In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of ∠CAQ and ∠PAC. If ∠BAQ = 30°, prove that : BD is diameter of the circle.

In the given figure, PT touches the circle with center O at point R. Diameter SQ is produced to meet the tangent TR at P.

Given ∠SPR = x° and ∠QRP = y°;

prove that :

(i) ∠ORS = y°

(ii) write an expression connecting x and y.

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate :

(i) ∠CBT

(ii) ∠BAT

(iii) ∠APT

In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.