How to Simplify a Geometrical Proof

Video Transcript

"In order to make your proof the best it can be, you of course, want to try to make it as short as possible, and this depends a lot on the audience. It depends on whether you're writing for an expert, who may only need one stepping stone to the finish, or for fellow students who may need lots of stepping stones, or of course, your teacher or professor, who needs to know that you know, how to get there. There are several alternatives to proving the original statement. Here's an original statement. If X, Y, and Z, are the angles of a triangle, then X Y Z, equals 180 degrees. Now, the first alternative to that, is the contrapositive. The contrapositive of a statement, is a statement which is exactly equivalent to the original, in whether it's true or false. To get to contrapositive, you do two things. One, you reverse the order of the original statement. You switch the if portion and the then portion. Second, you also have to negate each of those original parts. Here's what happens to our original statement. I say if, X Y Z does not equal 180 degrees. I've started with the finish, from the original, and I've also negated it, and made it not equal. Then, X, Y, and Z are not the three angles of a triangle.That's the original first part. Now, at the end, and negate it. The third option, is to write down exactly the opposite, of your original statement, and then show that the opposite of your original statement, is false. This is what we call reaching a contradiction, or reducing to absurdity. In our case, I've taken the original statement, and completely negated it, written down the opposite. Here's the opposite. There is a triangle, with angles X, Y, and Z, who's sum is not 180 degrees. I need to show that that is a false statement. I can do that by showing that, assuming this statement implies, something everyone agrees is false. Either it implies something like 1=2, or that it implies two completely contradictory things. For instance, it might imply that X=Y, and X does not equal Y. Everyone knows that that can't be true."