1. 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.

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, if the difference between the two acute angles is 72°, find their measures in radians.

4. 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.

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

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

7. If the sum of two angles is 135° and their difference is \(\frac{\pi^c}{12}\), find their sexagesimal and circular measures.

8. "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) \(\cfrac{π}{5},\cfrac{π}{15}\)

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

10. In a right-angled triangle, if one of the acute angles is 30°, determine the measure of the other acute angle in sexagesimal system.

11. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.

12. 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.

13. If the sum of two angles is 135° and their difference is \(\cfrac{π}{12}\), write their sexagesimal and circular measures.

14. If the sum of two angles is 135° and their difference is \(\cfrac{π}{12}\), write their sexagesimal and circular measures.

15. 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."**

16. The difference in measurement between two angles is 1°, and the sum of their radian measures is 1 radian. Express the angles in degrees.

17. 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.

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

19. If the ratio of the radii of two solid right circular cones is 2:3 and the ratio of their heights is 5:3, determine the ratio of their volumes.

(a) 27:20 (b) 4:9 (c) 9:4 (d) 20:27

20. The difference between two positive integers is 3, and the sum of their squares is 117. Determine the two numbers.

21. 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.

22. If the sum of two angles is 135° and their difference is \( \cfrac{\pi}{12} \), write the sexagesimal and circular measures of the two angles.

23. 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.

24. 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.

25. 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.

26. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), then calculate and write the measures of the two angles in both sexagesimal (degree-minute-second) and radian systems.

27. 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 \( \sec^2\theta = 1 + \tan^2\theta \).

28. If the measures of two angles of a triangle are 39°2′56″ and 35°57′4″, determine the circular (degree) measure of the third angle.

29. 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

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