So there’s hope for the Geometry community and for teachers everywhere.
This site and this site also get it right. The former definition would make such concepts as the trapezoidal approximation to a definite integral be ill-defined. The latter definition is consistent with its uses in higher mathematics such as calculus. Other authors 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). Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. There is also some disagreement on the allowed number of parallel sides in a trapezoid. Here’s a nice paragraph from the wikipedia entry on trapezoid: 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. It allows you to play with the trapezoid, moving its vertices and edges. I’ve prepared a simple GeoGebra applet and posted it here. (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 emphasized the words “ one and only one,” which is what I want to comment about in this post. “A quadrilateral with one and only one pair of parallel sides.” Here’s the usual definition of a trapezoid (taken from our school’s Geometry text book, by Holt Rinehart and Winston): In my humble opinion, it’s a major problem with Geometry as it’s currently taught. Or…not with trapezoids–I like trapezoids–but a bone of contention I have with the definition of trapezoids.
I should have made this post a long time ago, because it’s a bone of contention I’ve always had with trapezoids.
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