If the diagonals of a parallelogram are of equal lengths, the parallelogram is a rectangle. Prove it.

Given:

A parallelogram ABCD with diagonals AC and BD of equal lengths.

To prove:

The parallelogram ABCD is a rectangle.

Proof:

Consider triangles Δ ABC and Δ ABD:

• AB = AB (Common side)

• AC = BD (Given)

• BC = AD (opposite sides of parallelogram)

By Side Side Side congruency,

Δ ABC ≅ Δ ABD

By using Corresponding Parts of Congruent Triangles,

∠ A = ∠ B

But we know that adjacent angles of parallelogram are supplementary.

∠ A + ∠ B = 180°

∠ A = ∠ B = 90°

Similarly, ∠ C = ∠ D = 90°

Hence, if the diagonals of a parallelogram are of equal lengths, the parallelogram is a rectangle.