Which of the following statements are true and which of them are false?
(i) All squares are congruent.
(ii) If two squares have equal areas, they are congruent.
(iii) If two figures have equal areas, they are congruent.
(iv) If two triangles are equal in area, they are congruent.
(v) If two sides and one angle of a triangle are equal to the corresponding two sides and angle of another triangle, the triangles are congruent.
(vi) If two angles and any side of a triangle are equal to the corresponding angles and the side of another triangle then the triangles are congruent.
(vii) If three angles of a triangle are equal to the corresponding angles of another triangle then the triangles are congruent.
(viii) If the hypotenuse and an acute angle of a right triangle are equal to the hypotenuse and the corresponding acute angle of another right triangle then the triangles are congruent.
(ix) If the hypotenuse of a right triangle is equal to the hypotenuse of another right triangle then the triangles are congruent.

Answer

To determine the truth value of each statement regarding congruence, let's analyze them one by one:

### Step-by-Step Solution:

1. **Statement (i): All squares are congruent.**
- **Analysis:** This statement is **False**. Squares can have different side lengths, and only squares with the same side length are congruent.
- **Conclusion:** **False**

2. **Statement (ii): If two squares have equal areas, they are congruent.**
- **Analysis:** This statement is **True**. If two squares have equal areas, they must have the same side length, hence they are congruent.
- **Conclusion:** **True**

3. **Statement (iii): If two figures have equal areas, they are congruent.**
- **Analysis:** This statement is **False**. Different shapes can have the same area (e.g., a triangle and a rectangle), but they are not necessarily congruent.
- **Conclusion:** **False**

4. **Statement (iv): If two triangles are equal in area, they are congruent.**
- **Analysis:** This statement is **False**. Two triangles can have the same area but different shapes (e.g., one could be scalene and the other could be isosceles).
- **Conclusion:** **False**

5. **Statement (v): If two sides and one angle of a triangle are equal to the corresponding two sides and angle of another triangle, the triangles are congruent.**
- **Analysis:** This statement is **False**. The two triangles may not be congruent unless the angle is included between the two sides.
- **Conclusion:** **False**

6. **Statement (vi): If two angles and any side of a triangle are equal to the corresponding angles and the side of another triangle, then the triangles are congruent.**
- **Analysis:** This statement is **True**. This follows the Angle-Side-Angle (ASA) congruence criterion.
- **Conclusion:** **True**

7. **Statement (vii): If three angles of a triangle are equal to the corresponding angles of another triangle, then the triangles are congruent.**
- **Analysis:** This statement is **False**. While the triangles are similar, they are not necessarily congruent unless their sides are also equal.
- **Conclusion:** **False**

8. **Statement (viii): If the hypotenuse and an acute angle of a right triangle are equal to the hypotenuse and the corresponding acute angle of another right triangle, then the triangles are congruent.**
- **Analysis:** This statement is **True**. This follows the Right Angle-Hypotenuse-Angle (RHA) congruence criterion.
- **Conclusion:** **True**

9. **Statement (ix): If the hypotenuse of a right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.**
- **Analysis:** This statement is **False**. The hypotenuse alone does not guarantee congruence; the other sides must also be considered.
- **Conclusion:** **False**

### Summary of Truth Values:
- (i) False
- (ii) True
- (iii) False
- (iv) False
- (v) False
- (vi) True
- (vii) False
- (viii) True
- (ix) False