 Okay, we're still talking about geometry. Now, one aspect of geometry that really comes up a lot of places is the relationship between parallel lines. Now, whenever they draw something like this or two lines and they want to show that they're parallel, what they do is put arrows on them like this. This would mean these two lines are parallel. If you have multiple parallel lines, for example, if you have something like this and something like this and they want to say that this line is parallel to this, they'll use single arrows for this and if they want to say that line is parallel to this, they'll use double arrows just like the double ticks when we use the links of sides to say that these two lines are parallel. You can go on from there. You could have two more lines with three little arrows on them saying that those two lines are parallel. Basically, it's just visually organizing the information for you. Now, one of the most popular, one of the things they use the most is giving you two parallel lines and having a line go through them. And as soon as you have this, there's a few very important things that happen, relationship between the angles that come up. We've talked about this one before, which is alternate interior, which is vertically opposite angles where if you have two lines crossing, this angle will equal that angle and that angle will equal that angle. That's a given. It doesn't make a difference if there's all the parallel lines or not. That's just the two lines crossing each other. Now, the relationship between two parallel lines is this. There's three main ones. One of them is called alternate interior angles. You can think of this as the Z rule. The Z rule says, or alternate interior angle says, if you have two parallel lines, a line going through them, this angle is equal to this angle. This is your Z. That's why I sometimes refer to it as a Z rule, or the way I personally remember it is the Z rule. There are different words that they use in different places, different parts of the world for what this relationship is. Alternate interior angle is the most common for this one. Now, the Z rule can also work backwards. You could have a Z going this way, which would mean that this guy, whoops, use one tick to say that they're the same. So that guy equals that guy, and this guy is equal to this guy. So there is your backward Z, and here is your forward Z. That's the alternate interior angle. Now, there's a couple other relationships that come out with parallel lines. I'm not going to use the names because the names vary for those ones a lot more, but the relationships are these. When you have two parallel lines going across and a line crossing them, this angle plus that angle is equal to, they're not equal to each other, they're different angles, but if you add them up, they equal 180 degrees. So the insides of two parallel lines being crossed by another line, the insides, the two angles added up inside equal to 180 degrees. So that means if you knew what this was, let's say this was 80 degrees, this angle here was 80 degrees, then you would know that's 100 degrees because you could go 180 minus 8. So that's another very powerful relationship. Another rule that you have, so the other one is called a C rule, C rule because you have a C this way, so that plus that equals 180, and you also have a C this way, which is that plus that equals 180. So it works both ways. It's a mirror image of each other. The other rule that you have is called the F rule. Again, these aren't the official names for them, so depending on where you are, you should find out what they're called, what they refer to, and refer to the relationship as those, not the F rule, C rule, or the F rule. The F rule says, here's your F. Now remember, it works as a backwards F as well. The F rule says this angle is equal to this angle, which makes sense because they're parallel lines. Now another way you could get to this relationship is you could use a C rule. This angle plus that angle equals 180 degrees, right? Well, angles on a line equal 180. That's called supplementary angles. So if this plus this equals 180, and this plus this equals 180, then this guy must be equal to that guy. So that's called the F rule. The way it works is you can do it backwards as well, where you have this guy equals this guy. Equals that guy. Equals that guy. The other way also it works is the tops can equal these guys as well. So there's a lot of different angles here that equal each other. You could use the vertically opposite. So this guy must be equal to this guy. And that guy is equal to that guy because of the vertically opposite. So right away you get relationship at each crucial point between the two lines crossing each other, and you can move yourself around a drawing when they give it to you in a pattern. So some of the terminology that they use is alternate interior for saying the Z rule, where this angle equals that angle, and that angle equals that angle. You got the C rule where this plus this equals 180, and that plus that equals 180. Which also goes, if you take it one step further, that plus that must equal 180, and that guy plus that guy must equal 180. You got the F rule where that equals that, that equals that, that equals that, and that equals that. I'm doing pairs, all four of them don't equal each other. Now the other terminology that they use is supplementary, meaning the angles add up to 180. Now all angles on a line equal to 180. That plus that equals 180. That plus that equals 180, which goes to any point on it. The angle at a point, if you go all the way around, is 360, which is basically a circle, right? So if you cut it in half, each one's going to be 180. The other terminology that they use is complementary, which is two angles added up equal 90 degrees. Now over here we don't have one of those, but just keep that in mind. If they give you two angles, if they give you a right angle, triangle, and they said, okay, this guy's 50 degrees, and this is angle A and this is angle B, and angle A and B are complementary, then you know that they add up to 90 degrees. So this guy has to be 90 minus 50 is 40 degrees. So these are some of the terminologies you're going to encounter when you're dealing with lines and angles. You definitely have to memorize these, and you have to refer to them when you're doing proofs. Longer proofs or shorter proofs, you have to refer to each relationship according to what makes this angle equal that angle. And as long as you say, for example, if they give you W here and Q here, and if you say W equals Q is equal to Q, angle W is equal to angle Q, and the way you write this is, let's do this correctly. If you say angle W is equal to angle Q, and if you say the reason is alternate interior angles, that's a valid response, and you get a point for that. So you have to know your terminology because each term refers to something, and it's a tool for you to be able to make statements such as this. So learn these terminologies and apply them correctly. We will do proofs later on with this stuff a lot more, but for now they should clear things up.