The statement that proves a quadrilateral is a square is:

"Quadrilateral ABCD has diagonals that are congruent and perpendicular bisectors."

This is because, for a quadrilateral to be classified as a square:

1. It must have all sides equal (congruent sides).
2. All angles must be 90 degrees (right angles).
3. The diagonals must be equal in length (congruent) and also bisect each other at right angles (perpendicular bisectors).

This condition ensures both the properties of a rectangle (equal diagonals) and a rhombus (perpendicular bisectors), which combine to define a square.

Would you like further clarification on this or a detailed proof?

Here are 5 related questions to deepen your understanding:

1. What are the properties of diagonals in a rhombus?
2. How do you differentiate between a rhombus and a square?
3. What additional properties does a square share with rectangles?
4. Can a quadrilateral be both a trapezoid and a kite?
5. Why is it important that the diagonals of a square bisect at right angles?

Tip: In geometry, the properties of diagonals (like congruence and perpendicularity) often play a key role in classifying quadrilaterals.