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/