1. 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

2. In a circle, AB and AC are two chords. Tangents are drawn at points B and C, and they intersect at point P. If \(\angle\)BAC = 54°, then what is the measure of \(\angle\)BPC?

3. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

4. The perpendicular bisectors of the sides of \(\triangle\)ABC intersect at point O. If \(\angle\)OAB = 50°, find the measure of \(\angle\)ACB.

(a) 50° (b) 100° (c) 40° (d) 180°

5. ABCD is a cyclic quadrilateral. The bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y. Prove that XY is the diameter of the circle.

6. 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.

7. In the given figure, the angle between the two radii OA and OB of a circle centered at O is 130°. Tangents drawn at points A and B intersect at point T. Calculate and write the measures of \(\angle\)ATB and \(\angle\)ATO.

8. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

9. ABCD is a cyclic quadrilateral, and CD is extended up to point E. If \(\angle\)ADE = 92°, then what is the measure of \(\angle\)ABC?

(a) 88\(^o\) (b) 29\(^o\) (c) 92\(^o\) (d) 60\(^o\)

10. In triangle ABC, AB = AC. A line drawn from point C intersects the extended line BA at point D. If AC = AD, what is the measure of \(\angle\)BCD in radians?

(a) \(\cfrac{π}{2}\) (b) \(π\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{3}\)

11. O is the center of a circle. PQ is a diameter, and R is a point on the circumference. If \(\angle\)PQR = 40°, then what is the measure of \(\angle\)POR?

(a) 80° (b) 40° (c) 20° (d) 100°

12. AB is a diameter of a circle with center O. CD is a chord whose length is equal to the radius of the circle. When AC and BD are extended, they intersect at point P. What is the measure of ∠APB?

13. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

14. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?

15. Two radii, OA and OB, of a circle centered at O form an angle of 130°. Tangents are drawn at points A and B, and they intersect at point T. Find the measures of \(\angle\)ATB and \(\angle\)ATO.

16. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

17. 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.

18. AB is extended to point X in the cyclic quadrilateral ABCD. If \(\angle\)XBC = 98° and \(\angle\)ADB = 45°, then what is the measure of \(\angle\)BAC?

19. Two circles touch externally at point C. AB is a common tangent to both circles, touching them at points A and B. The measure of \(\angle\) ACB is?

(a) 60° (b) 45° (c) 90° (d) 75°

20. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

21. 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.

22. In a circle centered at O, PQ is a diameter; R is a point on the circle, and PR = RQ. Then, the measure of \(\angle\)RPQ is: ____?

(a) 30° (b) 90° (c) 60° (d) 45°

23. In the adjacent figure, AOB is the diameter and center of the circle. Radius OC is perpendicular to AB. If point P lies on the arc CB, calculate and write the measures of \(\angle\)BAC and \(\angle\)APC.

24. ABC is an acute-angled triangle. AP is the diameter of the circumcircle of triangle ABC. BE and CF are perpendiculars dropped to sides AC and AB respectively, and they intersect at point Q. Prove that BPCQ is a parallelogram.

25. The internal and external angle bisectors of the vertex angle of a triangle intersect the circumcircle of the triangle at points P and Q respectively. Prove that PQ is a diameter of the circle.

26. ABCD is a cyclic quadrilateral. The angle bisectors of ∠DAB and ∠BCD intersect the circle at points X and Y, respectively. Prove that XY is a diameter of the circle.

27. In the adjacent figure, O is the center of the circle, and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extension of BA at point T. If ∠PBO = 30°, then find the measure of ∠PTA.

28. 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.

29. In the cyclic quadrilateral ABCD, side AB is extended up to point X. If \(\angle\)XBC = 82° and \(\angle\)ADB = 47°, then what is the measure of \(\angle\)BAC?

(a) 45° (b) 45° (c) 35° (d) 60°