(Credit due to my teaching of a UCSMP Geometry course 20 years ago.) The more general names are toward the top of the chart and the more specific shapes are at the bottom. On the other hand, using an inclusive definition of trapezoids ( at leastone pair of parallel sides) creates the far more consistent relationship structure shown below. Interestingly, this same area formula also applies (with acknowledgement that ) to parallelograms, rhombi, rectangles, and squares, a “coincidence” ignored by proponents of the exclusive definition. With nothing guaranteed about the non-parallel or congruent sides, about all you can claim is its area formula: where and are the parallel bases and ht is the height.
The classification question has been discussed by Zalman Usiskin and others, but I’ll attempt a portion of my own micro-version around trapezoid classifications here.įirst, pure trapezoids (those that can’t be called anything more specific under either definition) have precious few properties. I agree completely because once definitions are accepted–especially in mathematics–they absolutely drive and constrain any and all logical conclusions one logically is able to reach.
#Isosceles trapezoid definition free
Who cares, some may ask? From one perspective, everyone is perfectly free to use either definition, but I agree with the contributor who argues that part of the discussion ought to be about what makes a definition in mathematics. There have been many discussions about the importance of definitions in mathematics, and in this thread, the contributors rightly noted that the inclusive definition makes parallelograms a type of trapezoid (exactly like squares are a type of rectangle), while the exclusive definition expressly forbids this because parallelograms don’t have exactly one pair of parallel sides.
There was a recent debate on the AP Calculus EDG on the definition of a trapezoid, specifically whether a trapezoid has “exactly one pair of parallel sides” (the exclusive definition) or “at least one pair of parallel sides” (the inclusive definition).
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