and 2y at the centre

Angles around a point add up to 360°.

2x + 2y = 360° so x + y = 180°


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