Yes it’s true. I’m writing about trapezoids again (having written passionately about them here and here previously). I’ve been taking a break from blogging, as I usually do in the summer. For us, school starts in just two weeks. So I thought I’d come out of my shell and post something…and of course I always have something to say about trapezoids :-).

Let’s start with the following easy test question. Don’t peek. See if you can answer the question without any help.

Which of the following quadrilaterals are trapezoids?

Before giving the answer, let me first just remind you about my very strongly held position. I believe that instead of this typical textbook definition (the “exclusive definition” we’ll call it) that reads:

“A quadrilateral with one and only one pair of parallel sides.”

the definition should be made inclusive, and read:

“A quadrilateral with at least one pair of parallel sides.”

So the test question above was easy, right? Quadrilaterals (A) and (C) are trapezoids, I hear you say.

Not so fast!! If you’re using the inclusive definition, then the correct answers are actually (A), (B), (C), (D), and (E). But it gets better: If you were using the the exclusive definition, then NONE of these are trapezoids. In order for (A) and (C) to be trapezoids, under the exclusive definition, you must prove that two sides are parallel AND the two remaining sides are not parallel (and you can’t assume that from the picture…especially for (C)!).

Can you see the absurdity of the exclusive definition now?

I finish by offering the following list of reasons why the inclusive definition is better (can you suggest more reasons?):

1. All other quadrilaterals are defined in the inclusive way, so that quadrilaterals “beneath” them inherit all the properties of their “parents.” A square is a rectangle because a square meets the definition of a rectangle. Likewise, parallelograms, rectangles, rhombuses, and squares should all be special cases of a trapezoid.
2. The area formula for a trapezoid still works, even if the legs are parallel. It’s true! The area formula works fine for a parallelogram, rectangle, rhombus, or square.
3. No other definitions break when you use the inclusive definition. With the exception of the definition that some texts use for an isosceles trapezoid. Those texts define an isosceles trapezoid has having both legs congruent, which would make a parallelogram an isosceles trapezoid. Instead, define an isosceles trapezoid as having base angles congruent, or equivalently, having a line of symmetry.
4. The trapezoidal approximation method in Calculus doesn’t fail when one of the trapezoids is actually a rectangle. But under the exclusive definition, you would have to change its name to the “trapezoidal and/or rectangular approximation method,” or perhaps ban people from doing the trapezoidal method on problems like this one: Approximate using the trapezoidal method with 5 equal intervals. (Note here that the center trapezoid is actually a rectangle…God forbid!!)
5. When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel (see the beginning of this post!).

I should have made this post a long time ago, because it’s a bone of contention I’ve always had with trapezoids. Or…not with trapezoids–I like trapezoids–but a bone of contention I have with the definition of trapezoids. In my humble opinion, it’s a major problem with Geometry as it’s currently taught. Here’s the usual definition of a trapezoid (taken from our school’s Geometry text book, by Holt Rinehart and Winston):

“A quadrilateral with one and only one pair of parallel sides.”

I’ve emphasized the words “one and only one,” which is what I want to comment about in this post. (Here’s another source and another source and another source that say it that way, too.) Sometimes it’s also said, “a quadrilateral with exactly one pair of parallel sides.”

I’ve prepared a simple GeoGebra applet and posted it here. It allows you to play with the trapezoid, moving its vertices and edges. As you drag it around, at all times, one pair of sides will be parallel. But wait, it’s not always a trapezoid, is it? According to the Geometry book, there’s one moment, as you’re dragging it around, that it stops being a trapezoid and for that one second is exclusively a parallelogram. Here’s the moment I’m talking about:

Is this still a trapezoid?

That’s right, using the Geometry textbook’s definition of a trapezoid, if both pairs of opposite sides of the quadrilateral happen to be parallel,  it’s not a trapezoid anymore. At this point, the mathematical reader should be crying, “Foul! How did we ever let this happen? This definition of a trapezoid is so inelegant!!” And I couldn’t agree more.

We don’t do this with the definition of any other quadrilateral. Why do it with a trapezoid? If I were to make another little applet that lets you drag around a rectangle, would we say “it’s not a rectangle” at the moment you make the four sides equal? No! That would be absurd.

The definition of a trapezoid, in my opinion (and thankfully in the opinion of some others) should read:

“A quadrilateral with at least one pair of parallel sides.”

And the hierarchical diagram should look like this one, I found online (taken from a mathematically enlightened author):

from mathisfun.com

Here’s a nice paragraph from the wikipedia entry on trapezoid:

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors^[2] define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors^[3] define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid (along with the rhombus, the rectangle and the square). The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral be ill-defined.

This site and this site also get it right. So there’s hope for the Geometry community and for teachers everywhere. But please, let’s work hard to eradicate the “exclusive” definition from ALL the textbooks. It’s hideous.

For more posts on this topic, visit here and here.