1. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument and the angle of depression to its base are complementary angles. If the height of the monument is four times the height of the building, find the measures of the elevation and depression angles.

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. If one angle of a cyclic quadrilateral is 75°, what is the measure of its opposite angle?

4. 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.

5. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

6. 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°

7. 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°

8. 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°

9. 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?

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

11. In an equilateral triangle ABC, the base BC is extended to a point E such that CE = BC. A is joined to E to form triangle ACE. Find the circular (radian) measures of the angles of triangle ACE.

12. 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?

13. The radian measure of the supplementary angle of an angle measuring \(\cfrac{3π}{8}\) is ____.

14. Give the definition of a radian. The angles of a triangle are in the ratio 2:5:3; what is the radian measure of the smallest angle of the triangle?

15. In the adjacent figure, O is the center of the circle and AB is the diameter. Given that \(\angle\)AOD = 140° and \(\angle\)CAB = 50°, find the measure of \(\angle\)BED.

16. 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

17. If one side of a rhombus is 6 cm and one of its angles measures 60°, then the area of the rhombus-shaped field is –

(a) \(9\sqrt{3}\) square cm (b) \(18\sqrt{3}\) square cm (c) \(36\sqrt{3}\) square cm (d) \(6\sqrt{3}\)square cm

18. 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?

19. In a circle centered at O, AOB is the diameter. Given that \(\angle\)BCE = 20° and \(\angle\)CAE = 25°, determine the measure of \(\angle\)AEC.

(a) 50° (b) 90° (c) 45° (d) 20°

20. In a circle centered at O, AC is the diameter, and DC || EB. Given that \(\angle\)AOB = 80° and \(\angle\)ACE = 10°, determine the measure of \(\angle\)BED.

21. 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°

22. In the given figure, QS is the bisector of \(\angle\)PQR. Given that \(\angle\)SQR = 35° and \(\angle\)PRQ = 32°, determine the measure of \(\angle\)QSR.

23. \(\triangle\)PQR is an equilateral triangle, and its circumcircle is centered at \(O\). The measure of \(\angle POQ\) is _____.

24. The radian measure of the supplementary angle of \( \cfrac{3\pi}{8} \) is _____.

25. The radius of a circle is 28 cm. What is the circular measure of the central angle subtended by an arc of length 5.5 cm in that circle?

26. The circular measure of the complementary angle of \(\frac{3\pi}{8}\) is _____.

27. An equilateral triangle where each side is 6 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

28. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

29. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

30. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).