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;

1. ∠DBC
2. ∠BCP
3. ∠ADB

Answer 1

 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°)

`=>` 130° + ∠BCP = 180°

∴ ∠BCP = 180° – 130° = 50°

iii. In ΔABD,

 ∠BAD = 90°, ∠ABD = 60°

∴  ∠ADB = 180° – (90° + 60°)

`=>` ∠ADB = 180° – 150° = 30°