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Geometry: High School15 chapters | 160 lessons

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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.

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.

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.

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.

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.

Now that we know the properties that make up a parallelogram, it becomes easy to identify a parallelogram. We don't have to check to make sure all the properties are there. If just a few are there, then we can say with confidence that the shape is a parallelogram. There are four ways to do this.

- You have two pairs of parallel opposite sides
- You have two pairs of equal opposite angles
- You have two pairs of equal and parallel opposite sides
- Both diagonals bisect each other

If any one of the above is satisfied, then you are looking at a parallelogram.

If you phrase any of the above in a statement such as, 'If a quadrilateral has two pairs of parallel opposite sides, then the quadrilateral is a parallelogram,' then you are using it as a proof theorem.

Now, let's recap what we've learned. **Parallelograms** are four-sided flat shapes whose opposite sides are both equal and parallel to each other. A parallelogram has two pairs of opposite sides that are parallel and equal in length. A parallelogram also has two pairs of opposite angles that are also equal to each other. The adjacent angles form supplementary angles that add up to 180 degrees. The diagonals of a parallelogram will always bisect each other or meet at exactly the midpoint of each diagonal. To identify a parallelogram, the shape has to meet one of the following criteria:

- It has two pairs of parallel opposite sides
- It has two pairs of equal opposite angles
- It has two pairs of equal and parallel opposite sides
- Its diagonals bisect each other

Phrase any of the above four items in a statement such as, 'If it has two pairs of equal opposite angles, then the quadrilateral is a parallelogram,' then you have yourself a proof theorem.

This lesson could provide you with the information necessary to:

- Discuss the particulars of parallelograms, supplementary angles and proof theorems
- Provide characteristics of the angles of parallelograms
- List the criteria required to identify a parallelogram

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Geometry: High School15 chapters | 160 lessons

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