### 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.

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°.

*α* + *c* = 180° and *b *+ *d *= 180°

**Example**

Calculate the angles *α* and *b*.

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*.

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

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

Angles around a point add up to 360°.

2*x* + 2*y* = 360° so *x + y *= 180°