Question

Which statement describes a parallelogram that must be a square? (a) A parallelogram with diagonals that bisect each other and opposite sides that are congruent. A parallelogram with perpendicular diagonals. A parallelogram with diagonals that are congruent and perpendicular. A parallelogram with diagonals that are congruent and opposite sides that are congruent.

Answer 1

A parallelogram with diagonals that are congruent and perpendicular.

Explanation:

I just did this question and this was the answer that was right for me. I hope this helps!

Answer 2

Consider a parallelogram with diagonals that are congruent and perpendicular.

In triangles Δ AOB and Δ AOD,

OA = OA (common)

OB = OD (Since the diagonals of a parallelogram bisect each other)

∠ AOB = ∠ AOD = 90° (diagonals are perpendicular)

Therefore, Δ AOB ≅ Δ AOD (By SAS postulate)

Since, corresponding parts of congruent triangles are equal,

AB = AD

Similarly, we can prove

BC = CD and CD = DA

So, AB = BC = CD = DA.

Also, it is given that AC = BD.

Hence, ABCD is a square.

Therefore, the correct statement is:

A parallelogram with diagonals that are congruent and perpendicular.