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A Trapezoid is a quadrilateral with at least one set of parallel sides.

An Isosceles Trapezoid is a Trapezoid where the legs are of equal length.

These definitions are called inclusive. This means that parallelograms (with two sets of parallel sides) are a type of trapezoid.

What is the most formal and authoritative definition of an Isosceles Trapezoid? Rarely do I see anybody make them more exclusive, thus requiring particular angles and lines of symmetry. I find those limiting, but I don't want to be teaching my students incorrectly.

Similar question for illustration: Is a Square a Rectangle? Is a rectangle exclusively a parallelogram where some sides must be of different length of some other side? I don't like exclusivity, I like inheritance.

EDIT

Follwoing the the comments below, I will go ahead and state my follow-up question:

enter image description here

Is this an Isosceles Trapezoid? Many prior discussions have led me to believe that it is, and it does indeed fit the above definition.

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    $\begingroup$ In “real” mathematics, ambiguous terms are defined as needed. Don’t worry too much about teaching precisely “correct” definitions. However, in my schooling I learned the inclusive definitions. I learned that squares are rectangles. $\endgroup$ Commented Jul 1, 2024 at 16:45
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    $\begingroup$ In these situations, I think back on the wisdom of Humpty Dumpty in Lewis Carroll's Through the Looking Glass: "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean — neither more nor less." ... There is no naming authority in mathematics, so the literature is replete with competing definitions and conventions. (It's worth noting that we can't even decide whether zero is a natural number.) A thoughtful author will take care to precisely define a term that might have multiple meanings. $\endgroup$ Commented Jul 1, 2024 at 16:54
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    $\begingroup$ You have to be careful with an inclusive definition because what are legs and bases of a parallelogram? Of course, you can further define that legs and bases apply only to trapezoids with one pair of parallel sides. $\endgroup$ Commented Jul 1, 2024 at 16:57
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    $\begingroup$ @Suamere define them in the way which is convenient for your interesting question. There is no council to appeal to. $\endgroup$ Commented Jul 1, 2024 at 17:05
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    $\begingroup$ @Suamere: "I need exacting rules." You won't find them. :) Definitions are context-dependent. I, for one, tend to allow vertices of a triangle to be collinear, so that my preferred rendition of the Triangle Inequality is $a+b\geq c$; others disagree ... and I'm often one of them. My usage can hinge on whether I'm considering a dynamic family of triangles that might reasonably include "flat" members, vs discussing a fixed geometric figure that simply doesn't make sense with flats and where the "strict" Triangle Inequality helps. $\endgroup$ Commented Jul 1, 2024 at 17:16

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An Isosceles Trapezoid is a trapezoid where the legs are of equal length.

This inclusive definition allows for an asymmetrical Isosceles Trapezoid.

It seems like added exclusivity that isn't widespread.

Actually, for completeness, please do cite a source of the above definition (Definition A), which disagrees with these three equivalent definitions (Definitions B)—the first from Wolfram MathWorld and the rest from Wikipedia—of an isosceles trapezoid/trapezium:

  • a trapezoid in which the base angles are equal
  • a trapezoid whose diagonals have equal length
  • a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides.

Now, an exclusive (non-inclusive) definition ousts a natural subset from a set; that is, it excludes objects by imposing some restriction (for example, excluding rectangles from the set of parallelograms or excluding equilateral triangles from the set of isosceles triangles). Definitions B are not in fact exclusive definitions, since they are not arbitrarily excluding unequal-base-angled trapezoids from some larger set. Converting Definitions B to an exclusive definition:

  • a trapezoid whose legs are of equal length but not parallel.

Although Definition A is purely etymological—the Greek roots of isosceles mean equal legs—and its objects are a strict superset of Definitions B's objects, neither of these competing classifications is necessarily more natural/intuitive than the other.

enter image description here

Is this an Isosceles Trapezoid?

Not going by Mathworld's sensible definition, which requires symmetry.

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  • $\begingroup$ Thanks Ryan! Only in comments did I ask for wikipedia to not be used, but your answer is good to have up here. Is it common in professional or collegiate settings to say that a Trapezoid has this requirement? It seems like added exclusivity that isn't wide spread. Though I do see a benefit in that "Isosceles Trapezoid Area Equations" have the expectation of symmetry. But other inclusive definitions allow for the aforementioned non-symmetrical illustration of an Isosceles Trapezoid. $\endgroup$ Commented Jul 1, 2024 at 17:43
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    $\begingroup$ It's worth noting that, as a wiki (it's right there in the name!), Wikipedia is editable by "anyone". While there is a review process to (hopefully) weed-out utter nonsense, one probably shouldn't take a given entry as particularly authoritative on a matter subject to differing opinions or conventions. $\endgroup$ Commented Jul 1, 2024 at 17:54
  • $\begingroup$ @ryang: I didn't mean to suggest that you suggested authoritativity. :) My comment was for readers (of which I've encountered quite a few) who don't realize that Wikipedia is publicly edited and might therefore presume authoritativity. $\endgroup$ Commented Jul 1, 2024 at 18:21
  • $\begingroup$ @Suamere I've just expanded the answer to reply to your comment. $\endgroup$ Commented Jul 3, 2024 at 4:59

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