From the Quadrilateral to the Parallelogram

In a quadrilateral where each pair of opposite sides are also parallel to each other, the quadrilateral is parallel.
We are asked, is every pair of sides in the quadrilateral also parallel? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

When:
$AB∥DC$
and also 
$AD∥BC$

Then:
$ABCD$ Parallelogram.

In a quadrilateral where each pair of opposite sides are also equal to each other, the quadrilateral is parallel.
We are asked, are each pair of opposite sides in the quadrilateral also equal? If the answer is yes, it is determined that the quadrilateral is parallel.

When:
$AB=DC$
And also  
$AD=BC$

Then:
$ABCD$ Parallelogram.

If a quadrilateral has a pair of opposite sides that are equal and parallel, the quadrilateral is a parallelogram.
We are asked, is there a pair of sides in the quadrilateral that are equal and parallel? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

When:
$AB=DC$
and also  
$AB∥DC$

Then:
$ABCD$ Parallelogram.

If in a quadrilateral, the diagonals intersect, the quadrilateral is a parallelogram.
We are asked, do the diagonals intersect in this quadrilateral? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

When:

$AE=CE$
and also 
$BE∥DE$

Then:
$ABCD$ Parallelogram.

If a quadrilateral has two pairs of equal opposite angles, the quadrilateral is a parallelogram.
We are asked, are there two pairs of equal opposite angles in this quadrilateral? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

When:

$∢A=∢C=α$
$∢B=∢D=β$

Then:
$ABCD$ Parallelogram.

We have known the beloved and familiar quadrilateral for a long time, but today we will reveal how to determine that the quadrilateral we have in front of us is actually a parallelogram!
First, we will remember the characteristics that we know about the quadrilateral

• A quadrilateral is a polygon with 4 sides.
• The sum of the interior angles in a quadrilateral is equal to $360$.

You might be wondering, what makes our quadrilateral a parallelogram or how can we determine for sure that a quadrilateral is a parallelogram?
Don't worry, we are here to teach you, we will learn some main ways to help you determine if your quadrilateral is also a parallelogram.

The first way to determine that a quadrilateral is a parallelogram is to examine the following statement:

How do we remember this?
The word parallelogram greatly reminds us of the word parallel.
That's how we remember that we are looking for sides that are parallel!

Please note: This argument makes a lot of sense since the definition of a parallelogram is a quadrilateral in which each pair of opposite sides is parallel.
We are asked, is each pair of sides also in a quadrilateral? Parallel? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

The second way to determine that the quadrilateral is a parallelogram is to examine the following statement:

We are asked, are each pair of opposite sides in the quadrilateral equal? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

To help you better understand why this argument is true, we will show you how this statement holds up as proof:

All we will need to prove this statement is to draw a diagonal line in our quadrilateral and superimpose the triangles formed by us according to $S.S.S$:
Given:

$AB=DC$
$AD=BC$

The third way to determine that the quadrilateral is a parallelogram is to examine the following statement:

Please note: a pair of sides where both properties exist: both parallel and equal, is enough to have a parallelogram.

We are asked, is there a pair of sides in a quadrilateral that are equal and also parallel? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

Let's test this statement:

First, we will draw a diagonal in the quadrilateral in front of us
and we can test an overlap between the two triangles created for us.

Given that:
$AB=DC$
$AB∥DC$

The fourth way to determine that the quadrilateral is a parallelogram is to examine the following statement:

Pay attention: the condition talks about diagonal intersections and not just intersections.
What is meant by diagonals intersecting each other?
If one diagonal crosses another in the middle - 2 equal halves and the same happens in reverse, the diagonals intersect each other.

We are asked, do the diagonals of this quadrilateral intersect? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

Let's see this in the figure:

We can see that in this quadrilateral there are two diagonals that intersect:
$AE=CE$
$BE=DE$

Let's prove this statement:
We will superimpose $⊿BEC$  and $⊿AED$
 according to $L.A.L$

The fifth way to determine that the quadrilateral is a parallelogram is to examine the following statement:

That is:

If  
$∢A=∢C=α$
$∢B=∢D=β$

Then the quadrilateral in front of you $ABCD$ is a parallelogram.
We are asked, are there two pairs of opposite angles equal in this quadrilateral? If the answer is yes, it is determined that the quadrilateral is a parallelogram.

To better understand the logic behind this statement, we will prove it here:
We will extend the sides of the quadrilateral a bit and obtain:

$∢C2$ Is an adjacent angle to $α$, it meets him on the same line and therefore complements to $180$.
It is determined that:
$∢C2=180-α$

What else do we know?
that the sum of the angles of a quadrilateral equals $360$.
Therefore we can determine that:
$α+α+β+β=360$
$2α+2β=360$
$α+β=180$
We can isolate β and obtain:
$β=180-α$

Now we will notice that the angle $β$ and the angle $∢C2$ Have the same size: $180-α$.
Note that we were asked for the corresponding angles.

If the two corresponding angles are also equal, then the two lines forming them are parallel.

The same can be proven with the second pair of corresponding angles and prove the parallelism of the second pair of sides in a quadrilateral.

Wonderful! Now you know all the ways to determine if a quadrilateral is also a parallelogram.