Converse, Inverse, Contrapositive

Date: 09/08/2001 at 23:57:03
From: Hana
Subject: Geometry - Converse,inverse and comtrapositive.

For the directions it says "Write the converse, inverse, and 
contrapositive of each conditional. Determine if the converse, 
inverse, and contapositive are true or false. If false, give a 
counterexample."

I can't seem to do these:

   All squares are quadrilaterals.

   If a ray bisects an angle, then the two angles formed are 
   congruent.

   Vertical angles are congruent.

Thank you!

Date: 09/09/2001 at 18:30:24
From: Doctor Achilles
Subject: Re: Geometry - Converse,inverse and comtrapositive.

Hi Hana,

Thanks for writing to Dr. Math.

Some basics about logic: When you're talking about converse, inverse, 
and contrapositive, then you're dealing with sentences of the form

  "if p, then q"

The converse of any such sentence is

  "if q, then p"

The inverse is

  "if not p, then not q"

And the contrapositive is

  "if not q, then not p"

The problem with your questions are that they don't neatly fit into 
the "if p, then q" format, so you need to first find EQUIVALENT 
sentences that are "if p, then q."

Your first example says "all squares are quadrilaterals."  That is the 
same as saying "if x is a square, then x is a quadrilateral."

Your second example says "if a ray bisects an angle, then the two 
angles formed are congruent." That is the same as saying "if a ray 
bisects an angle, then the ray that is drawn within an angle forms two 
congruent angles."

Your third example is just like your first example.

I hope this all gets you going. If you're still stuck, please write 
back and I'd love to help you more.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/