Question

In the given figure, AOB is a diameter of the circle and C, D, E are any three points on the semicircle. Find the value of ∠ACD + ∠BED.

Answer 1

Join AE.

AOB is the diameter of the circle.
C, D and E are the points on the semicircle.
ACDE is a cyclic quadrilateral.
Thus, ∠ACD + ∠DEA = 180° ...(i) (Sum of opposite angles of a cyclic quadrilateral = 180°)
Also, ∠AEB = 90° ...(ii) (Angle in a semicircle)
On adding (i) and (ii), we get:
(∠ACD + ∠ DEA) + ∠AEB = 180° + 90° = 270°
⇒ ∠ACD + (∠DEA + ∠AEB) = 270°
⇒ ∠ACD + ∠DEB = 270° [∵ ∠DEA + ∠AEB = ∠DEB]
Hence, ACD + ∠DEB = 270°