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# In the Given Figure, Ac is the Diameter of the Circle with Centre O. Cd and Be Are Parallel. Angle ∠Aob = 80° and ∠Ace = 10°. Calculate: (I) Angle Bec, (Ii) Angle Bcd, (Iii) Angle Ced - ICSE Class 10 - Mathematics

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, AC is the diameter of the circle with centre O. CD and BE are parallel. Angle
∠AOB = 80° and ∠ACE = 10°.
Calculate :
(i) Angle BEC,
(ii) Angle BCD,
(iii) Angle CED

#### Solution

(i) ∠BOC = 180° - 80° = 100° (Straight line)
And ∠BOC = 2∠BEC
(Angle at the centre is double the angle at the circumference subtended by the same chord)
⇒ ∠BEC = (100°)/2 = 50°

(ii) DC || EB
∴ DCE = ∠BEC = 50° (Alternate angles)
∴ ∠AOB = 80°
⇒ ∠ACB = 1 /2 ∠AOB = 40°
(Angle at the center is double the angle at the circumference subtended by the same chord) We have,
∠BCD = ∠ACB + ∠ACE + ∠DCE = 40° +10°+ 50° = 100°

(iii) ∠BED =180° - ∠BCD =180° -100° = 80°
(pair of opposite angles in a cyclic quadrilateral are supplementary)
⇒∠CED + 50° = 80°
⇒ ∠CED = 30°

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Solution In the Given Figure, Ac is the Diameter of the Circle with Centre O. Cd and Be Are Parallel. Angle ∠Aob = 80° and ∠Ace = 10°. Calculate: (I) Angle Bec, (Ii) Angle Bcd, (Iii) Angle Ced 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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