From a Quadrilateral to a Rectangle

How do we recognize that the quadrilateral in front of us is actually a rectangle?
In two quite simple ways!

First form: angle check

A rectangle is a quadrilateral whose angles are equal to $90^o$ degrees, if we can prove that this is also the case for our quadrilateral, we can prove that it is a rectangle.

Second form: parallelogram proof and then rectangle proof

This form is a bit more complicated, as it involves two steps.
So, why is it useful?
There are five ways to prove that a quadrilateral is a parallelogram, so many times (depending on the data) it will be easier to prove that the quadrilateral is a parallelogram.
Once we have been able to prove this, we can move on to the next step and prove why this parallelogram is a rectangle.
Remember, a rectangle is a special case of a parallelogram.

Detailed explanation

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Test yourself on from a quadrilateral to a rectangle!

It is possible to draw a quadrilateral that is not a rectangle, with the sum of its two adjacent angles equaling 180?

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From a Quadrilateral to a Rectangle

Many times we are asked to prove that the quadrilateral we see is a rectangle, or we will need it to continue with our solution.
To prove that a quadrilateral is a rectangle, we can proceed with the proof in one of two ways:

First form: angle check

If in the quadrilateral in front of you there are $3$ angles equal to $90^o$ degrees each, you can determine that this quadrilateral is a rectangle.
It is not necessary to verify the fourth angle since we know that the sum of the internal angles in the quadrilateral is $360^o$ degrees and equal to $90^o$ degrees.

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Test your knowledge

Question 1

It is possible to have a rectangle with different angles?

Question 2

It is possible to draw a quadrilateral that is not a rectangle and that has two equal opposite sides?

Question 3

It is possible to draw a quadrilateral that is not a rectangle and that has two opposite parallel sides?

Second form: from the quadrilateral to the parallelogram, from the parallelogram to the rectangle

This form is a bit more complex and first you must verify that the quadrilateral in front of you is a parallelogram.
We briefly remind you of the conditions to prove a parallelogram:

If in a quadrilateral each pair of opposite sides are also parallel to each other, the quadrilateral is a parallelogram.
If in a quadrilateral each pair of opposite sides are also equal to each other, the quadrilateral is a parallelogram.
If in a quadrilateral each pair of opposite sides are equal and parallel, the quadrilateral is a parallelogram.
If in a quadrilateral, the diagonals intersect the quadrilateral is a parallelogram.
If in a quadrilateral there are two pairs of equal opposite angles, the quadrilateral is a parallelogram.

Have you proven that the quadrilateral in front of you is a parallelogram using one of the previous conditions?
Excellent!
You can continue with the next step
Now, you must prove that the parallelogram in front of you is a rectangle using one of these two conditions:

If the parallelogram has a 90-degree angle, it is a rectangle.
If the diagonals are equal in the parallelogram, it is a rectangle.

Wonderful! Now you know all the ways to prove that this is not an ordinary quadrilateral, but a rectangle.

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

Rectangle Area

The Perimeter of a Rectangle

Rectangles with Equivalent Area and Perimeter

Area of a Right Triangle

Congruence of Right Triangles (in the context of the Pythagorean Theorem)

From a Quadrilateral to a Rectangle

From a Parallelogram to a Rectangle

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

Examples and exercises with solutions from a quadrilateral to a rectangle

Exercise #1

ABCD is a square with sides measuring 4 cm.

Is ABCD a rectangle?

Video Solution

Step-by-Step Solution

Since we are given that a square appears in the drawing, we will remember that every pair of opposite sides are parallel.

Let's think about the properties of a rectangle, and remember that in a rectangle, every pair of opposite sides are also parallel.

Therefore, the answer is correct and the quadrilateral ABCD is a rectangle.

Answer

Yes

Exercise #2

Given the quadrilateral ABCD whereby

AD||BC , AB||CD

Indicate if the quadrilateral is a rectangle.

Video Solution

Step-by-Step Solution

In a rectangle, it is known that all angles measure 90 degrees.

Since we know that angle B is equal to 100 degrees, the quadrilateral cannot be a rectangle.

Answer

Exercise #3

It is possible to draw a quadrilateral that is not a rectangle, with the sum of its two adjacent angles equaling 180?

Video Solution

Answer

Yes.

Exercise #4

It is possible to have a rectangle with different angles?

Video Solution

Answer

Exercise #5

It is possible to draw a quadrilateral that is not a rectangle and that has two equal opposite sides?

Video Solution

Answer

Yes.

Do you know what the answer is?

Question 1

It is possible to draw a quadrilateral that has opposite angles and is not a rectangle?

Question 2

There may be a rectangle with an acute angle.

Question 3

It is possible to draw a quadrilateral that is not a rectangle and that has diagonals which are not perpendicular to each other?

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