# A better hierarchy?

Every year in April (well it actually used to be October, but that’s a story for another day), I teach quadrilaterals to my geometry students. And every year we make a “family tree” or “hierarchy” of quadrilaterals. The one we make in class is pretty basic – this year, it looks like this:

Now, every year as I build this with my students I am hurt by the compromises here. A little background:

• Our textbook uses the exclusive definition of a trapezoid – an object with exactly one pair of parallel sides – but I decided several years ago I prefer the inclusive definition of at least one pair and use that in my class, despite the fact that it is a major difference between my classroom and others in the school. I disclaim it a lot, but stick by it.
• Our book doesn’t even mention kites, but I love them, so I always include them. Since I’m adding them myself, of course I can be inclusive with them (again, my preference!) and am. Two pairs of consecutive congruent sides clearly includes rhombi as well as non-rhombal kites.
• Our book DOES define an isosceles trapezoid, but of course since they use an “exactly one” definition for trapezoids, their isosceles trapezoid is also “exactly one pair of parallel sides”. I have always decided to just let that go – a major compromise. In an ideal world, I’d use a definition for isosceles trapezoid that includes rectangles (as the transformation-based definitions I’ve seen do) which would be nice since so many of the rectangle properties are matched by the isosceles trapezoid properties (congruent consecutive angles, congruent diagonals, etc).

Last year, I was ruminating about this on twitter, and Ben Hambrecht ran away with my idea to write one of my favorite blog posts about quadrilaterals. Go read it – it’s pretty awesome. He encapsulated so much of the beautiful symmetry of the hierarchy, and found several things I had never considered. And best of all, he invented a way to make the whole thing a bit more symmetric – a new quadrilateral he called the kitoid (all credit to him for the beautiful image below).

I’ve thought about this a lot since then, especially in how I might eventually introduce this absolutely amazingly symmetric and perfect hierarchy to students. Where I get hung up is on definitions, especially outside of the parallelogram land. Here are the definitions Ben came up with for the four non-parallelogram guys:

These have very nice symmetry with each other, but I cannot imagine introducing those as the definitions for ANY of these shapes. None of them are intuitive just from looking at the shape, at least not from a student perspective, and it’s crucial in my mind that a definition be almost obvious just from a picture. Luckily, I was able to prove that to have a kitoid, it is sufficient to prove that one pair of opposite angles are congruent. So I’ve been playing around with definitions, with some help from various folks on twitter, including Jonathan Osters and Corey Andreasen looking for something that works. Here’s one option:

Pros: Very nice symmetry between Kitoid <=> Trapezoid and Kite <=> Isosceles trapezoid

Cons: The definition for isosceles trapezoid is not particularly intuitive, and you’d need to prove that a kite is a kitoid AND that an isosceles trapezoid is a trapezoid here – that fact is not included in the definition.

I don’t think the “prove it” thing is a dealbreaker – we already prove that a rectangle and a rhombus are parallelograms, as their definitions don’t include that in them – but it might be nice to avoid. The non-obvious isosceles trapezoid definition is a bit more annoying. Let’s try…

Pros: Still has nice symmetry. The diagonal definitions connect nicely to rectangle / rhombus properties with an easier “seeing it” path than the consecutive angles.

Cons: Diagonals in the definition are studiously avoided elsewhere in the “typical” parallelogram group. That could be changed, and an entirely diagonal-based definition system written, but it will be necessity leave out the kiteoid and trapezoid (which have no special diagonal relationships).

Pros: Again, symmetry. And the kite is clearly a sub of the kitoid, while the isosceles trap is clearly a sub of the trapezoid. And the “isosceles” part connects nicely to the isosceles triangle theorem in this definition despite not specifically mentioning the congruent sids.

Cons: Isn’t actually sufficient as written! There are counterexamples to both the kite and isosceles trapezoid definitions I wrote here (imagine a trapezoid with one side like a rectangle – two right angles – for the isosceles trapezoid). So this needs to be modified a bit…

Pros: Solves the problem of the previous one. Introduces the common term “base angles”.

Cons: Very unwieldy, and almost necessitates a term for kites that I’m pretty sure isn’t standard or common. “base sides” is pretty bad, so I’d probably want to come up with something else.

Overall, I’m not perfectly satisfied with any of these definition sets. What do you think?