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Measures of angles formed by a transversal

Sal solves an equation to find missing angles given two parallel lines and a transversal. Created by Sal Khan.
Video transcript
So I've got two parallel lines. So that's the first line right over there and then the second line right over here. Let me denote that these are parallel. These are parallel lines. Actually, I can do that a little bit neater. And let me draw a transversal, so a line that intersects both of these parallel lines, so something like that. And now let's say that we are told that this angle right over here is 9x plus 88. And this is in degrees. And we're also told that this angle right over here is 6x plus 182, once again, in degrees. So my goal here and my question for you is, can we figure out what these angles actually are, given that these are parallel lines and this is a transversal line? And I encourage you to pause this video to try this on your own. Well, the key here to realize is that these right over here are related by the fact that they're formed from a transversal intersecting parallel lines. And we know, for example, that this angle corresponds to this angle right over here. They're going to be congruent angles. And so this is 6x plus 182. This is also going to be 6x plus 182. And then that helps us realize that this blue angle and this orange angle are actually going to be supplementary. They're going to add up to 180 degrees, because put together, when you make them adjacent, their outer rays form a line right over here. So we know that 6x plus 182 plus 9x plus 88 is going to be equal to 180 degrees. And now we just have to simplify this thing. So 6x plus 9x is going to give us 15x. And then we have 182 plus 88. Let's see, 182 plus 8, would get us to 190. And then we add another 80. It gets us to 270-- plus 270-- is equal to 180. If we subtract 270 from both sides, we get 15x is equal to negative 90. And now we can divide both sides by 15. And we get x is equal to-- what is this? Let's see, 6 times 15 is 60 plus 30 is 90. So x is going to be equal to negative 6. So far, we've made a lot of progress. We figured out what x is equal to. x is equal to negative 6, but we still haven't figured out what these angles are equal to. So this angle right over here, 9x plus 88, this is going to be equal to 9 times negative 6 plus 88. 9 times negative 6 is negative 54. Let me write this down before I make a mistake. Negative 54 plus 88 is going to be-- let's see, to go from 88 minus 54 will give us 34 degrees. So this is equal to 34, and it's in degrees. So this orange angle right here is 34 degrees. The blue angle is going to be 180 minus that. But we can verify that by actually evaluating 6x plus 182. So this is going to be equal to 6 times negative 6 is negative 36 plus 182. So this is going to be equal to-- let's see, if I subtract the 6 first, I get to 176. So this gets us to 146 degrees. And you can verify-- 146 plus 34 is equal to 180 degrees. Now, we could also figure out the other angles from this as well. We know that if this is 34 degrees, then this must be 34 degrees as well. Those are opposite angles. This angle also corresponds to this angle so it must also be 34 degrees, which is opposite to this angle, which is going to be 34 degrees. Similarly, if this one right over here is 146 degrees, we already know that this one is going to be 146. This one's going to be 146 since it's opposite. And that's going to be 146 degrees as well.