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

2. 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.

3. 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.

4. Match the items on the left with those on the right (any two):

5. In parallelogram ABCD, the two angles adjacent to side BC are in the ratio 4 : 5. The measures of those two angles are —

(a) 40°, 50° (b) 80°, 100° (c) 45°, 135° (d) None of the above

6. 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 \(64^\circ 3' 16''\)?

7. 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.

8. If the difference between the measures of the two acute angles in a right-angled triangle is 10°, determine their circular measures.

9. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

10. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

11. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.

12. If a perpendicular is drawn from the right-angled vertex of any right triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other, and each of them is also similar to the original triangle.

13. In a right-angled triangle, if the difference between the two acute angles is \(\cfrac{2\pi}{5}\), then write the values of those two angles in sexagesimal (degree-minute-second) system.

14. If the volumes of two solid right circular cylinders are equal and their heights are in the ratio 1:2, what will be the ratio of the lengths of their radii?

(a) 1:√2 (b) √2:1 (c) 1:2 (d) 2:1

15. If the volumes of two right circular cylinders of equal height are in the ratio 16:25, then what is the ratio of their radii?

(a) 4:5 (b) 5:4 (c) 3:5 (d) 5:3

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

17. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

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

19. If the three angles of a triangle are in the ratio 3:4:5, then the triangle will always be a right-angled triangle.

20. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)

21. 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.

22. 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?

23. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

24. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

25. 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?

26. 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.

27. Draw a right-angled triangle whose two sides adjacent to the right angle are 8 cm and 6 cm respectively, and draw an incircle of the triangle. (Only construction marks are required)

28. Draw a right-angled triangle whose two sides adjacent to the right angle are 4.5 cm and 6 cm. Then draw the incircle of that triangle. (Only construction marks are required.)

29. 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?

30. The corresponding sides of two similar-angled triangles are ________.