From a Parallelogram to a Rectangle

If a parallelogram has an angle of $90^o$ degrees, it is a rectangle.
We are given a parallelogram $ABCD$ :

Given that:
$∢B=90$

We need to prove that:
$ABCD$ is a rectangle.

Solution:

We know that in a parallelogram, the opposite angles are equal. Therefore, we can affirm that:
$∢B=∢D=90$
Now, we can affirm that:
$∢C=180-90$
Since the adjacent parallel angles are equal to $180^o$.
We obtain that:
$∢C=90$

Wonderful. Now we can affirm that:
$∢A=∢C=90$
Since the opposite parallel angles are equal.

Magnificent! We proved that all angles in a parallelogram are equal to $90^o$ degrees.
Therefore, we can determine that the parallelogram is a rectangle.

Remember, the formal definition of a rectangle is a parallelogram where it has an angle of $90^o$ degrees.
Therefore, we will not have to prove that all angles are equal to $90^o$.

If the diagonals are equal in a parallelogram, it is a rectangle.
We are given a parallelogram $ABCD$ :

and it has equal diagonals:
$AC=BD$

It is necessary to prove that: $ABCD$ is a rectangle.

Solution:
We know that in the parallelogram the diagonals cross each other.
Therefore, we can determine that:
$AE=CE$
$BE=DE$

We also know from the given data that: $AC=BD$
Therefore, we can affirm that all halves are equal.
Why?
We can write that:
$AE=CE=\frac{AC}{2}$

$BE=DE=\frac{BD}{2}$

Since:
$AC=BD$
we can compare 
$\frac{AC}{2}=\frac{BD}{2}$
And according to the transitive rule we obtain:

$AE=CE=BE=DE$

As all these segments are equal, isosceles triangles are created.

We will identify the equal angles in the drawing:

Opposite equal sides, the angles are equal.
Moreover, in a parallelogram, the opposite sides are parallel
and the alternate angles between parallel lines are equal.

Wonderful.
Now, remember that in a quadrilateral the sum of the interior angles equals $360^o$.
Therefore:
$α+β+α+β+α+β+α+β=360$
$4α+4β=360$
we divide by $4$ and obtain:
$α+β=90$

Note that each of our parallel angles consists of $α+β$ and, therefore, each parallel angle equals $90^o$ degrees.
Therefore, the parallelogram we have in front of us is a rectangle, since a rectangle is a parallelogram with an angle of $90^o$ degrees.

If you are interested in this article, you might also be interested in the following articles:

Parallelogram - Checking the parallelogram

The area of the parallelogram: what is it and how is it calculated?

Rotational symmetry in parallelograms

Rectangle

Rectangle area

Rectangles with equivalent area and perimeter

From a quadrilateral to a rectangle

In the Tutorela blog, you will find a variety of articles on mathematics.