Identifying a Parallelogram

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 quadrilateral
• Diagonals of the quadrilateral
• Angles of the quadrilateral

Wonderful.
Let's start with the first category:

1) If both pairs of opposite sides are parallel to each other, the quadrilateral is a parallelogram.

Ask yourself: Are both pairs of opposite sides parallel in the quadrilateral you're examining? If the answer is yes, you can determine that it is a parallelogram.

2) If both pairs of opposite sides are equal in length, the quadrilateral is a parallelogram.

Ask yourself: Are both pairs of opposite sides equal in length? If the answer is yes, you can determine that it is a parallelogram. Note that this theorem requires data about both pairs of opposite sides being equal.

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 quadrilateral, we can use the third theorem of this key expression:

3) If one pair of opposite sides are both equal in length and parallel, the quadrilateral is a parallelogram.

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

Ask yourself: Is there at least one pair of opposite sides that are both equal in length and parallel? If the answer is yes, you can determine that it is a parallelogram.

Now, let's move on to the second category:

4) If the diagonals bisect each other (intersect at their midpoints), the quadrilateral is a parallelogram.

Remember: Bisecting diagonals means the diagonals cut each other exactly in half at their intersection point.

Ask yourself: Do the diagonals bisect each other in the quadrilateral you're examining? If the answer is yes, you can determine that it is a parallelogram.

Now, let's move on to the third category:

5) If both pairs of opposite angles are equal, the quadrilateral is a parallelogram.

Ask yourself: Are both pairs of opposite angles equal in the quadrilateral you're examining? If the answer is yes, you can determine that it is a parallelogram.

6) If consecutive angles are supplementary (add up to 180°), the quadrilateral is a parallelogram.

Ask yourself: Do any two adjacent angles add up to 180°? If this is true for all consecutive angle pairs, you can determine that it is 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.

Common Mistakes to Avoid

Mistake 1: Confusing "opposite" with "adjacent"

• Wrong: "Two adjacent sides are equal, so it's a parallelogram"
• Correct: You need opposite sides to be equal, not adjacent sides

Mistake 2: Thinking one pair of parallel sides is enough

• Wrong: "Sides AB and CD are parallel, so ABCD is a parallelogram"
• Correct: You need both pairs of opposite sides to be parallel (or use Method 3 with one pair being both equal and parallel)

Mistake 3: Misunderstanding the diagonal condition

• Wrong: "The diagonals intersect, so it's a parallelogram"
• Correct: The diagonals must bisect each other (cut each other exactly in half), not just intersect

Mistake 4: Assuming right angles make a parallelogram

• Wrong: "This quadrilateral has right angles, so it's a parallelogram"
• Correct: Right angles alone don't prove it's a parallelogram (though a rectangle IS a special parallelogram)

Mistake 5: Only checking one pair of opposite angles

• Wrong: "Angles A and C are equal, so it's a parallelogram"
• Correct: You need both pairs of opposite angles to be equal (∠A = ∠C AND ∠B = ∠D)

What Does NOT Prove a Parallelogram

❌ One pair of parallel sides only

• This creates a trapezoid, not necessarily a parallelogram
• Example: A shape where AB ∥ CD but BC is not parallel to AD

❌ Equal adjacent sides

• This could be a kite or other quadrilateral
• Example: AB = BC and CD = DA, but opposite sides aren't necessarily equal

❌ Equal diagonals

• Parallelograms don't require equal diagonals
• This is a property of rectangles and isosceles trapezoids, not all parallelograms

❌ Perpendicular diagonals

• This could be a kite or rhombus
• Parallelograms don't require perpendicular diagonals

❌ One pair of equal opposite angles

• You need both pairs of opposite angles to be equal
• Having just ∠A = ∠C doesn't guarantee ∠B = ∠D

❌ Four right angles mentioned without side information

• While this creates a rectangle (which IS a parallelogram), you need additional information about the sides to confirm it's not just any rectangle

❌ Congruent triangles formed by one diagonal

• A diagonal splitting a quadrilateral into two congruent triangles doesn't guarantee it's a parallelogram
• You need specific information about how they're congruent

Remember: To prove something is a parallelogram, you must satisfy one complete condition from the six methods listed above. Partial information or single properties are usually not sufficient!

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.