Answer :

Consider the parallelogram ABCD with diagonals AC and BD as shown

A parallelogram is a rectangle if its adjacent angles are 90° and opposite sides are equal

Now as it is given that it is a parallelogram which means opposite sides are equal so we need to only check for the adjacent angles

To prove parallelogram ABCD is rectangle we just need to prove the adjacent angles are 90°

Consider ΔBAD and ΔCDA

AC = BD … given

AB = DC … opposite sides of a parallelogram

AD is the common side

Therefore, ΔBAD ≅ ΔCDA … SSS test for congruency

⇒ ∠BAD = ∠CDA …corresponding angles of congruent triangles …(i)

As it is given that ABCD is parallelogram

⇒ ∠BAD + ∠CDA = 180° … sum of adjacent angles of a parallelogram is 180°

Using equation (i)

⇒ ∠CDA + ∠CDA = 180°

⇒ 2 × ∠CDA = 180°

⇒ ∠CDA = 90°

⇒ ∠BAD = 90°

Adjacent angles are 90° implies ABCD is a rectangle

Therefore, if the lengths of two diagonals of a parallelogram are equal, then the parallelogram will be a rectangle.

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