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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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.