Geometry Definition

Writing a definition is a common exercise during the early stages of Geometry. An excellent geometry definition will classify, quantify, and not have a counterexample. Once a term is defined, it can be used in subsequent definitions; for example, once parallel lines are defined, they can be used in the definition of a parallelogram.

In geometry, it's imperative that you can write a good definition, because it will help you to understand the properties of whatever it is you're talking about.

The three key components of a good definition.

The first one, it uses previously defined terms. So if you've already defined what parallel lines are, you can use that to define a parallelogram.

Secondly, it classifies and quantifies. That is, by classifying it, is it a polygon? Is it a line? What is it? And quantifies how many. So if you're talking about a polygon, you're going to want to say how many sides.

And, last, it has no counter-example. But what is a counter-example? A counter-example is something, an example, that will make a definition or conjecture incorrect. So if you can find a counter-example to your definition, you haven't written a good one.

So a short example is let's say I had a square and I said that a square is a quadrilateral. Which means that it has four sides. And I just left my definition like that. Turned it into Mr. McCall.

Well, I'm going to say a quadrilateral, well that could be a trapezoid, where I could draw in one pair of parallel sides. It could be a kite where we have two pairs of congruent consecutive sides. I could draw in a rhombus. I could draw in a parallelogram. I could draw in lots of counter-examples that would make this definition not true or it wouldn't make it specific enough for just a square.

Let's look at two other ones. Let's say something that's not related to geometry, directly, a skateboard. Let's say I define a skateboard as something with wheels that you ride. Well, that's not very descriptive. This is not a good definition. First and foremost because I could say that this could be a bike, because a bike is something that has wheels that you ride.

What about a good definition? A good definition for a parallelogram is a quadrilateral with two pairs of parallel congruent sides. Notice that we're using words that we probably already defined. So quadrilateral, we would have defined before we started defining a parallelogram. Quadrilateral has four sides. Parallel lines we say never intersect. Two lines in the same plane that never intersect. And congruent means having the same measure or same length.

Notice I was able to write this definition of a parallelogram using three words that I've already previously defined and there's no other counter-example I could draw or come up with that would make this not apply to a parallelogram.

So keep that in mind when you're writing good definitions and it will help you even on your test and quizzes.