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 :

1. ∠COB,
2. ∠DOC,
3. ∠DAC,
4. ∠ADC.

Answer 1

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