Final answer:
If the diagonals of a parallelogram are equal and perpendicular bisectors of each other, then each interior angle is a right angle, and all sides are of equal length, confirming that the parallelogram is indeed a square. Therefore, the statement is true.
Step-by-step explanation:
If the diagonals of a parallelogram are equal and perpendicular bisectors of each other, it implies that each diagonal divides the parallelogram into two congruent triangles. Furthermore, since the diagonals are perpendicular bisectors, each vertex angle is bisected into two right angles, implying that all angles in the parallelogram are right angles. Consequently, with equal diagonals, all sides must also be equal, hence showing that the parallelogram is a square.
To show a parallelogram is a square, we can see that the condition that diagonals are equal and perpendicular bisectors of each other satisfies the definition of a square. Every square has equal diagonals that bisect each other perpendicularly, and a parallelogram with these properties must have all sides of equal length and all interior angles as right angles (90°). Thus, the statement is true: if a parallelogram has diagonals that are equal and perpendicular bisectors of each other, it is a square.