Parallelogram for 9th grade

The definition of a parallelogram is a quadrilateral that has two pairs of parallel opposite sides.

As shown in the illustration, you can see that $AB$ is parallel to $DC$
and $AD$ is parallel to $BC$.
To prove that a shape is a parallelogram, you can use 5 proof theorems (but of course 1 is enough):
1. If a quadrilateral has two pairs of equal opposite angles - it is a parallelogram.
2. If a quadrilateral has two pairs of opposite sides parallel to each other - it is a parallelogram (as in the definition).
3. If a quadrilateral has two pairs of equal opposite sides - it is a parallelogram.
4. If the diagonals of a quadrilateral bisect each other - it is a parallelogram.
5. If a quadrilateral has one pair of sides that are both parallel and equal - it is a parallelogram.

Click here to read more about parallelogram definitions!

Let's go over the five ways to prove that the quadrilateral before us is a parallelogram!

Let's ask, are all pairs of opposite sides in the quadrilateral also parallel? If the answer is yes, we will determine that the quadrilateral is a parallelogram.

When:
$AB$ is parallel and opposite to $DC$
and
$AD$ is parallel and opposite to $BC$
then:
$ABCD$ is a parallelogram.

Let's ask, are each pair of opposite sides in the quadrilateral equal? If the answer is yes, we will determine that the quadrilateral is a parallelogram.

When:
$AB=DC$ and opposite
And
$AD=BC$ and opposite
Then:
$ABCD$ is a parallelogram.

When:
$AB =DC$ and also parallel
Then:
$ABCD$ is a parallelogram.

Let's ask, do the diagonals of this quadrilateral bisect each other? If the answer is yes, we will determine that the quadrilateral is a parallelogram.

When:
$AE=EC$
and
$DE=BE$
then:
$ABCD$ is a parallelogram.

Let's ask, does this quadrilateral have two pairs of equal opposite angles? If the answer is yes, we will determine that the quadrilateral is a parallelogram.

When:
angle $B$ is equal and opposite to $D$
and angle $A$ is equal and opposite to $C$
then:
$ABCD$ is a parallelogram.

Click here to learn more about quadrilateral to parallelogram!

The easiest way to remember the theorems for identifying a parallelogram is to divide them into categories: sides, angles, and diagonals.
Remember - it's enough that only condition $1$ is met and the quadrilateral before you is a parallelogram.

Let's look at the sides and check if one of the conditions is met:

1. If in a quadrilateral every pair of opposite sides are parallel to each other, the quadrilateral is a parallelogram.
2. If in a quadrilateral every pair of opposite sides are equal to each other, the quadrilateral is a parallelogram.
3. If in a quadrilateral there is one pair of opposite sides that are both equal and parallel, the quadrilateral is a parallelogram.

Let's look at the diagonals and check if the following condition is met:

1. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Let's look at the angles and check if the following condition is met:

1. If a quadrilateral has two pairs of equal opposite angles, the quadrilateral is a parallelogram.

Click here to learn more about ways to identify a parallelogram!

The parallelogram manages to coincide with itself more than once during a complete rotation and therefore has rotational symmetry.
The order of rotation of a parallelogram is $2$ - the parallelogram manages to coincide with itself twice during a rotation.
Click here to learn more about rotational symmetry!