Question

Prove: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Short answer

A parallelogram is a rectangle if the diagonals of the parallelogram are congruent.

Step-by-step solution

Step 1. Consider the diagram.

The figure is made having diagonals AC and BD

Here, ABCD is a parallelogram and $\overrightarrow{AC}\cong \overrightarrow{BD}$

Step 2. State the proof.

In $\Delta ABC$and $\Delta ABD$.

$AB=AB$ (Common)

$AC=BD$ (Given)

$BC=AD$ (Sides of a parallelogram)

So, $\Delta ABC\cong \Delta ABD$ by SSS postulate.

Therefore, by CPCT $\angle ABC\cong \angle BAD$.

Using the property of co-interior angles in a parallelogram.

$\begin{array}{c}\angle ABC+\angle BAD=180\\ 2\angle ABC=180\left(\text{As}\angle ABC\cong \angle BAD\right)\\ \angle ABC=90\end{array}$

Therefore, parallelogram ABCD is a rectangle.

Step 3. State the conclusion.

Therefore, parallelogram ABCD is a rectangle if the diagonals of the parallelogram are congruent (proved).