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### 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.

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

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&amp;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/
```
Associated Topics:
College Definitions
College Triangles and Other Polygons
High School Definitions
High School Triangles and Other Polygons

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