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Question
In the figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of ∠ACD+∠BED.
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Solution
In the given figure, join AE.
Since, A, C, D and E are four points on a circle, then ACDE is a cyclic quadrilateral.
∠ACD+∠AED=180∘...........(i) [sum of opposite angles in a cyclic quadrilateral is 180∘]
Now, ∠AEB=90∘..............(ii) [The diameter subtends a right angle to the circle.]
On adding Eqs. (i) and (ii), we get
(∠ACD+∠AED)+∠AEB=180∘+90∘=270∘
⇒ ∠ACD+∠BED=270∘
Since, A, C, D and E are four points on a circle, then ACDE is a cyclic quadrilateral.
∠ACD+∠AED=180∘...........(i) [sum of opposite angles in a cyclic quadrilateral is 180∘]
Now, ∠AEB=90∘..............(ii) [The diameter subtends a right angle to the circle.]
On adding Eqs. (i) and (ii), we get
(∠ACD+∠AED)+∠AEB=180∘+90∘=270∘
⇒ ∠ACD+∠BED=270∘
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