Inclusive Definitions: Trapezoids
Date: 11/04/2004 at 20:48:55 From: Peter Subject: The true definition of a trapezoid. As far as I know, a trapezoid is defined as a quadrilateral with exactly one set of parallel sides. Most textbooks and websites will confirm this definition. However, a very highly regarded educator and textbook author recently argued that this definition is incorrect. His definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid. He even included this definition in the glossary of a newly published textbook. Is he correct or are thousands of books going to be published with the wrong definition? As a teacher looking to buy new books for my school, I would really like to know. Thanks.
Date: 11/04/2004 at 23:01:15 From: Doctor Peterson Subject: Re: The true definition of a trapezoid. Hi, Peter. Both definitions are in use, so neither is wrong! That does lead to confusion, but each author has to choose the definition that makes most sense in his context. Here are a couple pages from our site about this matter: Quadrilateral Classification: Definition of a Trapezoid http://mathforum.org/library/drmath/view/54901.html Inclusive and Exclusive Definitions http://mathforum.org/library/drmath/view/55295.html The same sort of issue arises with other shapes, such as the rectangle. Is a square a rectangle? Not to a child; we tell them "This is a square, and that is a rectangle," and they learn that a rectangle is like a square but doesn't have equal sides. Yet to a mathematician, such exclusive definitions are awkward, because everything that is true of a rectangle is true of a square, and we'd like to use one word to cover both when we write a theorem. For example, any quadrilateral with three right angles is a rectangle --why should we have to add "or a square"? And if we prove something is true of any parallelogram, we don't want to have to add "or rhombus, or rectangle, or square." So although even mathematicians find the exclusive definition useful when we want to point out objects (we generally use the most specific term we can, so that we wouldn't call a square a rectangle when we are trying to ask for one), for technical purposes we prefer the inclusive definition, and would prefer that it be taught in schools. It's a little more subtle with trapezoids, because there are fewer theorems about them, so we have less commitment to an inclusive definition. There are probably mathematicians, and certainly educators, who don't use the inclusive definition in this case. But as you'll see in the links above, the inclusive definition makes the relationships among quadrilaterals clearer. I should also mention that when a mathematician says "a trapezoid is a quadrilateral with two sides parallel," he probably means "at least two sides," not "exactly two sides"; that is the usual understanding of such a phrase, because we get used to speaking that way. It may not always be clear to non-mathematicians! It is well known that there are two definitions used in different textbooks; it is not a new thing. Here are a couple references I found: http://www.e-zgeometry.com/class/class6/6.5/6.5.htm Trapezoids: Definition #1 A trapezoid is a quadrilateral with exactly one pair of parallel sides. Definition #2 A trapezoid is a quadrilateral with at least one pair of parallel sides. This particular book uses definition #1. The two defintions look alike but they have quite serious implications as to classifications of quadrilaterals. It is quite rare to have the mathematical community undecided as to a definition. http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html Believe it or not, there is no general agreement on the definition of a trapezoid. In B&B and the handout from Jacobs you got the Exclusive Definition. Exclusive Definition of Trapezoid A quadrilateral having two and only two sides parallel is called a trapezoid. However, most mathematicians would probably define the concept with the Inclusive Definition. Inclusive Definition of Trapezoid A quadrilateral having at least two sides parallel is called a trapezoid. The difference is that under the second definition parallelograms are trapezoids and under the first, they are not. The advantage of the first definition is that it allows a verbal distinction between parallelograms and other quadrilaterals with some parallel sides. This seems to have been most important in earlier times. The advantage of the inclusive definition is that any theorem proved for trapezoids is automatically a theorem about parallelograms. This fits best with the nature of twentieth-century mathematics. It is possible to function perfectly well with either definition. However, it is important to have agreement in a math class on the definition used in the class. In Math 444 the official definition of a trapezoid is the Inclusive Definition. In the event we wish to distinguish trapezoids with exactly two parallel sides, we will call such trapezoids "strict trapezoids". Again, each definition has its place, and should be used in the appropriate context. The inclusive definition fits well into the context of geometry, and I recommend it. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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