In the following figure, O is the centre of the circle and ΔABC is equilateral.

Find:

(i) ∠ADB, (ii) ∠AEB.

(i) We know that each angle in an equilateral triangle = 60°.

∴ ∠ACB = 60°

As angles in same segment are equal.

∴ ∠ADB = ∠ACB = 60°.

Hence, ∠ADB = 60°.

(ii) Join OA and OB.

We know that,

Angle at the center is double the angle at the circumference subtended by the same chord.

∴ ∠AOB = 2∠ACB = 2 x 60° = 120°.

∴ ∠AEB = $\dfrac{1}{2}$ Reflex ∠AOB

= $\dfrac{1}{2}$(360° - 120°) = $\dfrac{1}{2} \times 240°$

= 120°.

Hence, ∠AEB = 120°.