1. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

2. In a circle with center O, two chords AB and CD intersect at point P. If \(\angle\)AOD = 100° and \(\angle\)BOC = 70°, find the measure of \(\angle\)APC.

3. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.

4. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

5. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

6. AT is a tangent at point A on a circle with center O. BC, the extended diameter, intersects the tangent at point T. Given that \(\angle\)ABC = 25°, determine the value of \(\angle\)ATB.

7. AOB is the diameter of a circle. AC and BD are two chords that intersect at point E. If \(\angle\)COD = 40°, find the value of \(\angle\)CED.

8. Two chords AB and CD of a circle, centered at O, intersect each other internally at point P. Prove that \(\angle\)AOD + \(\angle\)BOC = 2\(\angle\)BPC.

9. In a circle centered at O, PQ and PR are two chords. The tangents drawn at points Q and R intersect at point S. If \(\angle\)QSR = 70°, determine the value of \(\angle\)QPR.

10. AB and CD are two chords of a circle with center O. When extended, they intersect at point P. Prove that \(\angle AOC - \angle BOD = 2\angle BPC\).

11. ABCD is a cyclic quadrilateral. The angle bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y respectively. If O is the center of the circle, find the value of \(\angle\)XOY.

12. When the chords AB and CD of a circle with center O are extended, they intersect each other at point P. Prove that \(\angle\)AOC − \(\angle\)BOD = 2\(\angle\)BPC.

13. In a circle with center O, where BOC is the diameter and ABCD is a cyclic quadrilateral, if \(\angle ADC = 110^\circ\), then find the value of \(\angle ACB\).

14. Two circles intersect at points A and B. Two straight lines passing through points A and B intersect the circles at points P, Q and R, S respectively. If \(\angle\)APR = 80° and \(\angle\)BRP = 78°, then find the values of \(\angle\)AQS and \(\angle\)BSQ.

15. QR is a chord of a circle with center O. Tangents are drawn at points Q and R, and they intersect each other at point P. QM is the diameter of the circle. Prove that \(\angle\)QPR = 2\(\angle\)RQM.