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# In the Given Figure, Aob is a Diameter and Dc is Parallel to Ab. If ∠Cab = X°; Find (In Terms of X) the Values of ; (I) ∠Cob, (Ii) ∠Doc, (Iii) ∠Dac (Iv) ∠Adc. - 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, AOB is a diameter and DC is parallel to AB. If ∠CAB = x°; find (in terms of x) the values of ;
(i) ∠COB, (ii) ∠DOC, (iii) ∠DAC (iv) ∠ADC. #### Solution (i) ∠COB = 2 ∠CAB = 2x
(Angle ate the centre is double the angler at the circumference subtended by the same order)

(ii) ∠OCD = ∠COB = 2x  (Alternate angles)
In ∠OCD, OC = OD
∴ ∠ODC = ∠OCD = 2x

By angle sum property of ∆OCD,
∠DOC = 180° - 2x - 2x = 180° - 4x

(iii) ∠DAC = 1 /2∠DOC = 1/2 (180° - 4x ) = 90° - 2x
(Angle at the centre is double the angle at the circumference subtended by the same chord)

(iv) DC || AO
∴ ∠ACD = ∠OAC = x (Alternate angles) By angle sum property,
∠ADC = 180° - ∠DAC - ∠ACD = 180° - (90° - 2x ) - x = 90° + x

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Solution In the Given Figure, Aob is a Diameter and Dc is Parallel to Ab. If ∠Cab = X°; Find (In Terms of X) the Values of ; (I) ∠Cob, (Ii) ∠Doc, (Iii) ∠Dac (Iv) ∠Adc. 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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