# Understanding Proofs

This page is all about understanding proofs. First of all, a proof is a series of sequential steps of justified conclusions. The conclusions are justified by either a definition, a postulate or a previously proved theorem.

In a proof the first thing you see (other than the name of the theorem) is the given. The given is the information that you have to start with. This usually includes a picture of a figure. Often it is something like "a b with transversal t." From there, you must figure out what information applies in terms of applicable definitions, postulates, and previously proved theorems.

Sometimes teachers will ask you to include what you are trying to prove under the given. I've left that step out on all of my proofs, as it is unnecessary. If you really want to know what is going to be proved, read the theorem listed at the top of that proof's page.

In the actual proof itself, the information is divided into two categories: Statements/Conclusion and Justifications. Since a proof is written like a table, the columns are usually given these labels. While the most common way to write a proof is to have the categories divided between two columns, it can be written the other way (horizontally). (Confused yet?) In HTML, I usually label them as Table Headers (<TH>).

The first step you write down is the given. After that you just go through the routine of writing down the different statements and their justifications. At the end, you write your conclusion - what you're trying to prove. Sometimes you'll come across a step that seems redundant. While you don't always have to include them in your proof, you should until your teacher tells you otherwise. Many teachers I know would mark off points for excluding something small like that.

One thing that seems to catch many people is the similarity between the Substitution Property and the Transitive Property. However, there is a distinct difference. Check them out for yourself. Sometimes either can be used in a proof, but most often its either one or the other. Substitution is for replacing parts that are known to be equal and Transitive is for showing equality between two previously unrelated things.

My tables usually look like this:
Statements/Conclusion Justification
1) Insert given here. 1) The justification is that it is given.
2) First point. 2) First justification.
3) Second point. 3) Second justification.
And so on until...
9) The conclusion. 9) The final justification.
I usually have ALIGN=CENTER, BORDER=1, FRAME=BOX, and RULES=COLS. Some of that is probably redundant, but that's OK if it gets the job done. CELLPADDING doesn't seem to work yet, but put that at one anyways. It's your choice; use what you want.

There are eight completed theorem proofs available at this website. They are:

- Jaime III
"One can never consent to crawl when one feels the impulse to soar." - Helen Keller