1. An angle in a segment larger than a semicircle is an obtuse angle.

2. An angle in a segment greater than a semicircle is an obtuse angle.

3. An angle subtended by a segment smaller than a semicircle is an obtuse angle.

4. Prove that an angle in a segment greater than a semicircle is an acute angle.

5. Prove with reasoning that an angle in a segment greater than a semicircle is an acute angle.

6. An angle in a segment larger than a semicircle.

(a) Acute angle. (b) Obtuse angle. (c) Right angle. (d) Straight angle.

7. The angle in a segment smaller than a semicircle is _____.

8. An angle in a segment smaller than a semicircle is an ____.

9. The angle subtended by a segment of a circle greater than a semicircle is _____.

10. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

11. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

12. The measure of all angles in a circular segment is ______.

13. The angle subtended by a segment smaller than a semicircle is — —.

14. Prove that an angle in a semicircle is a right angle.

15. An angle subtended by a larger segment of a circle is an acute angle compared to the angle subtended by a semicircular segment.

16. A rectangle has a length of \( a \) units and a breadth of \( b \) units. When the rectangle is wrapped to form a right circular cylinder, the circumference of the base of the cylinder is equal to the length of the rectangle. The lateral surface area of the cylinder will be _____.

17. Points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\) of triangle \(∆PQR\), respectively. Given: \(PQ = 8\) units, \(YR = 12\) units, \(PY = 4\) units, and the length of \(PY\) is 2 units less than the length of \(XQ\). Determine, with reasoning, whether segment \(XY\) is parallel to side \(QR\).

18. Angle in a segment greater than a semicircle.

(a) right angle. (b) acute angle. (c) straight angle (d) obtuse angle.

19. The circular measure of each interior angle of a regular hexagon is –

(a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

20. ABC is a triangle. Find the value of \(\sin\left(\cfrac{B+C}{2}\right)\)

(a) sin⁡\(\cfrac{A}{2}\) (b) cos⁡\(\cfrac{A}{2}\) (c) sinA (d) cosA

21. The sum of two angles is \(\cfrac{13π}{12}\). If one of the angles is \(58°\), find the other angle.

(a) 117° (b) 127° (c) 137° (d) 147°

22. 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

23. 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°

24. ABCD is a rectangle. O is the point where the diagonals intersect. If AB = 4 cm and OD = 2.5 cm, then what is the length of BC?

(a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

25. 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) \(\cfrac{8}{\sqrt{15}}\)

26. In triangle ABC, ∠A is a right angle. From point A, a line is drawn to point D, the midpoint of the hypotenuse BC. If BC = 10 cm, then the length of AD is —

(a) 5 cm (b) 6 cm (c) 7 cm (d) 8 cm

27. In a right-angled isosceles triangle ABC, ∠B is the right angle. The bisector of ∠BAC intersects BC at point D. If BD = 2 cm, then what is the length of CD?

28. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

29. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

30. 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.