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
Conjectures are statements that use an if, then structure and are commonly presented throughout Geometry (for example, if a triangle has two congruent base angles, then that triangle is isosceles). The math converse of a statement switches the if and then, resulting in a statement that may or may not be true; verifying the truth value of a converse is a common exercise in Geometry.
The converse is when you switch the if
and then of a conditional statement.
Well, conditional statement means if something
happens then something else must
be true.
But you could think of it as
a hypothesis and conclusion.
So a converse is not always true.
So let's look at two examples.
Here we're being asked find the converse
of the statement, then ask yourself
is it true.
So this first statement says if it
is Monday, then it is a weekday.
Well, that's true.
If today's Monday then it's a weekday.
So the converse is going to take the
if and the then and switch them.
Or another way of thinking about it is
we're going to take what comes after
then and write it after if.
So I'm going to say if it is a weekday --
so I'm going to take that second part
which was our conclusion, if it is
a weekday, now I need to switch it
again. Then I'm going to say the first part of
my statement here, which says it is Monday.
So the converse, again, takes a hypothesis
in the conclusion and switches them.
Well, if it's a weekday, then
Monday is not always true.
What if today was Tuesday.
Tuesday is a weekday.
So not every weekday is Monday.
So the statement here is not true.
The converse is not true.
Let's look at one more and
apply it to geometry.
If an angle measures 88 degrees,
then it is acute.
That's true by definition an acute angle
is any angle that measures less
than 90 degrees but more
than 0 degrees.
So let's find our converse.
So I'm going to take the if, and instead of
saying if an angle measures 88 degrees,
I'm going to take the second
part of this statement.
So I'm going to write that instead of
saying if it's acute, doesn't tell me
anything, if an angle is acute, okay.
So there I had to add in a couple of
words to make sure it made sense.
Then now I'm going to say the second part.
The angle measures 88 degrees.
Then the angle measures 88 degrees.
So if we look at this statement, let's say
I had an angle right here that measured
75 degrees.
Well, it's an acute angle, but it's
not equal to exactly 88 degrees.
So the converse of this statement is not
true as well but not every statement
in geometry whose converse
is going to be false.
So that's not always going to happen.
I just gave two examples here where if you
take the if and the then statement,
switch them and evaluate them, you can
find counter examples which makes
the converse not true.
