Building Two Column Proofs

Date: 09/12/98 at 22:00:01
From: Crystal 
Subject: Geometry - Proving

My geometry teacher has just taught us how to do proving, but I don't 
get it. I know how a certain theorem backs up each statement, but what 
I don't understand is, how do you know what order to write your 
statements?

Here's an example:

   <----|----|----|----|---->   
        A    B    C    D

   Given: AB = CD   A-B-C-D
   Prove: AC = BD

   Statements          Reason
   AB = CD             given
   BC = BC             reflexive property
   AB + BC = BC + CD   addition property of equality 
   A-B-C-D             given
   AC = AB + BC        definition of between
   AC = BC + BD        transitive property of equality
   BC+CD = BD          definition of between 
   AC = BD             transitive property of equality

See, I don't understand how you know what statements to put and in 
what order, if there is any. Please help.

Date: 09/14/98 at 12:55:26
From: Doctor Peterson
Subject: Re: Geometry - Proving

Hi, Crystal. A proof is just an orderly way to show that something is 
true, by building on other things you know are true. The only way that 
order matters is that each thing you say must be based on something 
you've already said. Often it will be based on the previous statement, 
but sometimes you will have to use earlier statements as well.

Think of it as building a tower to reach a high goal. Your "givens" 
are the foundation someone laid for you, and the theorems you have are 
the girders and rivets you have to put together to make the tower.

Let's try drawing your sample proof as a building, to show how its 
parts are connected. (I've added line numbers.)

    Statements             Reasons

    1. AB = CD             given
    2. BC = BC             reflexive property
    3. AB + BC = BC + CD   addition property of equality
    4. A-B-C-D             given
    5. AC = AB + BC        definition of between
    6. AC = BC + CD        transitive property of equality
    7. BC + CD = BD        definition of between
    8. AC = BD             transitive property of equality

Here is the foundation at the bottom and the goal at the top:

        ??AC=BD??


    AB=CD     A-B-C-D
    =================

Statement 2 gives me a new fact I can build on that doesn't depend on 
the givens, but has been applied to a specific value, BC:

            ??AC=BD??


      2       1
    BC=BC   AB=CD     A-B-C-D
    =========================

Statement 3 uses statements 1 and 2, by adding them together:

            ??AC=BD??


          3
     AB+BC=BC+CD
       /     \
      2       1
    BC=BC   AB=CD     A-B-C-D
    =========================

Statement 4 is a given. Statement 5 uses its definition:

            ??AC=BD??


          3          5
     AB+BC=BC+CD AC=AB+BC
       /     \          \
      2       1          4
    BC=BC   AB=CD     A-B-C-D
    =========================

Statement 6 combines statements 3 and 5 using transitivity:

            ??AC=BD??


                6
            AC=BC+CD
           /        \
          3          5
     AB+BC=BC+CD AC=AB+BC
       /     \          \
      2       1          4
    BC=BC   AB=CD     A-B-C-D
    =========================

Statement 7 is again based on the definition of statement 4:

            ??AC=BD??


                6
            AC=BC+CD
           /        \
          3          5     7
     AB+BC=BC+CD AC=AB+BC BC+CD=BD
       /     \          \ /
      2       1          4
    BC=BC   AB=CD     A-B-C-D
    =========================

Now finally we can join statements 6 and 7 to get to the goal
                    
                  AC=BD
                 /     \
                6       \
            AC=BC+CD     \
           /        \     \
          3          5     7
     AB+BC=BC+CD AC=AB+BC BC+CD=BD
       /     \          \ /
      2       1          4
    BC=BC   AB=CD     A-B-C-D
    =========================

Do you see how each statement we put in is supported by some reason 
(the girder) using one or more previous statements (the rivets)? I 
could have built some parts in a different order (such as 7 before 5), 
but nothing can come before what it depends on.

We never diagram this way, but it can be very helpful to write after 
each reason which steps it depends on, if any, like this:

    Statements       Reasons
    1. AB=CD         given
    2. BC=BC         reflexive property
    3. AB+BC=BC+CD   addition property of equality (1,2)
    4. A-B-C-D       given
    5. AC=AB+BC      definition of between (4)
    6. AC=BC+CD      transitive property of equality (3,5)
    7. BC+CD=BD      definition of between (4)
    8. AC=BD         transitive property of equality (6,7)

I hope that helps. There's a lot more to learn about proofs, like how 
you can figure out how to get from a given to a goal, but that will 
come with experience. Write back if you need more ideas.

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

Date: 09/15/98 at 23:28:23
From: Crystal Lee
Subject: Re: Geometry - Proving

Dr.Math,

Thanks for answering my question. There's one thing I want to make 
sure. There really isn't any correct order of statements?

Crystal

Date: 09/16/98 at 08:58:56
From: Doctor Peterson
Subject: Re: Geometry - Proving

Hi, again, Crystal. There isn't a single correct order, but often 
there is a best order, and there are many incorrect orders. In your 
example, I like the order it is in because most statements follow from 
the previous statement, possibly combined with an earlier one. But 
here is another way it could have been done that would be equally 
valid:

         Statements             Reasons

     [4] 1. A-B-C-D             given
     [5] 2. AC = AB + BC        definition of between (1)
     [7] 3. BC + CD = BD        definition of between (1)
     [1] 4. AB = CD             given
     [2] 5. BC = BC             reflexive property
     [3] 6. AB + BC = BC + CD   addition property of equality (2,5)
     [6] 7. AC = BC + CD        transitive property of equality (2,6)
     [8] 8. AC = BD             transitive property of equality (3,7)

Everything still depends only on what came before, so it is a valid 
proof. 

Another point to make is that some people prefer to put all the givens 
as the first steps, rather than saving some for later, as in step 4 
here.

You will also find that often there are several very different paths 
you can take to prove something, so that your proof may look very 
different from someone else's, yet be completely correct.

I hope you'll have fun working on proofs. As you should see from this, 
proofs are really a very creative thing, more like writing (or even 
art) than the kinds of math you have done before.

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