The Conditional Map: A Summary of Quadrilateral Theorems

While churning out material for summer school lessons- some repackaged, some brand new- I came up with a graphic I’d like to share:

This is what I call a “Conditional Map.” (There’s probably already some other name for what this is.) It basically sums up all the quadrilateral theorems in a typical geometry curriculum. As you can see in the legend, each of the boxed-in phrases represents a condition; an arrow, as it ordinarily would, represents a causal relationship. So an arrow from “Rectangle” points to “diagonals congruent,” but no arrow points the other way, since congruent diagonals don’t imply a rectangle. Also, paths that meet at a floating node and continue together to a third condition indicate “If A and B, then C.”

As a summative review exercise, I gave an arrow-less version of this graphic to my students, showed them a couple of simple examples of where arrows could be placed, and watched to see how many arrows they could get on their own.

I prefer this to the usual Quadrilateral family tree diagram. The classic diagram has the broadest definition at the top, and the paths downward indicate inclusion (i.e. parallelogram leads to rhombus and rectangle because it includes both of these). Here, the square is at the top, and the condition of a shape being a square leads to all other conditions; at the same time, we see how combinations of conditions let us flow back up the tree to the square.