Do you want to know how to prove that the parallelogram in front of you is actually a rectangle? First, you should know that the formal definition of a rectangle is a parallelogram whose angle is $90^o$ degrees. Additionally, if the diagonals in parallelograms are equal, it is a rectangle.

That is, if you are given a parallelogram, you can prove it is a rectangle using one of the following theorems:

If a parallelogram has an angle of $90^o$ degrees, it is a rectangle.

If the diagonals are equal in a parallelogram, it is a rectangle

We briefly remind you of the conditions for a parallelogram check:

If in a quadrilateral where each pair of opposite sides are also parallel to each other, the quadrilateral is a parallelogram.

If in a quadrilateral where each pair of opposite sides are also equal to each other, the quadrilateral is a parallelogram.

If a quadrilateral has a pair of opposite sides that are equal and parallel, the quadrilateral is a parallelogram.

If in a quadrilateral the diagonals cross each other, the quadrilateral is a parallelogram.

If a quadrilateral has two pairs of equal opposite angles, the quadrilateral is a parallelogram.

Let's demonstrate that a parallelogram is a rectangle using the first theorem.

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

From Parallelogram to Rectangle

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 $90^o$.

Now, let's demonstrate that a parallelogram is a rectangle using the second theorem:

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

Parallelogram

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:

Parallelogram

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 theTutorelablog, you will find a variety of articles on mathematics.

Examples and exercises with solutions from the parallelogram to the rectangle

Exercise #1

Is this parallelogram a rectangle

Video Solution

Answer

No

Exercise #2

Is this parallelogram a rectangle?

Video Solution

Answer

Yes

Exercise #3

The quadrilateral ABCD is a parallelogram.

AD = CB

Is the parallelogram a rectangle?

Video Solution

Answer

_{Yes, a parallelogram with equal diagonals is a rectangle.}

Exercise #4

The quadrilateral ABCD is a parallelogram.

$∢B=90°$

Is it a rectangle?

Video Solution

Answer

Yes.

Exercise #5

Perhaps all parallelogram is also a rectangle?

Video Solution

Answer

_{No, a rectangle necessarily has angles of 90°.}

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