Sorry, I thought I got it all out of my system in my first post about trapezoids last week :-). Allow me to rant a bit more about trapezoids. First let me remind you of the problem. Many Geometry books, our school district’s book included, state the definition of a trapezoid this way:
“A quadrilateral with one and only one pair of parallel sides.”
In case you didn’t catch the point of my first post: I think this is a poor definition and should be abolished from all Geometry curriculum everywhere. Here are some pictures I recently came across on the internet depicting the hierarchy of quadrilaterals. These picture agree with the above definition. Let me just say once more, I completely and totally disagree with these pictures, and I think you should too. That is to say, all of the following pictures are WRONG.
And I could go on and on. Now here are two good ones.
To be fair, the first set of pictures are only partially wrong. They have good intentions. Typically, the first breakdown of quadrilaterals in those pictures is by “number of parallel sides.” The first lines that come off of the word ‘quadrilateral’ divide quadrilaterals into three categories usually:
- No parallel sides (i.e. the kite)
- Exactly one set of parallel sides (i.e. the trapezoid)
- Two sets of parallel sides (i.e. the parallelogram)
So the pictures aren’t wrong, per say. They just depict different information. The problem comes when teachers ask, “Look at this diagram and tell me: Is every rectangle a trapezoid? Is every rhombus a kite?” The answer to both questions is ‘yes.’ But students instinctively answer ‘no’ when using the first set pictures, and you can see why.
The problem is a historic one. If you go back to Euclid’s Elements, Definition 22 in Book 1, you can see the origin of this problem right away (a translation from the Greek):
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
In the above definition from Euclid, here are the (not perfect) translations of each figure:
- Euclid’s square –> Our square
- Euclid’s oblong –> Our rectangle
- Euclid’s rhombus –> Our rhombus
- Euclid’s rhomboid –> Our parallelogram
- Euclid’s trapezia –> Our…trapezia/trapezium?
The last definition is a bit confusing, since we don’t have a very well-agreed upon name for this figure. But notice that ALL of Euclid’s definitions are exclusive. A square is never an oblong. A square is never a rhombus. Each of the above quadrilaterals, according to Euclid, is mutually exclusive. These exclusive definitions persisted into the 19th century.
But sorry Euclid, no one likes your definitions anymore. I hate to say it, because everyone loves Euclid.
In his defense, he wasn’t using these names for the same purpose we do. Nothing about his language is very technical and he doesn’t say ANYTHING else substantial about these definitions. He doesn’t use them to make categorical statements about quadrilaterals or to give properties that might be inherited. The names he uses are of little consequence to the rest of his work.
Can we lay this issue to rest yet? A parallelogram is always a trapezoid. Say it with me,
A parallelogram is a trapezoid.
A parallelogram is a trapezoid.
A parallelogram is a trapezoid.
Anything you can say about a trapezoid will be true about a parallelogram (area formulas, cyclic properties, properties about the diagonals). A parallelogram is a trapezoid.
Thanks for your thoughts on the importance of definitions. As further support of your premise, I offer up the area formulas for the trapezoid and the parallelogram. Specifically, you can calculate the area for a parallelogram using the trapezoid formula: A = 1/2 * (b_1 + b_2) * h. Of course it will work since the “bases” of a parallelogram are equal and the formula reduces to A = b * h, where b = b_1 = b_2.
If the formula for the trapezoid’s area also calculates the parallelograms area, the parallelogram is a trapezoid.
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I’ve been told that the inclusive definition of trapezoid is the usual one in Canada (whereas the exclusive definition is the usual one in the US–which isn’t to say you can’t find perfectly good US sources for the inclusive definition). It’s almost enough to persuade one to move north.
@Andy: Thanks for the added insights!
@LSquared: Okay, we’re moving to Canada!
John – I admire your passion and tenacity on this issue. It takes a certain David v. Goliath form of courage to stand for what you believe against Euclid and the current majority consensus of folks who think about polygons. Plus it makes for entertaining debates in the math office! I think I’m starting to see the light about your preferred hierarchy and the logic of viewing parallelograms as specific cases of trapezoids. It makes sense to go from zero parallel sides to (at least) 1 pair of parallel sides to 2 pairs of parallel sides. Keep up the good fight!
