In the given figure, ∆ABC is equilateral.

(i)
Given: ΔABC is an equilateral triangle.
i.e., each of its angle = 60°
⇒ ∠BAC = ∠ABC = ∠ACB = 60°
Angles in the same segment of a circle are equal.
i.e., ∠BDC = ∠BAC = 60°
∴ ∠BDC = 60°
(ii)
The opposite angles of a cyclic quadrilateral are supplementary.
Then in cyclic quadrilateral ABEC, we have:
∠BAC + ∠BEC = 180°
⇒ 60° + ∠BEC = 180°
⇒ ∠BEC = (180° – 60°) = 120°
∴ ∠BDC = 60° and ∠BEC = 120°