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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.

Geoffrey Trang
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Suamere
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    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. – Malady Jul 01 '24 at 16:45
  • @Malady Thanks! I do plan to use the accepted answer/reference as a proof for a different follow up question. Assuming the accepted answer agrees with what you and I said. In a way, I hope or expect the accepted answer to have some exclusivity, which will stop me from doing a different follow up question. :/ – Suamere Jul 01 '24 at 16:47
  • Maybe write your follow-up question in your original post! That might give some context for what you need these definitions for. – Malady Jul 01 '24 at 16:51
  • I love that, but I feel it's leading the witness. Or tainting the results. I'm extremely torn about doing that. I'd rather have people say Oh, in THAT case yeah, the definition is wrong. But only after nailing down (or allowing ambiguousness) is solidly determined here. – Suamere Jul 01 '24 at 16:53
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    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. – Blue Jul 01 '24 at 16:54
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    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. – Vasili Jul 01 '24 at 16:57
  • @Blue Very fair point. As an Engineer, I need exacting rules, if not explicitly stated ambiguity. But you could be right that not enough people agree. Love the analogy. Hoping for any respectable answer at all though, with any source from something beyond an elementary teaching website or wikipedia. Anything collegiate or professional. – Suamere Jul 01 '24 at 16:57
  • @Vasili Agreed. And I'd say that the bases (top and bottom) are the two sides that meet the original requirement of "at least one set of parallel sides". The "legs" then are the left and right sides, whether they be parallel or not. So inclusivity works fine here. – Suamere Jul 01 '24 at 16:59
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    @Suamere define them in the way which is convenient for your interesting question. There is no council to appeal to. – Malady Jul 01 '24 at 17:05
  • @Malady I suppose you're all right. This is a lot of discussion just to get to a definition, for which it appears to be far too fluid. In that case, I've added my followup question as you previously suggested. – Suamere Jul 01 '24 at 17:09
  • If "the legs" are defined to mean "choose any pair of parallel sides, then the other two sides are called the legs" in the case of a parallelogram, then any parallelogram would meet the "equal legs" definition of an isosceles trapezoid. An isosceles trapezoid should instead be defined to mean either an exclusive trapezoid with equal legs or a rectangle. A non-rectangular parallelogram is not an isosceles trapezoid. – Geoffrey Trang Jul 01 '24 at 17:13
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    @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. – Blue Jul 01 '24 at 17:16
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    Opinion-wise, I tend to be an inclusionist, so that I can name things based on minimal info. If I have a quad with one pair of parallel opposite sides, but don't (or can't) know how their lengths compare, or what's up w/the other sides, I may feel comfortable in calling it a trapezoid ... especially if the unknown properties are subject to change (as in a "dynamic" family). So, is a parallelogram an isos trap? Sure! Well, maybe. After all, in this old answer, I derive a trap area formula using an argument that fails for p-grams. Go figure. – Blue Jul 01 '24 at 17:40
  • @Blue Thanks again! I was hoping somebody would bring up the area formula. And your tone seems to agree with my stance that "my formula doesn't work if you define the shape that way" isn't a good reason not to define a shape some way. The shape is the shape, and formulas should take the shape's properties into account. Perhaps a non-symmetric (p-gram) isos trap should simply have a different formula. That doesn't preclude it from matching the definition of an isos trap. Yes? – Suamere Jul 01 '24 at 17:45
  • @GeoffreyTrang But that's where my illustrative question comes into play. By your reasoning, a Square is not a Rectangle. It's a Square only. – Suamere Jul 01 '24 at 17:48
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    I've seen trapezoid with exactly one pair of parallel sides defined as strict trapezoid and the isosceles trapezoid definition is given for strict trapezoids only. Your example would not be an isosceles trapezoid under these definitions. – Vasili Jul 01 '24 at 17:54
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    @Suamere Of course, a square is a rectangle. The only problem is with the "equal legs" definition of an isosceles trapezoid. With that definition, any parallelogram (with either pair of parallel sides chosen to be the "bases" and the other pair the "legs") would be an isosceles trapezoid. But in fact, rectangles are the only parallelograms that should be called isosceles trapezoids. – Geoffrey Trang Jul 01 '24 at 18:44
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    @Suamere: Comments aren't for discussion, so this may be my last reply ... I'm just gonna say "It depends". Besides, even when consensus exists "in academia", an author shouldn't assume every reader will be familiar with it, so it helps to be explicit about what you mean in the here-and-now. You could do this w/disclaimers: "This trap formula works for parallel sides $a\neq b$." You can also explicitly limit the scope of usage: "For the purposes of this discussion (or even this portion of it, or even just this formula), a 'trap' is taken to mean blah-blah-blah". Etc. ... Good luck! – Blue Jul 01 '24 at 19:04

1 Answers1

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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.

ryang
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  • 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. – Suamere Jul 01 '24 at 17:43
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    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. – Blue Jul 01 '24 at 17:54
  • @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. – Blue Jul 01 '24 at 18:21
  • @Suamere I've just expanded the answer to reply to your comment. – ryang Jul 03 '24 at 04:59