A hierarchy should go from less restrictive (more inclusive) to more restrictive (less inclusive) classifications. Your preferred hierarchy seems to follow that principle more closely than the conventional one.
We define it inclusively in Britain too, though of course we call them by a slightly different name. I assume that the alternative definition is so you can define an isosceles trapezium as “the other two sides equal” and get all the properties straight out without having to worry about it actually being a parallelogram, but for me at least that’s not worth the loss of elegance.
Two things: First, trapezoid means one thing and the opposite depending where you are. In some countries, it means zero parallel sides, and in US it means exactly one pair of parallel sides, or as you would prefer, at least one pair of parallel sides.
Second: There is no hierarchy in the terms, so that’s why you can’t draw a hierarchy that makes you happy. If trapezoid simply means “4 sides, at least 1 pair parallel” then you can draw a hierarchy based on the number of parallel sides (0 pairs, 1 pairs, 2 pairs). But if you also mix in the dimensions of equality of angles, lengths of sides, etc. then you will fail. Each of these is a dimension by itself and the definitions of the various shapes don’t exclude others.
That’s why a square is both a kite and a parallelogram, even though neither kite nor parallelogram is a subset of the other. It isn’t a hierarchy.
First thing: I know there’s disagreement on the definition of ‘trapezium’ but didn’t think anyone used ‘trapezoid’ to name a quadrilateral with no parallel sides. And you portray the US definition of trapezoid as monolithic. I don’t think it is. I’ve shown there are plenty (perhaps even the majority) of people who agree with me.
Second thing: Perhaps hierarchy isn’t the right word. You might be right that we should use a better word to describe it. I think ‘inheritance’ captures the definitions correctly and is a better term. Here’s the inheritance for a few common quadrilaterals. For each quadrilateral, I’ve included its “parents” in parentheses.
-Square (is a rhombus, rectangle, kite, trapezoid, parallelogram, etc)
-Rhombus (is a parallelogram, trapezoid, kite, etc)
-Rectangle (is a parallelogram, trapezoid, etc)
-Parallelogram (is a trapezoid)
-Kite (no parents except the quadrilateral)
Whether we call it a hierarchy or not, I think the last two diagrams capture the information I just listed. Would you agree?
I thought the same thing and I asked a textbook author about it. The reason that they do this is apparently so that the statement “in an isosceles trapezoid the base angles are congruent” is true.
Hmm…interesting! I’ve never thought of that. What a lousy reason to prefer the exclusive definition, though. Wow! I just did a bit of google research.
Here are some sites that agree with that textbook (the exclusive definition):
Here are some sites that use a definition consistent with the inclusive definition:
Interestingly, the second set of definitions either make a requirement about symmetry or about base angles.
Hi Mr. Chase. I like your post.
I am a Maths teacher in the Netherlands and we use the inclusive method. So I like the ‘GOOD’ set of pictures.
I do not like the Euclid’s definition for the Trapezia too. However this is just a case of ‘Lost in translation’. The greek word used by Euclid means actualy something like table. Use google translate to translate ‘table’ into greek: τραπέζι ‘trapezi’. You probably know your greek letters.
If you now fill in trapezium, it will translate it to τραπέζιο.
There are people who think that he wanted to use a word as table (or ‘little table’) for irregular (scalene) quadrilaterals. That would mean that the current trapezium with one pair of parallel lines never crossed Euclid’s mind or he did not think it was special enough to give it a name.
You seem hung up on the fact that Euclid’s definitions are exclusive. Why is this a bad thing. Of course, a square is a special kind of rectangle. But when I give a test to my pupils and draw a square and ask the pupils what the name of that figure is, I do not want them to write down rectangle…
Because of the properties of the square it is a special kind of rectangle, rhombus, etc. but in the first place, it is a square. Not a rectangle.
how do you draw a Irregular quadrilateral trapezoid with fixed dimensions for the two parallel bases and the two legs with no angles given using geometry tools?
top base= 328
bottom base= 223
left leg =220
right leg= 215
Best solution I can think of is this, using a geo triangle and a pair of compassess.
Start with the longest side (328). Set the distance of you pair of compasses to 220 and draw a circle with a centre that is the left tip of side 328 and a circle with a radius of 215 that has as a centre the right tip of side 328. Now you use the parallel lines on your geo triangle to draw a line parallel to side 328 with a length of 223 between the two circles.
