Question

Prove that if the diagonals of parallelograms are congruent, then the parallelogram is a rectangle.

Short answer

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

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Step description.

The CPCTC theorem states that when two triangles are congruent, then every corresponding part of one triangle is congruent to the other.

Step 3. Step description.

From the figure, $\Delta ABC$and $\Delta ABD$.

$AB=AB$ (Sides are common)

As the diagonals are congruent such that $AC=BD$and the diagonals of a parallelogram bisect such that $BC=AD$.

Therefore, by side-side-side postulate $\Delta ABC\cong \Delta BAD$.

Now, by CPCT theorem $\angle ABC\cong \angle BAD$.

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

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

Therefore, the parallelogram is a rectangle.