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.

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.

Philoid · Accepted Answer

Construction: Join AE

Consider cyclic quadrilateral ACDEA

Here,

∠ACD + ∠DEA = 180° (opposite angles in cyclic quadrilateral are supplementary)

Also,

∠AEB = 90° (angle in semicircle)

∴∠ACD + ∠DEA + ∠AEB = 180° + 90°

∠ACD + ∠BED = 270° (∠DEA + ∠AEB = ∠BED)

∴∠ACD + ∠BED = 270°

Hence proved.