A discussion about this problem can be found here: https://mrchasemath.wordpress.com/2011/11/05/constructing-a-trapezoid-using-the-side-lengths/
and here: https://mrchasemath.wordpress.com/2011/11/05/trapezoid-problem-take-2/
But you cannot do this with a parallelogram. Knowing the lengths of the sides of a parallelogram is not enough to tell you its shape, i.e., what are the angles at the various vertices. It is enough for trapezoids in the exclusive sense. There is a theorem that allows us to calculate the area enclosed by a trapezoid knowing the four lengths (and which two lengths are the bases) when trapezoid is used in the exclusive sense, but it does not work at all for parallelograms. Thus, here are two things that can be done for strict trapezoids but not for parallelograms. Others have already mentioned theorems involving the traditional definition of isosceles trapezoids. With an inclusive definition, everything that applies to trapezoids should apply to parallelograms as well.
I very much sympathize with inclusive definitions (other inter-brandings of quadrilaterals, circles as ellipses, and equilateral triangles as isosceles are no problem to me), but this issue of parallelograms as trapezoids is not so easy for me to swallow, because I seem to be giving up too much.
You’re the first to raise this point. Thank you!
Can you give the name of the theorem, or perhaps a link to a source? Do you have a sense of how important or useful this result is?
It seems like you could just adjust the hypothesis of the theorem to say, instead, “Given the four side lengths of a trapezoid that is not a parallelogram…” or something like that.
This makes the statement of the theorem less elegant. But if this is a rarely used result, maybe we don’t care that it’s less elegant. That’s why I asked how important or widely-used this theorem is.
Thanks for the great contribution!
This topic can be a brain warper. There is quite a bit of information on mensuration of trapezoids at http://mathworld.wolfram.com/Trapezoid.html and at https://en.wikipedia.org/wiki/Trapezoid. There is some overlapping information between the two sites but some is complementary.
I have never seen a name for the area formula, but there are several derivations. One is from Bretschneider’s formula for the area of a general quadrilateral, but you must be given not only the four side lengths, but also about sum of one pair of opposite angles. The trapezoid area formula needs only the four lengths (although, in general, you are required to be given which two of the lengths correspond to the bases, just as in Bretschneider’s formula you are required to be given the lengths in order around the quadrilateral). For a trapezoid all four angles are derived from the lengths of the bases and the lengths of the legs; for a parallelogram, you may have any value for the measure for any one of the angles, from which the other 3 angles are derived. The formulas for the area and for the angles involve division by 0 when the two bases are congruent–in other words, whenever you have a parallelogram. You asked me how frequently is this formula used? I don’t know–how often do people need to calculate the area of a trapezoid given the two base lengths and the two leg lengths? When one of the legs is set to 0 for the formula, it reduces to Heron’s formula for the area of an arbitrary triangle given the length of the 3 sides. I know most geometry classes spend more time on triangles, and even then Heron’s formula is not at the top of the list for finding the area of a triangle. I am heavily involved in MATHCOUNTS (a mathematics enrichment program for middle school students) and occasionally we have problems involving trapezoids closely related to this concept.
The main point to me, though, is not how much the formula is used but rather a key property that the formula implies: Given the lengths of the two legs and of the two sides (assuming they satisfy certain relationships similar to the triangle inequality theorem for triangles for such a trapezoid to exist), the trapezoid that results is unique up to reflection (place the longer base horizontally below the shorter base, then either of the two legs can be on the left and the other on the right, so two possible orientations of the trapezoid that are mirror images of each other). Uniqueness is a very important property of mathematics. For parallelograms, the angles are completely independent of the side lengths, with the result of having absolutely no idea of the area of the parallelogram without knowing the measure of any one of the angles. There is an uncountable infinity of parallelograms that can be generated knowing only the side lengths–ranging from a rectangle for maximum area to being squished arbitrarily flat approaching an area of 0. That does not happen with trapezoids.
