Question

Mary states, If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Decide if her statement is true or false?

Answer 1

True

Explanation:

Let us consider a parallelogram ABCD in which AB=CD and AD=BC.

Now, from ΔABC and ΔBAD, we have

AD=BC (Opposite sides of parallelogram)

BD=AC (Given)

AB=BA (common)

Thus, by SSS rule of congruency,

ΔABC≅ΔBAD.

Now, by corresponding parts of congruent triangles, we have

∠ABC=∠BAD.

But, we know that ∠ABC and ∠BAD forms the corresponding angle pair, thus ∠ABC+∠BAD=180°

⇒2∠ABC=180

⇒∠ABC=90°

Since they are interior angles of parallel lines AC and BC on the same side of their common secant AB. They are therefore both right, and ABCD is a rectangle.

Hence, the statement If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle is true.