Angles between intersecting lines
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Let's say that we have two parallel lines. So that's one line right over there, and then this is the other line that is parallel to the first one. I'll draw it as parallel as I can. So these two lines are parallel. This is the symbol right over here to show that these two lines are parallel. And then let me draw a transversal here. So let me draw a transversal. This is also a line. Now, let's say that we know that this angle right over here is 110 degrees. What other angles can we figure out here? Well, the first thing that we might realize is that, look, corresponding angles are equivalent. This angle, the angle between this parallel line and the transversal, is going to be the same as the angle between this parallel line and the transversal. So this right over here is also going to be 110 degrees. Now, we also know that vertical angles are equivalent. So if this is 110 degrees, then this angle right over here on the opposite side of the intersection is also going to be 110 degrees. And we could use that same logic right over here to say that if this is 110 degrees, then this is also 110 degrees. We could've also said that, look, this angle right over here corresponds to this angle right over here so that they also will have to be the same. Now, what about these other angles? So this angle right over here, its outside ray, I guess you could say, forms a line with this angle right over here. This pink angle is supplementary to this 110 degree angle. So this pink angle plus 110 is going to be equal to 180. Or we know that this pink angle is going to be 70 degrees. And then we know that it's a vertical angle with this angle right over here, so this is also 70 degrees. This angle that's kind of right below this parallel line with the transversal, the bottom left, I guess you could say, corresponds to this bottom left angle right over here. So this is also 70 degrees. And we could've also figured that out by saying, hey, this angle is supplementary to this angle right over here. And then we could use multiple arguments. The vertical angle argument, the supplementary argument two ways, or the corresponding angle argument to say that, hey, this must be 70 degrees as well.