Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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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.