The usefulness of classification

Why is understanding the relationships between sets useful? Consider the following shape:

What do we know about this particular shape?

Notice that some of these properties are redundant. For example, if I know something is a trapezoid, I also know it's a quadrilateral. Here are the relevant set relations, for this particular list of properties.

Notice that some set relations are established by definition, while others are established by proof. Now we have a list of set relations, which we can apply to our initial list of properties. We find our initial list boils down, to just

The beauty of knowing the essential properties of this shape, is that now we can ask: "Are there any other isosceles trapezoid rep-tiles, or is this the only one? I will leave that for you to discover, in case you don't already know the answer.