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 2x and x respectively.

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