Proof: Angle formed by intersecting secants

0:00- [Person] Just let me apologize, there's a leaf blower outside of my window, hopefully, you can't hear it, but I can, but I will try to focus. So what do I have going on here? I have two secants that intersect at a point outside of the circle, and what I wanna do in this video is come up with a relationship between the angle measure where those two secants intersect outside of the circle, maybe we call that "measure X" right over here, a relationship between that and the measure of the arcs that it intersects, so this arc right over here, maybe it has a measure of theta, and then, this arc over here, maybe it has a measure of phi, and what we're going to prove is that X is going to be equal

0:47to one-half the difference of those arc measures, so one-half the difference, the way I've drawn it here, phi is larger, so I'll put phi minus theta. Minus theta. So let's get about proving that, and so, one thing that we've seen multiple times is we know the relationship between an inscribed angle and the arc...

1:10The measure of the arc that it actually intersects, and we might be able to use that. So for example, if I were to draw a line here, if I were to draw a line here, we have this angle, BED, it intersects... It's an inscribed angle, and it intersects this arc of measure theta, we know an inscribed angle has half the measure of the arc that it intersects, so this is going to be...

1:38Let me write it over here, this is going to have a measure of theta over two, I know that's a little bit hard to see, so that's interesting, it still doesn't quite get us to X yet, but we have another inscribed angle that intersects an arc whose measure we know, we have this angle right over here, angle CBE, that angle, it...

2:00As we just said, it intersects phi, so this is going to be phi over two. Now, why is this useful? Well, this angle, angle CBE, it's colinear with angle EBA, and so, they are supplementary, so this angle... Let me do this in a new color, this angle right over here, this whole thing, I'll do it with two arcs so you don't get confused with that, I'll...

2:26It looks really messy. Let me see, this whole angle right over here is going to be 180 minus phi over two, let me be careful, minus, do it in that same color, phi over two. Well, now, we know two angles in the triangle, we can figure out the third. We can write that X is going to be equal to 180 minus these two, I could have just said that X plus this plus that...

2:54Actually, let me just do it that way, so X plus theta over two, plus theta over two plus 180, plus 180, minus phi over two, minus phi over two, is going to be equal to, is going to be equal to 180. Well, the first thing that I could do is subtract 180 from both sides, so let's do that, and then, I could add...

3:27Or I could subtract each of these from both sides, so let me subtract theta over two, subtract theta over two... Let me do it this way. You'll see why I'm doing it in a second. Subtract theta... Subtract theta over two, and then, over here... Well, I'm not gonna subtract each, I'm gonna add the opposite, I should say, and let's add phi over two, I'll do it here, so I have a little bit of more space, so plus phi over two to both sides, and then, I'm gonna put a phi-over-two on that side, and then, what I'm going to be left with is that X...

4:07All I have left on the left-hand side is an X, so I'll get X is equal to phi over two minus theta over two, which is the result that we wanted, phi over two minus theta over two, which is exactly this over here if you distribute the one-half, I could factor out a one-half and I would get exactly that, and we are done.