Question

Find the value of x in the following figure, where O is the centre of the circle: 

Answer 1

Use the central angle

The given 160° O is the central angle ∠AOC.

A central angle equals the measure of the arc it subtends, so

AC = 160°

Find ∠ADC

∠ADC is an inscribed angle standing on arc AC.

For an inscribed angle:

inscribed angle = `1/2` × (intercepted arc)

x = ∠ADC = `1/2`AC

= `1/2`(160°)

= 80°

Quadrilateral ABCD is cyclic (all vertices on the circle), so opposite angles are supplementary:

∠ABC + ∠ADC = 180°.

At B, x is the exterior angle between the tangent and chord BC.

This exterior angle is supplementary to the interior angle ∠ABC

x = 180° − ∠ABC.

From the cyclic quadrilateral relation,

180° − ∠ABC = ∠ADC.

x = ∠ADC = 80°.