Introducing the Step-by-Step Geometrical Proof

Video Transcript

"Suppose you've discovered a pattern, perhaps a relationship of cause of effect in Geometry. A proof is just an explanation of factuality of a statement that you have that's your result. Now to get a proof, what you're looking for is to build a bridge for your audience. All right, so let's say this stream is the river of missing knowledge that we have to cross from our givens to our result. Well we are allowed to build our series of stepping stones anywhere we like. We might just look for stepping stones that are near the givens and other stones perhaps that are near the result. Stepping stones are facts and theorems that we know and we'd like to connect them in order to get across. The proof, when it's written down should be a nice root to take. A step at a time from our givens to our result. Here's an example, say that we're suppose to show that given X,Y, and Z are numbers which are the angles of a triangle. Then, we must show X plus Y plus Z if they're in degrees total to 180 degrees. All right, there's a lot of facts about triangles that we know and a lot of facts about angles. Here's a few facts about angles and triangles and straight lines. One is that if you draw a straight line then the angle made by thinking that of it as two lines extending from a dot in the middle is 180 degrees. If the two lines cross then the opposite angles are equal. In fact, we can use the same letter to denote both of those angles. If two parallel lines, denoted by the little arrows showing that they're parallel, cross a third straight line then the two corresponding angles are equal. Again, we could just use the same letter to denote both of them. These are stepping stones and, you can image there may be a lot more stepping stones just by opening up your Geometry book.Now, this top one is closer to your finish. Since 180 degrees is what we're trying to show these angles sum to. The other two, we can try and start with either of them but actually this one is closer to the beginning. So, our proof is going to go in this order. All right, let me show you how that works on the original triangle. All right first, I can use the fact that those opposite angles are equal. This angle X is exactly the same as this angle. I will call it X. This angle Y is exactly the same as that angle. I will call it Y. Now, I will use my second fact about corresponding angles. To do that, I'm going to have to add a line to my picture. Notice that I've already extended these lines a little bit to create the angles, the new angles opposite to the original ones. Now I'll need an entirely new line. It's going to go through the point at which the Z is the angle. And, it's going to be parallel to the opposite side from Z. Now I have parallel lines and first, I can note that this side of the triangle is the line crossing those parallel lines. So, this angle is also X. Again, I can note that this side of the triangle is a straight line crossing parallel lines so that this angle is Y. But now, look what's happened. I have three angles X,Y, and Z making up a full 180 degrees. So X plus Y plus Z equals 180."