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प्रश्न
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 :
- ∠COB,
- ∠DOC,
- ∠DAC,
- ∠ADC.
उत्तर
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 ΔOCD, 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^circ - 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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