The Roots of the Equation

Stressful as it is, I am loving teaching new courses. When I first start teaching, I felt like I was learning new stuff all the time, stuff about algebra (and how it connects to other courses) that I didn’t know I didn’t know, and now it keeps happening with geometry, especially with the more transformational tinge CC geometry has.

One of the things that struck me was, last week, when I used this Illustrative Mathematics task as a follow-up to my lesson about the diagonals of quadrilaterals. I feel like the understanding I had internalized that you can prove triangles congruent with less information because they are rigid structures, but quadrilaterals are not, so there are no quadrilateral congruence theorems. But I realized that’s not true.

Last time, we constructed all of the special quadrilaterals by taking a triangle and applying a rigid motion transformation. That meant that every special quadrilateral can be split into two congruent triangles. Therefore, if you had enough information to prove one pair of triangles is congruent, you could prove the whole quadrilaterals are congruent.

So if we’re looking at SSSS in terms of the triangles, we really only know two sides of the triangles. Since that’s not information to prove the triangles congruent, then it’s not enough for the parallelograms. But SAS is enough for the triangles, so it’s enough for the parallelograms.

Here’s a non-parallelogram example. Here are two isosceles trapezoids with the same diagonals, same legs, and the same angle between the diagonals and one of the bases, but the trapezoids are not congruent. But that’s because, when you look at the triangles, we have Angle-Side-Side, which we all know is not a congruence theorem. If, instead, we had had SSS (a leg, a base, and a diagonal), then they would be congruent.