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In the Given Figure, Ab = Bc = Cd and ∠Abc = 132 . Calcualte: (I) ∠Aeb, (Ii) ∠Aed, (Iii) ∠Cod. - 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, AB = BC = CD and ∠ABC = 132 . Calcualte:

(i) ∠AEB,  (ii) ∠AED,  (iii) ∠COD.

Solution

In the figure, O is the centre of circle, with AB = BC = CD.
Also, ∠ABC = 132°

(i) In cyclic quadrilateral ABCE
∠ABC + ∠AEC = 180°               [sum of opposite angles]
→ ∠132 + ∠AEC = 180°
→ ∠AEC = 180° -132°
→ ∠AEC = 48°

Since, AB = BC, ∠AEB = ∠BEC          [equal chords subtends equal angles]
∴ `∠AEB = 1 /2∠AEC`
           = `1 /2xx 48° `
           = 24°

(ii) Similarly, AB = BC = CD
∠AEB = ∠BEC = ∠CED = 24°
∠AED = ∠AEB + ∠BEC + ∠CED
= 24° + 24° + 24°= 72°

(iii) Arc CD subtends ∠COD at the centre and
∠CED at the remaining part of the circle.
∴ COD = 2∠CED
= 2 × 24°
= 48°

  Is there an error in this question or solution?
Solution In the Given Figure, Ab = Bc = Cd and ∠Abc = 132 . Calcualte: (I) ∠Aeb, (Ii) ∠Aed, (Iii) ∠Cod. 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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