The only other case I am aware of regarding inclusive versus exclusive definitions with any funny business is circles being ellipses. Ellipses are often defined in terms of a directrix. The concept of directrix is at best really stretched for circles. However, there are alternative, equivalents ways of defining ellipses that work for circles. Because of the alternatives and because a directrix is an augmented construct and not part of the figure in question, I have absolutely no concerns about regarding circles as ellipses. Every property and formula not involving directices that applies to ellipses applies also to circles and one can apply limits of arbitrarily distant directrices with the consequent ellipses approaching arbitrarily close to eccentricity 0 and a circle. Therefore, I expect that an inclusive definition of ellipse would be used. Similarly, if a parallelogram is to be regarded as a trapezoid, my expectation is that every significant property, formula, and theorem that applies to trapezoids would apply equally to parallelograms, since all parallelograms would be trapezoids and all trapezoids have these characteristics.
There are two choices:
(1). Implement your idea under consideration to adopt the inclusive definition of trapezoids. Then the set of trapezoids would effectively be partitioned into two subsets: the set of trapezoids that are not parallelograms and the set of trapezoids that are parallelograms (which is the same as the set of parallelograms). Many theorems and formulas would apply to both, so you would start of with “If quadrilateral ABCD is a trapezoid, then ….” Some theorems and formulas would apply to all trapezoids except parallelograms, so you would start of with “If quadrilateral ABCD is a trapezoid that is not a parallelogram, then ….” You could leave that way with the explicit exception or you could give a special name to trapezoids that are not parallelograms, say strict trapezoids (in line with the French, who regard 0 as a positive number and use the term strictly positive when they want to deal with only 1, 2, 3, ,,,).
(2). Use an exclusive definition of trapezoids. Many theorems and formulas would apply to both trapezoids and parallelograms, so you would start of with “If quadrilateral ABCD is a trapezoid or parallelogram, then ….” Some theorems and formulas would apply to only trapezoids in the exclusive sense, so you would start of with “If quadrilateral ABCD is a trapezoid, then ….”
Either option affects the description of some property, formula, or theorem. In option (1) an explicit exception must be stated in some cases. In option (2) the opposite cases are affected and those cases can be handled by: stating the characteristic separately for parallelograms and for trapezoids; stating the combining as “parallelograms or trapezoids”, or ignoring one of the two categories when it does not seem that important.
You favor option (1), and I sympathize with you in that regard. I share your striving in that direction, namely that inclusive definitions are to be favored over exclusive definitions unless there is a significant characteristic that does not flow down from the general category cleanly to a specific subcategory.
I currently favor option (2) because I see what I regard as a significant characteristic that applies to all trapezoids in the exclusive sense but does not apply at all to parallelograms, the concept proposed to be merged into trapezoids as inclusion. All of the discussion that has taken place so far has not convinced me that the characteristic I am concerned about is unimportant enough to support this case of inclusion. I am certainly open for you to keep trying–I recognize quite well how woefully short my knowledge is, and something I do not yet know might convince me.
I’m loving your thoughtful comments. You’re making this page a more comprehensive resource with each of your replies, and the mathematical community thanks you!
It might be time for another blog post about trapezoids. It seems like I might need to summarize the support each side of the debate receives.
I still think that the “uniqueness” property that you like so much is not as elegant as the elegance achieved by the assertion that “every quadrilateral definition is inclusive.” It seems strange to me that we would make trapezoids the only exclusive definition.
I would definitely be in favor of your suggested solution: Use the inclusive definition, but when we want to refer to a trapezoid that is not a parallelogram, we use the term strict trapezoid. I like that. Shall we start a revolution?
Perhaps I was brainwashed in my courses in ordinary and partial differential equations, abstract algebra, etc. when professors, for seemingly important reasons, would spend a lot of time first proving for general cases that solutions existed and then that those solutions were unique. All kinds of controversies arise, especially in blogging sites, pertaining to exponents and powers simply because of the multivaluedness (non-uniqueness) of functions on complex domains and codomains (and I respond to a lot of these). So, yes, uniqueness is important to me.
The importance of the uniqueness for [strict] trapezoids can be seen from a practical (real-world) perspective in the old engineering of bridges. I remember in my young years seeing bridges with trapezoidal frames with cross-beams either on the diagonals or as triangles to enhance the strength. However, it is the uniqueness of solution for the trapezoid that provides the basic rigidity and consequent strength and support for the bridge that would be much more problematic for rectangles or more general parallelograms.
