1. If three angles of a quadrilateral are \(\frac{π}{3}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the measure of the fourth angle in both sexagesimal (degree) and circular (radian) units.

2. 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?

3. A rotating ray turns counterclockwise from a certain position, completing two full revolutions and then an additional angle of \(30^\circ\). Calculate and write the angle in trigonometric measurement in both sexagesimal (degree) and circular (radian) systems.

4. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?

(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)

5. If the sum of two angles is 135° and their difference is \(\frac{π}{12}\), write the angles in sexagesimal and circular measure.

6. If the sum of two angles is 135° and their difference is \(\frac{\pi^c}{12}\), find their sexagesimal and circular measures.

7. The English translation is: **"If the sum of two angles is 135° and their difference is \(\frac{\pi}{12}\), determine the angles in both sexagesimal and circular measure."**

8. If the measurements of three angles of a quadrilateral are respectively \( \frac{π}{3} \), \( \frac{5π}{6} \), and 90°, then calculate and write the sexagesimal (degree) and circular (radian) value of the fourth angle.

9. If the measure of an angle in degrees is \(D\) and its measure in radians is \(R\), then find the value of \(\cfrac{R}{D}\).

10. If in parallelogram ABCD, \(\angle A - \angle D = 50^\circ\), then what is the measure of \(\angle C\)?

(a) 115\(^o\) (b) 110\(^o\) (c) 90\(^o\) (d) 65\(^o\)

11. In triangle ABC, angle B is a right angle. The hypotenuse is \(\sqrt{15}\), and the sum of the other two sides is 4. What is the value of \((\cos A + \cos C)\)?

(a) \(\cfrac{8}{\sqrt{13}}\) (b) \(\cfrac{-8}{\sqrt{15}}\) (c) \(\cfrac{-8}{\sqrt{13}}\) (d) \(\cfrac{8}{\sqrt{15}}\)

12. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

13. 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°

14. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

15. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

16. 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?

17. If three angles of a quadrilateral are \(\frac{π}{5}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the sexagesimal (degree) and circular (radian) measure of the fourth angle.

18. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

19. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

20. 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 \(\angle PTA\)?

21. In triangle \(\triangle ABC\), O is the circumcenter. If \(\angle OAB = 35^\circ\), then what is the measure of \(\angle ACB\)?

(a) 45° (b) 50° (c) 55° (d) 60°

22. If the volume of a right circular cone is \(V\), the area of its base is \(A\), and its height is \(H\), then find the value of \(\left(\frac{AH}{V}\right)^2\).

(a) 1 (b) 4 (c) 9 (d) 3

23. If the sides of a triangle are in the ratio \(3x : 4x : 5x\), what is the measure of the largest angle of the triangle?

(a) 105\(^o\) (b) 75\(^o\) (c) 60\(^o\) (d) None of the above

24. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

25. 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?

26. In \( \triangle PQR \), ST is parallel to QR and intersects PQ at point S and PR at point T. Given that \( PQ = (X+1) \) units, \( PS = 3 \) units, \( PR = (X+6) \) units, and \( PT = 6 \) units, determine the value of \( x \).

27. If the curved surface area of a right circular cylinder is \(S\), its volume is \(V\), and its radius is \(r\), then the value of \(\cfrac{Sr}{V}\) is -

(a) 2 (b) 4 (c) 8 (d) none of the above

28. The sum of two angles is 135°, and their difference is \(\cfrac{\pi}{12}\). Determine the sexagesimal and circular measures of both angles.

29. If the sum of two angles is 135° and their difference is \( \cfrac{\pi}{12} \), write the sexagesimal and circular measures of the two angles.

30. If the sum of two angles is 135° and their difference is \(\cfrac{π}{12}\), write their sexagesimal and circular measures.