Question

prove a parallelogram is a square if and only if its diagnols are both congruent and perpendicular.

Answer 1

A parallelogram has congruent diagonals, which takes the shape of a rectangle. If a parallelogram has perpendicular diagonals, you know it is a rhombus. So thinking of a parallelogram that is both a rectangle and a rhombus, the only parallelogram that satisfies such description is a square.