#### Question

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

(i) ∠ABD, (ii) ∠DBC, (iii) ∠ADC.

#### Solution

Join BD.

(i) `∠ABD = 1/2∠AOD = 1/2xx 60° = 30° `

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

(ii) ∠BDA = 90°

(Angle in a semicircle)

Also, ∠OAD 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)

Is there an error in this question or solution?

Solution Ab is the Diameter of the Circle with Centre O. Od is Parallel to Bc and ∠Aod = 60°. Calculate the Numerical Values of : (I) ∠Abd, (Ii) ∠Dbc, (Iii) ∠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.