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Parallelograms: Definition, Properties, and Proof Theorems

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  • 0:07 Parallelogram Basics
  • 0:37 Sides
  • 2:09 Angles
  • 3:09 Identifying a Parallelogram
  • 4:05 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can identify parallelograms. You will see the unique properties that belong to the parallelogram. You will see the special side and angle characteristics needed to prove a certain shape is a parallelogram.

A Parallelogram

Mathematically defined, a parallelogram is a four-sided flat shape whose opposite sides are both equal and parallel. What this means is that a parallelogram has two pairs of opposite sides that are parallel to each other and are the same length. However, each pair can be a different length than the other pair.

Visually defined, a parallelogram looks like a leaning rectangle. It's as if a rectangle had a long, busy day and is now just resting and leaning up against a wall.

Sides

Our mathematical definition of a parallelogram includes two inherent properties.

One property that is included in the definition is that the opposite sides are parallel. If you look at each pair of opposite sides and drew the lines out, you would find that these lines will never meet. Try it for yourself. Draw a parallelogram, and use a ruler to draw out first the bottom and top lines. Do they look like they will meet? Now do the same for the left and right sides of the parallelogram. Do these lines meet? As you can see, neither of these pairs of lines will ever meet each other.

Another property that we gather from the definition is that the opposite sides are also equal to each other in length. If you took a ruler and measured each pair, you will see that each pair is the same length. Why don't you try it? You will find, however, that the pairs are not necessarily equal to each other. One pair can be longer than the other. But as long as both lines in each individual pair are separately equal to each other, that is all that matters.

This next property is not specified in the definition but comes about because of it. It has to do with the diagonals and not the sides of the parallelogram. Because we have two pairs of equal and parallel opposite sides, the diagonals will bisect each other. The diagonals are the lines that connect the opposite corners to each other. The point where they bisect is exactly the halfway point of each diagonal.

Angles

Parallelograms have two properties related to their angles.

The first is that the opposite angles are equal to each other. Just like we have two pairs of opposite sides, we have two pairs of opposite angles. The two angles making up each pair have to be equal, but the two pairs don't have to be equal. Below, we can label one pair of angles with a and the other pair with b. So, if we went around clockwise starting from the top left angle, we would see a, b, a, and then b again.

In a parallelogram, opposite angles are equal.
image showing parallelogram with labeled angles

The second is that a pair of adjacent angles will always add up to 180 degrees. So, if we kept our a and b labels for our angles, then when you add up angle a with angle b, you will always get 180 degrees. These are called supplementary angles. You can combine any two angles that are next to each other in this way. You can combine the top two, the bottom two, the left two, or even the right two.

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