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# Writing a Good Definition - Concept

###### Brian McCall

###### 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

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.

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###### Brian McCall

B.S. in Chemical Engineering, University of Wisconsin

J.D. University of Wisconsin Law School (magna cum laude)

He doesn't beat around the bush. His straightforward teaching style is effective and his subtle midwestern accent is engaging. There's never a dull moment with him.

##### Concept (1)

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