Question

ABCD is a cyclic quadrilateral and PQ is a tangent to the circle at C. If BD is a diameter, angle DCQ=40 and angle ABD =60 then find the measure of,

i) angle DBC
ii) angle BCP
iii)angle DBC
iv)angle ADB

Answer 1

please see above for the solution

and sorry for I am unable to find the (iii) one.

Answer 2

(i)     ∠DBC = 40°

(ii)    ∠BCP = 50°

(iii)    ∠ADB = 30°

Step-by-step explanation:

Given, ABCD is a cyclic quadrilateral

          PQ is tangent and BD is diameter

           ∠DCQ = 40°  and ∠ABD = 60°

(i) Angle DBC

PQ is tangnet and CD is chord

∴ ∠DCQ = ∠DBC....... [ angles in the alternate segment]

∴ ∠DBC = 40°

(ii) Angle BCP

∠DCQ + ∠DCB + ∠BCP = 180°

40° + 90° + ∠BCP =  180°.....[∵ DCB = 90°]

∠BCP = 180° - 130°

∠BCP = 50°

(iii) Angle DBC

Two times repeated it is same as (i)

(iv) Angle ADB

In ΔABD

∠ABD + ∠DAB + ∠ADB = 180°

60° + 90° + ∠ADB = 180°

∠ADB = 180° - (60 +90)°

∠ADB = 30°