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Topic: Trapezoid definition
Replies: 26   Last Post: Oct 7, 2004 11:51 PM

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Posts: 26
Registered: 12/4/04
Re: Is a rectangle a square?
Posted: Sep 28, 2004 4:09 PM
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On 28 Sep 04 05:21:38 -0400 (EDT), Kit wrote:
>Pamela Paramour wrote:
>>Is a square a rectangle? ...
>> If you refer to Webster, ...

>Here is a nice story realy happend in german tv, sorry for my bad
>In the german quiz-show "Wer wird Millionär" (Who becomes a
>millionaire) from January, 31 2003 the 8000-Euro question was:
>Every rectangle is:
>(a) a rhombus
>(b) a square
>(c) a trapezoid
>(d) a parallelogram.
>In this show _allways_ exactly one answer is (has to be) correct.
>The candidate was so confused, she didn't know if c or d is thw right
>answer, so she skipped the question and went home (with "just" 4000
>Euro). In the following days the broadcast station got tons of mails,
>letters and phone calls. Nearly all "mathematicians" regarded c _and_
>d as correct. The broadcast station told, that they looked up in

>different encyclopaedias, all three saying that trapezoids have only
>one pair of parallel sides. Taking this definition only d is correct.
>That's the problem. Who is right: More than 90 percent of the
>mathematicians saying a parallelogram is also an trapezoid or three
>encyclopeadias saying the opposite?
>The Solomonian solution. In the next week the candidate got a "new"

You can't really say "who's right." It's just a question of how one
defines "trapezoid." In all American textbooks (except the University
of Chicago geometry textbook), a trapezoid is defined as a
quadrilateral with exactly one (or at most one) pair of parallel
sides. A parallelogram is defined as a quadrilateral with 2 pairs of
parallel sides.

Professor Conway, on the other hand, defines a trapezoid as a
quadrilateral with "at least" on pair of parallel sides. Using that
definition, the set of parallelograms is clearly a subset of the set
of trapezoids. Thus, using that definition, every rectangle is a
trapezoid, and also a parallelogram (as is the case with the usual

A similar problem exists with the definition of the kite. Most
writers say it is a quadrilateral in which AT MOST one diagonal is the
perpendicular bisector of the other. Conway would say that it is a
quadrilateral in which AT LEAST one diagonal is the perpendicular
bisector of the other. So using Conway's definition, every rhombus is
a kite.

There have been many, many message threads here on this issue.
Logically, there is a great deal to be said for Prof. Conway's

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