The Quadrilateral Zoo: Why Trapezoids Don’t Belong

I’m a resident teacher this year, and I’m working alongside an experienced teacher in a 10th grade Geometry class. During the unit where we discussed polygons and their properties, I came across this definition in the textbook:

A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent

Pearson Geometry Common Core

This definition seemed…off. Why have that second condition?

I think the idea is to exclude the case where 3 sides are all congruent to each other and the other side is not.

Ceci n’est pas un cerf-volant

But why not just say “two disjoint sets of consecutive sides congruent” instead of “no opposite sides congruent”? The problem is that according to the Pearson definition, a rhombus is not a kite

Kite or no kite?

There is a famous debate in the definition of trapezoid, which is whether to use the exclusive or inclusive definition.

A trapezoid is a quadrilateral with at least one pair of opposite parallel sides

Inclusive definition

A trapezoid is a quadrilateral with exactly one pair of opposite parallel sides

Exclusive definition

The inclusive definition implies that a parallelogram is a trapezoid. In other words, the set of parallelograms is included within the set of trapezoids. And the inverse holds for the exclusive definition.

I decided to scientifically study this question, so turned to that time-honored rigorous methodology of…The Twitter Poll

No consensus apparently.

Many geometry books and educational resources have some kind of comprehensive picture of the classification of quadrilaterals. For example, the one from Wikipedia looks like this:

I do like this diagram, if only for the combination of a Venn diagram and actual examples of the quadrilaterals themselves. I notice two things: that the author is using inclusive definitions for kite and trapezoid, and the absence of the isosceles trapezoid.

Ready for it? Here’s my diagram:

What do you notice or wonder?

I arranged this diagram specifically to correspond to the subgroup lattice for the dihedral group of order 8.

Huh?

Basically, I’m classifying the quadrilaterals based on what portion of the full set of symmetries of a square that they exhibit.

Here’s the subgroup lattice diagram:

The notation used in this diagram is that “e” is the identity, “a” is a rotation by 90 degrees, and “x” is a reflection through two vertices, which fixes a pair of vertices and exchanges the other pair. There are two differences between this lattice and the one for quadrilaterals. First, there are two different subgroups of order two each corresponding to the kite and isosceles trapezoid. This is because the reflection symmetry that each exhibits can also be characterized as the other reflection that either goes through vertices or midpoints for the kite or isosceles trapezoid respectively. These two reflections are related by the rotation by 180 degrees, a.k.a. a^2. The other difference is that there is a cyclic subgroup of order 4 generated by a which doesn’t have a corresponding invariant set of quadrilaterals. This is because as soon as a quadrilateral has a 90 degree rotation symmetry, it is automatically a square. The reflections come for free!

So those teachers out there are probably asking themselves: where does this debate fit in the curriculum? Well, consider:

CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Common Core State Standards for Mathematics

Ok, the standard doesn’t mention kites or rhombuses, which is a bit strange, but it still is clearly trying to get at a symmetry viewpoint of these shapes. In fact, the CCSSM has a thoroughly transformational perspective on geometry. The equivalence relations of congruence and similarity are consistently based in the transformation groups of rigid motions and dilations.

Interestingly, the standard says “trapezoid” even though a generic trapezoid has no symmetries at all.

In fact, what is up with trapezoids? As we saw, there’s some debate about the exact definition, but it always involves this parallel opposite sides idea. But we don’t need parallelism to define any of the other quadrilaterals. In fact, this points to a deeper fact: trapezoids don’t even have to exist! In spherical geometry, there are no parallel lines. So therefore, there are no trapezoids in spherical geometry.

Except, there are isosceles trapezoids. Because if we define an isosceles trapezoid as a quadrilateral with a reflection symmetry through the midpoints of opposite sides, then it exists perfectly well in spherical geometry. So here’s my hot take:

Isosceles Trapezoids are a more natural subset of quadrilaterals than Trapezoids.

When I say natural, I mean that it applies in a more general context, and fits more neatly in the symmetry classification. I consider the symmetry classification to be more consistent with the modern, transformational geometrical understanding than a classification based on sides and angles.

My friend Doug O’Roark pointed out that Zalman Usiskin has written an entire book on this subject, The Classification of Quadrilaterals : A Study of Definition. So this is not the end of this story. But for now, my current opinions are that trapezoids are weird, inclusive definitions are just better, and symmetry is a powerful and modern way to look at quadrilateral classification.

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