In each problem below, draw a circle and a chord to divide it into two parts such that the parts are as specified;

i) All angles on one part 80°.

ii) All angles on one part 110°.

iii) All angles on one part half of all angles on the other.

iv) All angles on one part, one and a half times the angles on the other.

Theorem:

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

(i) In the circle, let us draw a minor arc PR, which subtends an angle ∠ POR = 160° at the centre. Then, from the theorem, the angle at any other part will be

(ii) Here, AB is the major arc. It subtends an ∠ 220° at the centre, then, in its minor segment, the angle at any point will be half of 220°

(iii) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

From the figure (i), (ii) and (iii), ∠APB = 1/2 ∠AOB

(iv) Draw an angle 144° at the centre. Then, all angles on one part, one and a half times the angles on the other.