I'll Go There

Every parallelogram is a trapezoid.

Remember in school when your teacher told you that every square is a rectangle but not every rectangle is a square? Do you? DO YOU? Because it's literally the same thing. Every parallelogram is a trapezoid but not every trapezoid is a parallelogram.

For some reason this is a debate that periodically rages in many Mathematics teacher circles. Why? Because textbooks disagree on the definition of a trapezoid. The definition of a parallelogram is basically standard. A parallelogram is a quadrilateral with two pairs of parallel sides. The definition of a trapezoid, however, is not as standard. Some textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides. While other textbooks define a trapezoid as a quadrilateral with exactly one pair of parallel sides. 

It is this second definition that poses the problem for parallelograms. If a trapezoid is defined to have exactly one pair of parallel sides, then a parallelogram cannot be a trapezoid because a parallelogram has two pairs of parallel sides and two is bigger than one.

And this is usually how the arguments go. Some one claims that every book they've ever used has defined trapezoid one way. Then someone else claims that every book they've ever used defined trapezoid the other way.

Here is the problem as I see it. Who cares what the textbooks say? Think mathematically about it. What other properties do trapezoids have? Which of these properties are dependent on the definition chosen? Which of these properties, if any, are independent of the definition chosen? 

To my knowledge, there is not a single property of trapezoids that is dependent on the definition of trapezoid excluding parallelograms. Not one. Which means that literally every property of trapezoids is also true of parallelograms. This makes the reason for choosing a definition that specifically seeks to exclude parallelograms that much more puzzling. What mathematical reason is there for doing so?

Personally, I find this whole debate rather frustrating. I have never heard a mathematical reason for wanting to exclude parallelograms from being trapezoids. To my knowledge there is no sound mathematical reason for doing so. Rather, it always seems to come down to people wanting desperately to believe that they have managed to read from only the correct books, and those that disagree are rubes who have managed to read nothing but wrong books. That's ridiculous.

Maybe this post isn't about trapezoids.