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?):
- 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.
- 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.
- 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.
- 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!!)
- 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!).