1. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?

(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

2. 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

3. The angles of a triangle are in the ratio \(2:5:3\); calculate and write the circular (radian) measure of the smallest angle.

4. What will be the measure of the third angle of a triangle in radians if the other two angles are \(65^\circ 56' 44''\) and \(64^\circ 3' 16''\)?

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

6. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.

7. 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?

8. If the angles of a triangle are in the ratio \(1 : 1 : 2\), then what will be the ratio of the sides of the triangle?

(a) \(1:\sqrt{2}:1 \) (b) \(1:1:\sqrt{2}\) (c) \(1:1:2\) (d) \(1:2:1\)

9. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?

10. The angles of a triangle are in the ratio 2:5:3. Determine the circular measure of the smallest angle.

11. If the three angles of a triangle are in the ratio 2:3:4, then calculate and write the radian measure of the largest angle.

12. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

13. If the angles of a triangle are in the ratio 1:2:3, find the radian measure of its largest angle.

14. In a circle, two arcs of unequal lengths are in the ratio 5:2. If the central angle corresponding to the second arc is 30°, what is the radian measure of the central angle corresponding to the first arc?

15. 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

16. If the sides of a triangle are in the ratio \(\sqrt{7} : \sqrt{3} : 2\), then what type of triangle is it?

(a) right angle. (b) equilateral (c) isosceles (d) obtuse-angle

17. 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°

18. In triangle ABC, AB = AC. A line drawn from point C intersects the extended line BA at point D. If AC = AD, what is the measure of \(\angle\)BCD in radians?

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

19. 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) \(\cfrac{10}{13}\)

20. 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.

21. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.

22. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

23. If the area of the incircle of an equilateral triangle is 49 ???? cm 2 , what is the perimeter of the triangle?

(a) 21\(\sqrt3\) (b) 7\(\sqrt3\) (c) 14\(\sqrt3\) (d) 42\(\sqrt3\)

24. 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.

25. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?

26. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.

27. 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?

28. 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?

29. In a right-angled triangle, the difference between the two acute angles is \(\frac{2\pi}{5}\). Express the measures of these two angles in both radians and degrees.

30. In a right-angled triangle, the difference between the two acute angles is 30°. Express the measures of those two angles in both radians and degrees.