Vertical, complementary, and supplementary angles
Let's say I have two intersecting line segments. So let's call that segment AB. And then I have segment CD. So that is C and that is D. And they intersect right over here at point E. And let's say we know, we're given, that this angle right over here, that the measure of angle-- That B is kind of, I don't know why I wrote it so far away. So let me make that a little bit closer. Let me make that B a little bit closer. So let's say-- I'll do that in yellow. Let's say that we know that the measure of this angle right over here, angle BED, let's say that we know that measure is 70 degrees. Given that information, what I want to do, based only on what we know so far and not using a protractor, what I want to do is figure out what the other angles in this picture are. So what's the measure of angle CEB, the measure of angle AEC, and the measure of angle AED? So the first thing that you might notice when you look at this, I've already told you that this is a line segment and that this is a line segment. You see that angle BED and angle CEB are adjacent. And we also see that if you take the outer sides of those angles, it forms a straight angle. And we also see that angle CED is a straight angle. So we know that these two angles must also be supplementary. They're next to each other and they form a straight angle when you take their outer sides. So we know that angle BED and angle CEB are supplementary, which means they add up to, or that their measures add up to 180 degrees. Supplementary angles. Which tells us that the measure of angle BED plus the measure of angle CEB-- and I keep writing measure here. Sometimes you'll just see people write, angle BED plus angle CEB is equal to 180 degrees. Now we already know the measure of angle BED is 70 degrees. So we already know that this thing right over here is 70 degrees. And so 70 degrees plus the measure of angle CEB is 180 degrees. You subtract 70 from both sides, and we get the measure of angle CEB is equal to 110 degrees. I just subtracted 70 from both sides of that. So we figured out that this right over here is 110 degrees. Well, that's interesting. And I went through more steps than you would if you were doing this problem quickly. If you did this problem quickly in your head, you'd say, look this is 70 degrees, this angle plus this angle would be 180 degrees, so this has to be 110 degrees. So now let's use the same logic to figure out what angle CEA is. So now we care about the measure of angle CEA. And we can use the exact same logic that we used over here. Angle CEA and angle CEB, they are adjacent. They form a straight angle, if you look at their outsides, so they must be supplementary. They form a straight angle right over here. So they're supplementary. So they must add up to 180 degrees. So the measure of angle CEA plus the measure of angle CEB, which is 110 degrees, must be equal to 180 degrees. So once again, subtract 110 from both sides. You get the measure of angle CEA is equal to 70 degrees. So this one right over here is also 70 degrees. And what we'll learn in the next video is that this is no coincidence. These two angles, angle CEA and angle BED, sometimes they're called opposite angles-- well, I have often called them opposite angles, but the more correct term for them is vertical angles. And we haven't proved it. We've just seen a special case here where these vertical angles are equal. But it actually turns out that vertical angles are always equal. But we haven't proved it to ourselves for the general case. But let me just write down this word since it's a nice new word. So angle CEA and angle BED are vertical. And you might say, wait, they look like they're horizontal, they're next to each other. And the vertical really just means that they're across from each other, across an intersection from each other. Angle CEB and angle AED are also vertical. So let me write that down. Angle CEB and angle AED are also vertical. And that might even make a little bit more sense, because it literally is, one is on top and one is on bottom. They're kind of vertically opposite from each other. But these horizontally opposite angles are also called vertical angles. So now we have one angle left to figure out, angle AED. And based on what I already told you, vertical angles tend to be, or they are always, equal. But we haven't proven that to ourselves yet so we can't just use that property to say that this is 110 degrees. So what we're going to do is use the exact same logic. CEA and AED are clearly supplementary. Their outsides form a straight angle. They're clearly supplementary, so CEA and AED must add up to 180 degrees. Or we could say the measure of angle AED plus the measure of angle CEA must be equal to 180 degrees. We know the measure of CEA is 70 degrees. We know it is 70 degrees. So you subtract 70 from both sides. You get the measure of angle AED is equal to 110 degrees. So we got the exact result that we expected. So this angle right over here is 110 degrees. And so if you take any of the adjacent angles that their outer sides form a straight angle, you see they add up to 180. This one and that one add up to 180. This one and that one add up to 180. This one and that one add up to 180. And this one and that one add up to 180. If you go all the way around the circle, you'll see that they add up to 360 degrees. Because you literally are going all the way around. So 70 plus 110 is 180, plus 70 is 250, plus 110 is 360 degrees. I'll leave you there. This is the first time that we've kind of found some interesting results using the tool kit that we've built up so far. In the next video, we'll actually prove to ourselves using pretty much the exact same logic here, but we'll just do it with generalized numbers-- we won't use 70 degrees-- to prove that the measure of vertical angles are equal.