In the given figure, O is the center of the circumcircle ABC. Tangents A and C intersect at P. Given angle AOB = 140° and angle APC = 80°; find the angle BAC.

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.