# From the Quadrilateral to the Parallelogram

From now on, you'll be able to determine if the quadrilateral in front of you is a parallelogram!

We present to you 5 different ways to prove it

When:

## The first method

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.

## The second method

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.

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## The third method

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.

## The fourth method

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.

## The fifth method

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.

## From the Quadrilateral to the 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:

#### In a quadrilateral where each pair of opposite sides is also parallel to each other, the quadrilateral is a parallelogram.

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:

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

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:

$AB=DC$
$AD=BC$

 Argument Explanation $AD=BC$ Side Given $AB=DC$ Side Given $BD=BD$ Side One common side, each size equal to itself. Therefore $⊿BCD=⊿DAB$ The triangles overlap according to S.S.S. It is deduced that: It seems like you haven't provided any text for translation. Could you please provide the text you need translated? $∢D1=∢B1$ By the same superposition $∢D2=∢B2$ By the same superposition $∢D1=∢B1$ Alternate and also equal It seems like you haven't provided any text for translation. Could you please provide the content you need translated? Therefore $AB∥DC$ If alternate angles are equal, the lines are parallel $∢D2=∢B2$ Alternate and also equal It seems like you haven't provided any text for translation. Could you please provide the content you need translated? And therefore $AD∥BC$ If alternate angles are equal, the lines are parallel It is deduced that:  $ABCD$ It's a parallelogram! If each pair of opposite sides in the quadrilateral are parallel to each other, the quadrilateral is a parallelogram.

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

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

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:
$AB=DC$
$AB∥DC$

 Argument Explanation $AB=DC$ Side Given $AB∥DC$ Given It is deduced that:$∢B1=∢C1$ Angle Alternate angles between parallel lines are necessarily equal. $BD=BD$ Side One common side, each size equal to itself. Y por lo tanto $⊿ABD=⊿CDB$ According to L.A.L It follows that:$AB=DC$ and also$AD=BC$ According to angle superposition It follows that- $ABCD$ is a parallelogram! If each pair of opposite sides in the quadrilateral are parallel to each other, the quadrilateral is a parallelogram.

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

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

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 $L.A.L$

 Argument Explanation $AE=CE$ Side Given $∢AED=∢BEC$ Angle The angles of the vertices are equal to each other. $BE=DE$ Side Given It is deduced that:$⊿AED=⊿BEC$ According to L.A.L Therefore, we can say that:$∢DBC=∢ADB$ Equal and also alternate Superposition of triangles Therefore:$AD∥BC$ If alternate angles are equal, the lines are parallel $AD=BC$ Superposition of triangles And therefore: $ABCD$ is a parallelogram! If a quadrilateral has a pair of opposite sides that are equal and parallel, the quadrilateral is a parallelogram.

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

#### If in a quadrilateral there are two pairs of equal opposite angles, the quadrilateral is a parallelogram.

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.

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