Mathematics acquires great inherent value and usefulness from beauty, simplicity, and consistency of structure (you similarly used the word elegance). I agree that an inclusive classification scheme contributes very positively in that direction. On the other hand, I also think that knowing that solutions exist is important for understanding that we are discussing something meaningful (there is no point in discussing what doesn’t exist, except perhaps to understand why it doesn’t exist); it is also valuable to know at there is only one possible solution so that when we find a solution, we know we are done–this to me also enhances the beauty, simplicity, and consistency of mathematics. However, in this situation a conflict arises between two goals intended to achieve a common objective, so one must be abandoned in this one context. You clearly favor the inclusion side; I clearly favor the side of more easily aligning the qualitative characteristic solutions with the terminology, though I suspect you favor your position more strongly than I favor mine because I am very sympathetic with he inclusion viewpoint. I wish we could have both, but it seems we cannot. Like I wrote before, I am certainly open to being convinced the other way. Perhaps my mathematics professors brainwashed me regarding the importance of uniqueness of solutions–I know that my physics professors brainwashed me into thinking that weight is properly only a type of force and never a synonym for mass, but I have long since been cured of that one. I have had a rather diverse technical career (as perhaps you have noticed, given my descriptions and examples). I have been involved with mathematics, physics, radar engineering (with some dabbling in other engineering fields), computer science, and software development, and I enjoy finding interrelationships drawing on facts and techniques from one field and applying them to another. I like finding beauty, simplicity, and consistency among all these fields and merging them together, and it has been very helpful, enjoyable, and inspiring to me. Keep laying it on me and see if you can convince me to cross over to the other side of the force.
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I completely agree that we need to resolve the problem with the definitions of trapezoid. If you consult the premier mathematics sites (Dr. Math, Wolfram, etc) they agree with “at least one pair of parallel sides”. Unfortunately most textbooks and standardized exams posit the “2 pair of parallel sides”
I posted this on your first post about this as well, but wasn’t sure which you would respond to first, so I will post it here as well.
Just a question then – what do you do with the following two theorems regarding isosceles trapezoids when the trapezoid becomes a parallelogram?
Base Angles Theorem for Isosceles Trapezoids – the base angles of an isosceles trapezoid are congruent – not true for basic parallelograms (only rectangles).
Theorem (I don’t think it has a specific name) – the diagonals of an isosceles trapezoid are congruent – not true for basic parallelograms (only rectangles).
Do you then say that these theorems only apply for non-parallelogram isosceles trapezoids or that isosceles trapezoids can’t be parallelograms?
Seems to be an issue. If you have a solution for it, I would love to hear it because this is the one thing that seems to be a problem for the difference in definitions.
If you look at the hierarchy listed first after the good label, you will see that the isosceles trapezoid (it) does not have to be a parallelogram. The rectangle does. The property that the (it) transfers to the rectangle is the congruent base angles. The property that the p-gram transfers to the rectangle is both pairs of opposite sides being parallel. This is the same idea as the rectangle giving the square its congruent angles and the rhombus giving it its congruent sides.
The same is true for the congruent diagonals. It is a common property for (it)s, rectangles, and squares. None of the other shapes have this property, so they have to line up from the most general form to the most specific.
I hope this explanation helps. I have always been bothered whenever I have taught the (it). I never figured out why until I tried to answer your questions. The issue was the congruent diagonals. Thank you for making me reach this conclusion.
Wow, been a while since I was on this post, but I just got an e-mail indicating a follow-up reply had been posted.
Anyways, a few quick thoughts without trying to start a full-on debate.
You emphatically say that “a parallelogram is a trapezoid.” If that is true, and if an isosceles trapezoid is by definition a trapezoid that has a set of congruent opposite sides (which is the typical use of the term “isosceles” as used in triangles – in triangles, 2 or more (could be 3) congruent sides, and in isosceles trapezoids, there could be 2 or 3 congruent sides as well), then a parallelogram must also be considered an isosceles trapezoid unless a parallelogram is defined as a trapezoid with 2 sets of opposite congruent sides while an isosceles trapezoid is prohibited from the 4th side being side being congruent to its opposite side. I think you still run into a few definitional issues somewhere along the way.
