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Subject:   RE: More on What is a Trapezoid
Author: Mathman
Date: Jan 10 2005
On Jan 10 2005, proofpad wrote:

Sorry, but how do your reach
> that conclusion?  An Isosceles
> trapezoid [although I've never heard
> the term applied to other
> than the triangle], isn't *any* trapezoid,
> but one in particular,
> as the square isn't any rectangle, but one in
> particular.  If by
> Isosceles trapezoid you mean one with base angles
> equal, "any
> parallelogram" does not have two base angles equal, but
> only the
> "rectangular parallelogram" has that property.  So all
> rectangles
> would be "isosceles trapezoids", but not all
> parallelograms.
> David.

When the geometry texts (as mine, the McDougal-Littell)
> define a Trapezoid as a quadrilateral with exactly one pair of
> parallel sides, it will then define an isosceles trapezoid as a
> trapezoid with the legs (non-parallel sides) congruent.
> -Gabriel Edge

I can draw a trapezoid with the other two sides of equal length, but the base
agles not equal.  Is it an isosceles trapezoid?  Thsi is why I suggested thatthe
parallelogram may not be included perhaps in the definition.  That is; is the
trapezoid isosceles because of equal base angles, or because of equal
non-parallel sides?  I'm thinking it is one with equal base angles, which
would make those sides equal.  in orderto prove that a figure was then an
isosceles trapezoid, it would not be sufficientto show that the two
non-parallel sides are equal, but would be sufficient to show that the two
base angles are equal, or what amounts to the same thing thatthe two top angles
are equal.

No matter the opinion of one author, if we study the triangle for example, we
start with the scalene triangle and extract all of its properties.  Then we have
the isosceles triangle with all of the properties of the scalene and then some
more specific.  Then the equilateral triangle with its properties, retaining
those of the isosceles triangle.  The parallelogram [and so the reactangle and
square and rhombus] is a trapezoid since it retains *all* of the properties of
the trapezoid.  The trapezoid is a quadrilateral with *at least* two sides
parallel for that reason.  The isosceles triangle is still a [scalene] triangle
in that it retains all of the properties common to both. ...and so on.

If I can show that a quadrilateral has two sides parallel, I can then state that
it is [at least] a trapezoid.  If there is none further, then it remains a
trapezoid.   If the other pair are shown to be parallel, then it is a
parallelogram, and I can say that it is so. ...but I can not yet say if it is a
rectangle, or a square.  But at least it is a parallelogram, and further, it is
a trapezoid.  This is in as much as I can say that a polygon is a triangle if
its angles add to 180.  Showing that it may also be isosceles does not then make
it *not* a triangle.  Please understand, I'm not trying to be funny there, just
trying to make an analogy.


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