Question

AB is the diameter of the circle with centre O. OD is parallel to BC and ∠AOD = 60°. Calculate the numerical values of : 

1. ∠ABD,
2. ∠DBC, 
3. ∠ADC. 

Answer 1

Join BD.

i. `∠ABD = 1/2 ∠AOD = 1/2xx 60^circ = 30^circ` 

(Angle at the first is double the angle at the circumference subtended by the same chord)

ii. ∠BDA = 90°

(Angle in a semicircle)

Also, ΔOAD is equilateral  (∵ ∠OAD = 60°)

∴ ∠ODB = 90° – ∠ODA

= 90° – 60°

= 30°

Also, OD || BC

∴ ∠DBC = ∠ODB = 30° (Alternate angles)

iii. ∠ABC = ∠ABD + ∠DBC

= 30° + 30°

= 60°

In cyclic quadrilateral ABCD,

∠ADC = 180° – ∠ABC

= 180° – 60°

= 120°

(Pair of opposite angles in a cyclic quadrilateral are supplementary)