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  Math B

What is a Proof?

A proof is a written account of the complete thought process that is used
to reach a conclusion.  Each step of the process is supported by a theorem, postulate or definition verifying why the step is possible.
In formal Euclidean proofs,  no steps can be left out. 

If you think about the numerical problems you are used to solving in geometry,
you will realize that your mind often does a "fast-forward" through some of the logical steps needed to reach a valid answer.  In other words, you quickly "go right to the answer."
Check out the numerical problem below:

Example of a Number
Problem
(your thinking is in "fast-forward")

Answer:  Since bisects , D is the midpoint of ,
forming two congruent segments whose measures are equal . 
If AD = 6, then DC = 6 as well.

You probably arrived at the answer of 6 long before you finished
reading the explanation of the answer.  Right? 


When developing a proof of this same problem, we must be careful to include ALL of the steps that led to our answer.  We cannot "fast-forward" over steps when writing a proof.  Check out the "proof" of this same problem:

Proof of the
Same
Problem
(slow down your thinking)
 

Statements Reasons
1.  1.  Given
2.  2. Segment bisector intersects a segment
     at its midpoint.
3.  3. Midpoint divides a segment into 2
     congruent parts.
4. 

4. Congruent segments have equal
     measures.
5. 

5.  Substitution
                                                     QED

A proof requires that you document all of the little steps that you mentally
 "fast-forwarded" through in the numerical problem.

 

What's in a proof?

A formal 2-column proof contains the following:

 Statement of the original problem.

Example
 Diagram, marked with "Given" information Example
 Re-statement of the "Given" information in the proof Example
 Complete supporting reasons for each step in the proof Example
 The "Prove" statement as the last statement Example

 

A Successful Strategy...
Backwards Looking

For most proof problems, it is very helpful to examine
the problem backwards -- from the "Prove" statement back
to the "Given" information.  Let's look at an example:
 


A Backwards Look -- Where does the conclusion come from?
           
           
           
           

By looking at the above steps, we can see how to proceed with the proof.

Statements Reasons
1.  Given
2.  Reflexive Property
3.  SAS:   If 2 sides and the include angle
     one triangle are congruent respectively
     to 2 sides and the included angles of a
     second triangle, then the 2 triangles are
     congruent.
4.  Corresponding parts of congruent
     triangles are congruent.
                                                      QED

You might want to consider jotting down your "backward"
thoughts before you begin writing your proof.

 


           F. Roberts