In the given figure, A is the center of the circle, ABCD is a parallelogram and CDE is a straight line. Prove that : ∠BCD = 2∠ABE.

∠BAD = 2∠BED [Angle at the center is double the angle at the circumference subtended by the same chord.]

Since, CDE is a straight line and CD || AB.

∴ AB || ED.

⇒ ∠BED = ∠ABE [Alternate angles are equal]

Multiplying above equation by 2 we get,

⇒ 2∠BED = 2∠ABE

⇒ ∠BAD = 2∠ABE ……………(1)

ABCD is a parallelogram.

⇒ ∠BAD = ∠BCD [Opposite angles of a paralellogram are equal] ……….(2)

From (1) and (2) we get,

⇒ ∠BCD = 2∠ABE.

Hence, proved that ∠BCD = 2∠ABE.