I teach geometry and most (maybe all) books that I have used have definitely separated trapezoids from parallelograms. I teach my students that some people’s definitions mix the two while others keep them completely separate and that this is one area of math on which mathematicians have not been able to come to complete agreement. I can follow both arguments and, to this day, haven’t been fully persuaded either direction since both seem to have limiting flaws or extra additions to definitions.
Yes, this seems to be the snag (see our discussion in the comments above here).
But I’m willing to sacrifice the isosceles trapezoid on the altar of mathematical elegance. You gain so much by using the inclusive definition. And, furthermore, I think we can fix the definition of the IT to make it work with the inclusive definition. You’re right, you can’t use the “two-legs congruent” definition to obtain our prototypical IT (unless you concede that a parallelogram is an IT…but I’m not sure we’re ready to do that).
The two ways of redefining the IT are as follows (either one is sufficient):
1. A trapezoid with congruent base angles.
2. A quadrilateral with a line of symmetry bisecting two sides.
The sources I linked to in my earlier comment mostly do (1) but wikipedia does (2). Either one is fine with me, since they entail each other.
You guys got me fired about about trapezoids again, and I made another post today, here. 🙂
We are starting a unit on polygons soon … and since PARCC defines a trapezoid as “a quadrilateral with at least one pair of parallel sides”, we are rewriting our questions and activities and pitching out our textbook. I get why to use the inclusive definition, but my question has been how to define isosceles trapezoid. Is there a problem with defining it as “a trapezoid with congruent diagonals”? Or some preference to leave the congruent diagonals as a result to be proved? And do we just not talk about legs of a trapezoid anymore?
Oh – and what’s your opinion on still defining a parallelogram as a quadrilateral with both pairs of opposite sides parallel? Or would you change it to a trapezoid with both pairs of opposite sides parallel just because we can?
Thanks so much for all three of yours posts. They are most helpful.
I don’t teach geometry, so I haven’t looked over the new PARCC test materials for geometry. I love that they’ve chosen to use the inclusive definition. Thanks for letting me know!
I think you’re fine using the equivalent definition of an isosceles trapezoid: “a trapezoid with congruent diagonals.” You’re right, you forfeit the ability to prove that later, but that’s no big deal. You could, instead, have students prove that the base angles are congruent (the converse of what’s usually proved). I see no loss here.
And I think defining a parallelogram as a trapezoid with both pairs of opposite sides parallel is awesome, and you should totally say it that way. It emphasizes inheritance, which I think is a powerful idea. Do you define a square to be a rectangle with congruent sides? Or perhaps as a rhombus with congruent angles? Then you could choose to do the same thing here if you like, for the exact same reason.
We usually define a square as a parallelogram that is a rhombus and a rectangle. Thank you for your thoughtful posts and for your reply.
Ooo…yes, even better way of saying it. The square “inherits” traits from both of its “parents,” the rhombus and the rectangle.
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A square is never an oblong. A square is never a rhombus. Each of the above quadrilaterals, according to Euclid, is mutually exclusive. These exclusive definitions persisted into the 19th century.
And we’ve successfully made them all inclusive definitions, with the exception of the trapezoid–the last one to fall. But fall it will :-).
Euclid wasn’t really concerned with these definitions. In fact, Def 22 in Book 1 is the only place these names are mentioned (please correct me if I’m wrong). That being said, because Euclid didn’t really use these definitions much, he didn’t take the time to define them very well. I think if he had to use them more, as we do in modern geometry courses, he would also agree that the inclusive definitions are much more light-weight, elegant, and flexible.
Your preferred hierarchy seems to follow that principle more closely than the conventional one.
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Why is the parallelogram not an isosceles trapezoid? Are the equal-length sides of an isosceles trapezoid required to be non-parallel? That seems to go against the principle you’re using for the rest of this piece.
One could pursue this definition of an isosceles trapezoid (equal legs), but I prefer the symmetry or base-angles definition of an isosceles trapezoid.
If you pursue the equal legs definition, you have to throw out a number of other results about isosceles trapezoids that we all know and love, like midline symmetry, congruent base angles, or congruent diagonals.
In what way does the symmetry definition of an isosceles trapezoid go against the rest of the piece? The legs of an isosceles trapezoid *may* be parallel, if the trapezoid is a rectangle, but this is the only case when this would be true.