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
So if you've already defined what parallel
lines are, you can use that to define
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
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
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
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.