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In the Given Figure, Ae is the Diameter of the Circle. Write Down the Numerical Value of ∠Abc+∠Cde. Give Reasons for Your Answer. - ICSE Class 10 - Mathematics

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ConceptArc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle

Question

In the given figure, AE is the diameter of the circle. Write down the numerical value of ∠ABC+∠CDE. Give reasons for your answer.

Solution

Join centre O and C and EC.
`∠AOC = (180°)/2 = 90°`
And ∠AOC = 2∠AEC
(Angle at the centre is double the angle at the circumference subtended by the same chord)

⇒`∠AEC = (90°)/2 = 45°`
Now, ABCE is a cyclic quadrilateral
∴ ∠ABC + ∠AEC = 180°
(pair of opposite angles in a cyclic quadrilateral are supplementary)
⇒ ∠ABC =180° - 45° =135°
Similarly, ∠CDE = 135°
∴ ∠ABC + CDE -135° +135° = 270°

  Is there an error in this question or solution?
Solution In the Given Figure, Ae is the Diameter of the Circle. Write Down the Numerical Value of ∠Abc+∠Cde. Give Reasons for Your Answer. Concept: Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle.
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