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Re: Proofs are so annoying, any shortcuts?
Posted:
Jan 19, 1998 1:02 PM


A proof is an argument that convinces the reader that what the author is trying to prove is true. There are no shortcuts because you do not know what the reader already knows. As you prove theorems  they become shortcuts. For example, if you prove right triangles are congruent if they have a hypotenuse and a leg congruent by using the Pythagorean theorem to show that the third leg is also congruent and then using SSS, you can now name that as the HL (hypotenuseleg) theorem and use it as a shortcut. Thus, most of the theorems you prove in a beginning geometry course are statements that you would like to use as reasons in subsequent proofs. If what you are trying to prove is not labeled as a theorem, but rather as an exercise, then it is probably not as important for later use or is merely an example of where the theorem you just learned is useful. Corollaries are minitheorems that are direct results of the theorem you just proved.
Now the main question is what consists of a valid argument. Most high school students learn the twocolumn method of proof where the steps are numbered in the left column and the reasons correspondingly numbered and aligned in the right column. The purpose of this is to organize your thinking. Each step must have a reason and must logically follow from what is given or a previous step in the proof. This is sort of like explaining something to a young child that asks why after every statement. Some people find it easier to think backward  that is start with what you want to prove and determine why that might be true (for example congruent triangles using SSS), and then determine why the reason for that would be true (for example the three sides congruent), and so on. And then they write the proof forwards.
No one is born with a talent for writing proofs. It is an acquired skill that one gets by reading and writing lots of proofs. Unfortunately, many students don't like to write proofs and so many teachers don't like to teach them. I am afraid that over the years, we teach less and less proof writing in the high schools and colleges.
My main suggestion  read your textbook carefully and then see if you can replicate the proofs given. Try to prove as many exercises as you can  even if they are not assigned. The better you are at proofs, the easier future math courses, logic courses, etc. will be for you. Unfortunately, there are no shortcuts to being good at proofs.
Eileen Schoaff exhigh school math teacher Buffalo State College Buffalo, NY



