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

2. If the three sides of a triangle are in the ratio 5 : 12 : 13, then the triangle will always be a right-angled triangle.

3. If the lengths of the three sides of a triangle are in the ratio 3:4:5, then the triangle is a right-angled triangle.

4. If the ratio of the three sides of a triangle is 3:4:5, then the triangle will be a right-angled triangle.

5. 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.

6. 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) \(1:2:1\)

7. If the angles of a triangle are in the ratio 1:1:2, then the sides of the triangle will be in the ratio ———.

8. 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.

9. 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.

10. 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.

11. If two triangles are equiangular (have equal corresponding angles), then the ratios of their corresponding sides will be equal; that is, their corresponding sides will be proportional.

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. "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) \(\cfrac{π}{5},\cfrac{π}{15}\)

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

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

16. 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.)

17. If the lengths of the sides of two triangles are in proportion, then the triangles will be ——.

18. 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.)

19. \(\triangle\)ABC ~ \(\triangle\)DEF; BC and EF are corresponding sides. If BE : EF = 1 : 3, then the ratio of the areas of \(\triangle\)ABC and \(\triangle\)DEF will be 1 : 27.

20. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)

21. 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.)

22. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of a side of the first triangle is 4 cm, then the length of the corresponding side of the second triangle will be ____ .

23. If a straight line divides two sides of a triangle in the same ratio, then that line will be _____ to the third side of the triangle.

24. If the three consecutive angles of a cyclic quadrilateral are in the ratio \(1:2:5\), then the measure of the fourth angle will be _____.

25. If three consecutive angles of a cyclic quadrilateral are in the ratio \(1:3:4\), then the measure of the fourth angle will be _____.

26. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”

27. Draw a right-angled triangle with the two sides adjacent to the right angle measuring 4 cm each. Then draw the circumcircle of the triangle. (Only construction marks are required.)

28. 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.

29. 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.

30. Draw a right-angled triangle where the two sides adjacent to the right angle are 5 cm and 6 cm. Then, draw an incircle (a circle inscribed inside the triangle) within that triangle.