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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

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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, 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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