1. ABCD is a cyclic quadrilateral and O is the center of the circle. Given: \(\angle COD = 130^\circ\), \(\angle BAC = 25^\circ\) Find the values of \(\angle BOC\) and \(\angle BCD\).

(a) 40\(^o\),90\(^o\) (b) 50\(^o\),90\(^o\) (c) 65\(^o\),50\(^o\) (d) None of the above

2. In the adjacent figure, O is the center of the circle and \(\angle APB = 80^\circ\). Determine the sum of the measures of \(\angle AOB\) and \(\angle COD\), and provide reasoning to support your answer.

3. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle COD = 120^\circ\) and \(\angle BAC = 30^\circ\), then calculate the values of \(\angle BOC\) and \(\angle BCD\).

4. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

5. If \( O \) is the circumcenter of triangle \( ABC \) and \( \angle BAC = 40^\circ \), determine the value of \( \angle OBC \).

(a) 100° (b) 30° (c) 50° (d) 40°

6. In the cyclic quadrilateral ABCD, \( AB = AD \); given \( \angle DAC = 60° \) and \( \angle BDC = 50° \), determine the value of \( \angle ACD \).

7. In the given figure, \(O\) is the center of the circle, and \(AB\) is the diameter. If \(\angle ADC = 120^\circ\), then determine the value of \(\angle BAC\).

(a) 50° (b) 60° (c) 40° (d) 30°

8. O is the circumcenter of triangle ABC. If \(\angle BAC = 85^\circ\) and \(\angle BCA = 55^\circ\), then determine the value of \(\angle OAC\).

9. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

10. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is:

(a) 70° (b) 110° (c) 35° (d) 55°

11. 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°

12. 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°

13. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\).

(a) 60° (b) 80° (c) 120° (d) None of these

14. In triangles \(ABC\) and \(DEF\), if \(\angle A = \angle F = 40^\circ\), \(AB:ED = AC:EF\), and \(\angle F = 65^\circ\), find the value of \(AB\).

(a) 35° (b) 65° (c) 75° (d) 85°

15. On triangle \( \triangle ABC \), points P and Q are such that \( \angle ABC = \angle APQ \). Given that \( AP = 3.6 \) cm, \( QC = 1.6 \) cm, and \( AQ = 4.8 \) cm, find the length of \( PB \).

(a) 1.2 cm (b) 2.4 cm (c) 6 cm (d) None of the above

16. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

17. ABCD is a cyclic quadrilateral. Side AB is extended to point X. If \(\angle XBC = 82^\circ\) and \(\angle ADB = 47^\circ\), find the value of \(\angle BAC\).

18. 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 \(\angle PTA\)?

19. In a circle centered at O, ABCD is a cyclic quadrilateral. Given: \(\angle ABD = 50^\circ\), \(\angle CAD = 28^\circ\), and \(\angle ADB = 32^\circ\) Find the measure of \(\angle BCD\).

(a) 72° (b) 52° (c) 62° (d) 82°

20. In triangle \(\triangle ABC\), the incenter is \(O\), and \(\angle BOC = 120^\circ\). Find the measure of \(\angle BAC\).

21. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

22. In \( \triangle ABP \), the circumcenter is \( O \). Given that \( \angle OAP = 25^\circ \) and \( \angle OBP = 35^\circ \), determine \( \angle OAB \).

23. If \(O\) is the circumcenter of \(\triangle ABC\) and \(\angle BAC = 50^\circ\), then find the measure of \(\angle OBC\).

(a) 100° (b) 30° (c) 40° (d) 50°

24. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\). Given \(PA = 9\) cm and \(\angle APB = 60°\), find the length of \(AB\).

25. \(O\) is the circumcenter of triangle \(ABC\). If \(\angle ABC = 65^\circ\) and \(\angle BCA = 40^\circ\), then the measure of \(\angle BOC\) is _____?

26. In the cyclic quadrilateral ABCD, the side \( AB \) is extended to point \( X \), forming \( \angle XBC = 82^\circ \) and \( \angle ADB = 47^\circ \). Find the measure of \( \angle BAC \).

27. ABCD is a cyclic quadrilateral with O as the center of the circle. Side DC is extended up to point P. If \(\angle BCP = 108^\circ\), calculate and write the measure of \(\angle BOD\).

28. In the adjacent figure, O is the center of the circle, \(\angle AOD = 40^\circ\), and \(\angle ACB = 35^\circ\). Calculate and write the measures of \(\angle BCO\) and \(\angle BOD\), and provide reasoning to support your answers.

29. In a circle, triangle ABC is inscribed. The angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) meet the circle at points X, Y, and Z respectively. Prove that in triangle ΔXYZ, \(\angle YXZ = 90^\circ - \frac{\angle BAC}{2}\).

30. In \(\triangle ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, then calculate and write the length of side \(BC\).