In order to prove that the parallelogram ABCD is a rhombus, the following condition must be met: All sides of the parallelogram must be congruent.

Let's examine each of the given statements:

1. ∠ABC ≅ ∠CDA: This condition indicates that two angles of the parallelogram are congruent. While this tells us that the parallelogram has specific angle relationships, it doesn't directly prove that all sides are congruent. Hence, this is not sufficient to prove that ABCD is a rhombus.

2. Side AC ≅ Side BD: This condition tells us that the diagonals of the parallelogram are congruent. However, the fact that diagonals are congruent does not imply that all sides of the parallelogram are congruent, which is required to prove that the parallelogram is a rhombus. Therefore, this is not sufficient either.

3. AC parallel BD: This condition suggests that the diagonals are parallel, but in a parallelogram, diagonals are generally not parallel to each other unless the parallelogram is a rectangle (which would then not be a rhombus). Therefore, this doesn't provide any support for proving that ABCD is a rhombus.

4. AB parallel CD: This statement merely states that the opposite sides of the parallelogram are parallel, which is always true for any parallelogram. This condition does not help in proving that ABCD is a rhombus.

Conclusion: None of these statements directly prove that ABCD is a rhombus. However, for a parallelogram to be a rhombus, the diagonals must be perpendicular and bisect each other at right angles. So, if there was a statement about the diagonals being perpendicular or bisecting each other at right angles, that would be the correct proof.

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Would you like further clarification or help with related concepts?

Here are some related questions for further exploration:

1. How do the diagonals of a rhombus behave differently from those of a general parallelogram?
2. What properties must be true for a quadrilateral to be classified as a rhombus?
3. Can you prove that if the diagonals of a parallelogram bisect each other at right angles, it must be a rhombus?
4. What additional properties would a rhombus share with a square?
5. How do we prove that a quadrilateral is a parallelogram based on its diagonals?

Tip: In a rhombus, the diagonals not only bisect each other, but they are also perpendicular to each other. This is a key feature that helps distinguish a rhombus from other types of parallelograms.