Proving That a Quadrilateral is a Parallelogram

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  • 0:00 A Quadrilateral and a…
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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.

In this lesson, you'll learn that there are five ways you can go about proving that a certain kind of quadrilateral is a parallelogram. You'll also get a demonstration of how knowing the ways can help you when you need to prove something.

A Quadrilateral and a Parallelogram

When it comes to math, you have to be able to prove that what you're doing is correct. When it comes to geometry, it's the same. In geometry, you'll often be asked to prove that a certain shape is, indeed, that certain shape. For example, you might be shown a quadrilateral and be asked to prove that it is a parallelogram. Remember that a quadrilateral is a four-sided flat shape. A parallelogram is a quadrilateral with two pairs of opposite, parallel sides.

Looking at this shape, you might think that it is a parallelogram, but unless the problem specifically tells you and/or you can prove that it is, you can't say for sure that it's a parallelogram.

prove parallelogram

This is where mathematical proofs are very important. You can only say for sure that this is a parallelogram with a mathematical proof. Most times, when you're asked to prove that a certain quadrilateral is a parallelogram, you'll be given information about just a few sides. It's then your job to prove that these sides have the right properties of a parallelogram. We'll be taking a closer look at this in a little bit.

A Quadrilateral Is a Parallelogram

But first, let's go over five ways you can use to prove that a quadrilateral is a parallelogram. Depending on what information you have to work with, you'll be using one of these five ways.

1. Prove that both pairs of opposite sides are parallel.

This one is simply the reverse of the definition of a parallelogram. If you can prove that the quadrilateral fits the definition of a parallelogram, then it is a parallelogram.

2. Prove that both pairs of opposite sides are congruent.

If both pairs of opposite sides of a quadrilateral are congruent, then you'll always have two opposite pairs of parallel sides. Congruent means that they measure the same. Think about it: two congruent sides separating the other pair of opposite sides must always keep those opposite lines the same distance apart. This means, then, that the opposite sides are also parallel.

3. Prove that one pair of opposite sides is both congruent and parallel.

This one is kind of similar to the method before. It just goes about proving the case in another way. You can actually try this out with four toothpicks. Try it by placing two of the toothpicks opposite and parallel to each other. Now connect these two toothpicks at both ends with the other two toothpicks. You'll notice that no matter how you place your first pair of toothpicks, your second pair of toothpicks will always be parallel.

4. Prove that the diagonals bisect each other (that they can divide each other into two equal parts where they cross).

This one is a bit harder to visualize. But you can play around with it by taking two different-sized sticks and crossing them in the middle of both sticks. These two sticks are the diagonals inside your parallelogram. You'll see that no matter how you cross your sticks, as long as they cross in the middle, you'll always get a parallelogram. Your two sticks are the blue and green lines. You can see that these are the diagonals inside the parallelogram.

prove parallelogram

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