1. If ABCD is a cyclic quadrilateral and \(\angle\)A=120°, what is the measure of \(\angle\)C ?

(a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)

2. The center of a circle is O, and AB is its diameter. ABCD is a cyclic quadrilateral. If \(\angle\)ABC = 65° and \(\angle\)DAC = 40°, then the measure of \(\angle\)BCD is—?

(a) 75° (b) 105° (c) 115° (d) 80°

3. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

4. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?

(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)

5. The measure of a supplementary angle of a certain angle is four times the measure of its complementary angle. What is the measure of that angle?

(a) 30° (b) 60° (c) 45° (d) None of the above

6. 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\(^o\)

7. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?

(a) 90° (b) 30° (c) 60° (d) 120°

8. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?

(a) 30° (b) 38° (c) 60° (d) None of the above

9. A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?

10. If the ratio of three consecutive angles of a cyclic quadrilateral is 1 : 2 : 3, what are the measures of the first and third angles?

11. If one side of a cyclic quadrilateral is extended, the exterior angle thus formed is equal to the interior opposite angle — prove it.

12. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?

(a) 105° (b) 90° (c) 150° (d) 75°

13. 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?

14. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?

15. In the cyclic quadrilateral ABCD, AB is the diameter and \(\angle\)ACD = 50°, then what is the measure of \(\angle\)BAD?

(a) 30° (b) 40° (c) 50° (d) 60°

16. O is the center of the circle, and AB is its diameter. ABCD is a cyclic quadrilateral. Given that \(\angle\)ABC = 65° and \(\angle\)DAC = 60°, find the measure of \(\angle\)BCD.

17. When one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.

18. When one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.

19. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle\)COD = 120° and \(\angle\)BAC = 30°, then what is the measure of \(\angle\)BOC?

20. 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°

21. If the circular (radian) measure of an angle is \( \frac{7\pi}{12} \), what is its value in the sexagesimal (degree) system?

(a) 90° (b) 105° (c) 135° (d) 160°

22. If two angles of a triangle are 75° and \( \frac{\pi^c}{6} \), then what is the measure of the third angle?

(a) 75° (b) 60° (c) 65° (d) 70°

23. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

24. ABCD is a cyclic quadrilateral. If ∠ ABD = 48°, what is the measure of ∠ ACD?

(a) 42° (b) 138° (c) 48° (d) 12°

25. 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

26. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

27. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.

28. If the sum of a central angle and an inscribed angle of a circle is 180°, then what is the measure of the central angle?

29. Two angles of a triangle are 35°57′4″ and 39°2′56″. What is the radian measure of the third angle?

30. Two unequal arcs of a circle subtend two angles at the center in the ratio 5:3, and the sexagesimal (degree) measure of the second angle is 45°. What is the radian measure of the first angle?