Identifying a Parallelogram

We can identify that the square in front of us is a parallelogram if at least one of the following conditions is met:

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

Diagram illustrating the properties of a parallelogram: opposite sides are parallel and equal, opposite angles are equal, and diagonals bisect each other. Visual aid for understanding ways to identify a parallelogram, featured in a geometry guide.

Detailed explanation

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Test yourself on identifying a parallelogram!

Below is the quadrilateral ABCD:

$$ ∢A=90° $$ $$

$$ ∢C=95° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

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Ways to Identify Parallelograms

Do you want to know how to identify a parallelogram from miles away?
After this article, you will be sure to immediately know when it refers to a parallelogram and when to another square..
In order to make it easier for you to identify a parallelogram, we will divide the five identification sentences into 3 key expressions:

Sides of the square
Diagonals of the square
Angles of the square

Wonderful.
Let's start with the first expression:

Sides of the square

1) In a square where each pair of opposite sides are parallel to each other, the square is a parallelogram.

Parallelogram

We ask ourselves, are each pair of opposite sides also parallel in the square we have in front of us?
If the answer is affirmative, we can determine that it is a parallelogram.

2) In a square where each pair of opposite sides are equal to each other, the square is a parallelogram.

We ask ourselves, are each pair of opposite sides equal in the square we have in front of us?
If the answer is yes, we can determine that it is a parallelogram. Pair of opposite sides.
Note that these are theorems that describe a condition that exists in $2$ pairs of opposite sides.
That is, if we have data on $2$ pairs of opposite sides, both equal or parallel - we can determine that it is a parallelogram.

If we have data on only one pair of opposite sides in a square, we can use the third theorem of this key expression:

3) If a square has a pair of opposite sides that are equal and parallel, the square is a parallelogram.

In this theorem, the pair of opposite sides must be parallel and equal, but only one of those pairs is sufficient.

We ask ourselves, is there a pair of opposite sides that are both equal and parallel in the square we have in front of us?
If the answer is affirmative, we can determine that it refers to a parallelogram.

3 - If a square has a pair of opposite sides that are equal and parallel

Now, let's move on to the second key expression:

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Test your knowledge

Question 1

Given the quadrilateral ABCD that:

$$ ∢A=100° $$ $$

y $$ ∢C=70° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

Question 2

Given the quadrilateral ABCD that:

$$ ∢A=100° $$ $$

y $$ ∢D=100° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

Question 3

Given the quadrilateral ABCD that:

$$ ∢A=80° $$ $$

y $$ ∢C=80° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

Diagonals of the Square

In this expression, there is only one theorem that talks about the diagonals of the square:
4) If in a square, the diagonals that intersect the square are parallel.
Remember: intersecting diagonals are diagonals that intersect each other. Exactly in the middle!

4 - If in a square, the diagonals that intersect the square are parallel

We wonder, do the diagonals intersect (exactly in the middle) in the square we have in front of us?
If the answer is yes, we can determine that it refers to a parallelogram.
Now, let's move on to the third key expression:

Angles of the Square

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

We ask ourselves, are there two pairs of equal opposite angles in the square we have in front of us?
If the answer is affirmative, it is determined to be a parallelogram.

Tip:
To remember all the theorems, try to remember the key expression and remember that there are a total of $5$ sentences that will help us identify that it is a parallelogram.

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

Parallelogram - Checking the parallelogram

The area of the parallelogram: What is it and how is it calculated?

Rotational symmetry in parallelograms

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

Examples and exercises with solutions for identifying parallelograms

Exercise #1

Given the quadrilateral ABCD where:

$∢A=100°$

y $∢C=80°$

Is it possible to conclude that this quadrilateral is a parallelogram?

Video Solution

Step-by-Step Solution

A parallelogram is a quadrilateral whose two pairs of sides are parallel.

Since we know that angles A and C add up to 180 degrees, we know that AB is parallel to CD.

We have no way to prove if AC is parallel to BD since we have no data regarding angle B or angle D.

Therefore, the quadrilateral is not a parallelogram.

Answer

Exercise #2

Given the quadrilateral ABCD where:

$∢A=110°$

y $∢D=110°$

Is it possible to conclude that this quadrilateral is a parallelogram?

Video Solution

Step-by-Step Solution

Since we do not have data regarding the other angles, we cannot prove whether the square has opposite sides equal to one other.

As a result, the quadrilateral is not a parallelogram.

Answer

Exercise #3

Given the quadrilateral ABCD where:

$∢A=115°$

y $∢D=115°$

Is it possible to conclude that this quadrilateral is a parallelogram?

Video Solution

Step-by-Step Solution

Given that a parallelogram is a quadrilateral whose two pairs of sides are parallel, and in the figure only two angles are given to us.

We do not have enough data to determine and prove whether angles C and B are equal to each other.

Therefore, the quadrilateral is not a parallelogram.

Answer

Exercise #4

Given the quadrilateral ABCD where:

$∢D=95°$

y $∢C=85°$

Is it possible to conclude that this quadrilateral is a parallelogram?

Video Solution

Step-by-Step Solution

A parallelogram is a quadrilateral whose two pairs of sides are parallel.

In the figure, we are shown that the sum of angles C and D is 180 degrees but nothing is shared about the other angles.

Therefore, we cannot determine whether or not the sides are parallel to one other.

As a result, this quadrilateral is not a parallelogram.

Answer

Exercise #5

Shown below is the quadrilateral ABCD.

AB = 15 and CD = 13.

BD = 6 and AC = 4

Is it possible to conclude that this quadrilateral is a parallelogram?

Video Solution

Step-by-Step Solution

According to the properties of a parallelogram, each pair of opposite sides are parallel and equal to each other.

Since the data shows that each pair of sides are not equal to each other, the quadrilateral is not a parallelogram.

$15\ne13$

$4\ne6$

Answer

No.

Do you know what the answer is?

Question 1

Given the quadrilateral ABCD that:

$$ ∢B=100° $$ $$

y $$ ∢C=80° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

Question 2

Given the quadrilateral ABCD that:

$$ ∢B=45° $$ $$

y $$ ∢D=135° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

Question 3

Given the quadrilateral ABCD that:

$$ ∢D=105° $$ $$

y $$ ∢C=75° $$

Is it possible to conclude that this quadrilateral is a parallelogram?

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