From a Parallelogram to a Rectangle

🏆Practice from a parallelogram to a rectangle

From the Parallelogram to the Rectangle

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 90o 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 90o 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:

  1. If in a quadrilateral where each pair of opposite sides are also parallel to each other, the quadrilateral is a parallelogram.
  2. If in a quadrilateral where each pair of opposite sides are also equal to each other, the quadrilateral is a parallelogram.
  3. If a quadrilateral has a pair of opposite sides that are equal and parallel, the quadrilateral is a parallelogram.
  4. If in a quadrilateral the diagonals cross each other, the quadrilateral is a parallelogram.
  5. If a quadrilateral has two pairs of equal opposite angles, the quadrilateral is a parallelogram.
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einstein

AAABBBDDDCCC120°

Is this parallelogram a rectangle

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

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

From Parallelogram to Rectangle

A1 - From parallelogram to rectangle

Given that:
B=90∢B=90

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

Solution:

We know that in a parallelogram, the opposite angles are equal. Therefore, we can affirm that:
B=D=90∢B=∢D=90
Now, we can affirm that:
C=18090∢C=180-90
Since the adjacent parallel angles are equal to 180o 180^o .
We obtain that:
C=90∢C=90

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

Magnificent! We proved that all angles in a parallelogram are equal to 90o 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 90o 90^o degrees.
Therefore, we will not have to prove that all angles are equal to 90o 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 ABCD :

Parallelogram

A2 - Parallelogram

and it has equal diagonals:
AC=BDAC=BD

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

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

We also know from the given data that: AC=BDAC=BD
Therefore, we can affirm that all halves are equal.
Why?
We can write that:
AE=CE=AC2AE=CE=\frac{AC}{2}

BE=DE=BD2BE=DE=\frac{BD}{2}

Since:
AC=BDAC=BD
we can compare 
AC2=BD2\frac{AC}{2}=\frac{BD}{2}
And according to the transitive rule we obtain:

AE=CE=BE=DEAE=CE=BE=DE

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

We will identify the equal angles in the drawing:

Parallelogram

A3 - 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 360o 360^o .
Therefore:
α+β+α+β+α+β+α+β=360α+β+α+β+α+β+α+β=360
4α+4β=3604α+4β=360
we divide by 4 4 and obtain:
α+β=90α+β=90

Note that each of our parallel angles consists of α+β α+β and, therefore, each parallel angle equals 90o 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 90o 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.


Examples and exercises with solutions from the parallelogram to the rectangle

Exercise #1

AAABBBDDDCCC120°

Is this parallelogram a rectangle

Video Solution

Answer

No

Exercise #2

AAABBBDDDCCC45°45°

Is this parallelogram a rectangle?

Video Solution

Answer

Yes

Exercise #3

AAABBBDDDCCC

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

AAABBBDDDCCC90°

The quadrilateral ABCD is a parallelogram.

B=90° ∢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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