From the Quadrilateral to the Parallelogram

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

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 check: is there a pair of opposite 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:
$ABCD$ is a Parallelogram.

If in a quadrilateral, the diagonals bisect each other, the quadrilateral is a parallelogram.
We check: do the diagonals bisect each other 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:
$ABCD$ is a 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:
$ABCD$ is a 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 check: are both pairs of opposite sides in the quadrilateral parallel? If the answer is yes, the quadrilateral is a parallelogram.

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

We check: are both pairs of opposite sides in the quadrilateral equal? If the answer is yes, 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 prove that the triangles formed are congruent 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 opposite where both properties exist: both parallel and equal, is enough to have a parallelogram.

We check: is there a pair of opposite 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 prove congruence between the two triangles created.

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 bisection and not just intersections.
What is meant by diagonals bisecting each other?
If one diagonal crosses another at its midpoint - creating 2 equal halves - and the same happens for the other diagonal, then the diagonals bisect each other.

We check: do the diagonals of this quadrilateral bisect each other? 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 check: 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.