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is parallelogram a type of trapezoid?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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is parallelogram a type of trapezoid?

by sadi28 » Sat Sep 01, 2012 7:57 am
one of my math resources says "Trapezoid is a quadrilateral which has at least one pair of parallel sides"
if this definition is true, we can assume that parallelogram is a special kind of trapezoid.
I googled about this and it seems that there is disagreement about the definition. some says trapezoid has exactly one pair of parallel sides, while other says it has at least one pair of parallel sides!
now I wonder what is the definition of GMAT?
I checked the math review of OG13 but it doesn't have anything about trapezoid!
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by adthedaddy » Sat Sep 01, 2012 8:40 am
Hi, I do not know what official OG defines this but in my opinion, a parallelogram qualifies the definition of a trapezoid.
Trapezoid: one pair of opp sides are parallel
Parallelogram: both the pairs of opp sides are parellel

e.g. For the following parallelogram ABCD,

Area of a trapezoid is given by [(a+b)/2]*h

For a parallelogram, a=b; thus, area of trapezoid becomes (2a/2)*h = ah

The definition of area of a parallelogram is = ah

In both the cases we get the same result, thus, mathematically a parallelogram qualifies the definition of a trapezoid.
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by machichi » Sun Sep 02, 2012 7:53 pm
I love this question. Haha--this should never matter in our lives and yet it does! The answer in short is no one knows but there is a preference for YES! :)

From the Trapezoid Wiki:

Definition

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors[2] define a trapezoid as a quadrilateral having exactly one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Other authors[3] define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition[4]), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.
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by Jim@StratusPrep » Thu Sep 06, 2012 6:54 pm
Parallel sides does not make it a parallelogram. 2 pairs of parallel side do.
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by Tommy Wallach » Fri Aug 29, 2014 12:50 pm
Ha! Jim always gives the worst answers. Cracks me up.

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by ceilidh.erickson » Wed Sep 10, 2014 2:45 pm
Truth.

In answer to this old thread... it doesn't really matter. The GMAT isn't going to get tricky with "ha! We said trapezoid but we really meant parallelogram! Tricked you! We said rectangle but we meant square, because a square is a type of a rectangle!" (Or I suppose they could, but I've never seen a question like that, and it's really not their style).

There is some semantic debate in the mathosphere about whether trapezoid means "at least one pair of parallel sides," in which case a parallelogram would be a trapezoid, or "exactly one pair of parallel sides," in which case it wouldn't. The GMAT, to the best of my knowledge, has never addressed this one way or the other.

What you would be likely to see is something like:

Is quadrilateral ABCD a parallelogram?

(1) AB is parallel to CD
(2) BC is equal to CD

etc.
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