1. 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.

2. 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.

3. 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.

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

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

6. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

7. "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) \(\cfrac{π}{5},\cfrac{π}{15}\)

8. In a right-angled triangle, if the difference between the two acute angles is 72°, find their measures in radians.

9. 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.

10. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

11. If one angle of a parallelogram is 67°30′, find the radian measures of the other three angles.

12. 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.

13. In a right-angled triangle, the hypotenuse is 15 cm, and the difference between the other two sides is 3 cm. Find the lengths of those two sides.

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

15. 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.

16. 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.

17. 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."**

18. 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]

19. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

20. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.

21. 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.

22. In a circle, two arcs of unequal length subtend angles at the center in the ratio 5:2. If the sexagesimal (degree) measure of the second angle is 30°, then calculate and write both the sexagesimal and radian measures of the first angle.

23. 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°

24. 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.

25. A passenger on an airplane once saw Howrah Station on one side and Shaheed Minar on the opposite side at angles of depression of 60° and 30°, respectively. At that moment, if the airplane was flying at a height of \(545\sqrt{3}\) meters, what is the distance between Howrah Station and Shaheed Minar?

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

27. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)

28. 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.

29. 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.

30. There is a palm tree on the bank of a river. Directly opposite it, on the other side of the river, a pole is fixed into the ground. If someone walks 7\(\sqrt{3}\) meters along the riverbank from the pole toward the tree, the base of the tree appears to form a 60° angle with the riverbank at that point. Determine the width of the river.