#### Question

In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40° and ∠ABD = 60°, find;

(i) ∠DBC (ii) ∠BCP (iii) ∠ADB

#### Solution

(i)PQ is tangent and CD is a chord

∴ `∠`DCQ = `∠` DBC (angles in the alternate segment)

^{∴}DBC = 40° (∵ `∠`DCQ = 40°)

ii) `∠`DCQ + `∠`DCB + `∠`BCP = 180°

⇒ 40° + 90° + `∠`BCP = 180° (∵ `∠`DCB = 90°)

⇒ `∠`BCP = 180° = 130° = 50°

iii) In Δ ABD

`∠`ADB = 180° , `∠`ABD = 60°

∴ `∠`ADB = 180° - ( 90° + 60°)

⇒ `∠`ADB = 180° - 150° = 30°

Is there an error in this question or solution?

Solution In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40° and ∠ABD = 60°, find; (i) ∠DBC (ii) ∠BCP (iii) ∠ADB Concept: Cyclic Properties.