Circle Theorems - Cyclic Quadrilaterals
# Circle Theorems - Cyclic Quadrilaterals

GCSE(H),

A quadrilateral where all four vertices touch the circumference of a circle is known as a **cyclic quadrilateral**.

The angle at the centre of a circle is twice that of an angle at the circumference when subtended by the same arc. For arc D-A-B, let the angles be 2*x* and *x* respectively.

For the arc D-C-B, let the angles be 2*y* and *y*.

At the centre of the circle, `360 = 2(x +y), text(or) 180 = x + y`

*Opposite angles in a cyclic quadrilateral add up to 180º.*

The exterior angle, ∠BCE, is 180 - *y* = *x*. *The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.*

## Examples

1. What is the value of the angle ∠DAB?

Answer: 78º

Opposite angles in a quadrilateral add up to 180º.

180 - 102 = 78º

2. Angle ∠ECB is 79º. What is the value of the angle ∠DAB?

Answer: 79º

The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

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