Worked example: Triangle angles (intersecting lines)

We're given a bunch of lines here that intersect in all different ways and form triangles. And what I want to do in this video, we've been given the measures of some of the angles, this angle, that angle, and that angle. And what we want to do in this video is figure out what the measure of this angle is. And we're going to call that measure x. And so I encourage you to pause the video right now and try it yourself. And then I'm going to give you the solution. So I'm assuming you've unpaused it. And you've solved it or you've given it at least a good shot of it. So let's try to do it. And what's fun about these is there's multiple ways to solve these. And you kind of just have to keep figuring out what you can figure out. So let's say you start on the left-hand side right over here. If this is 121 degrees, then you'd say, well look, this angle right over here is supplementary to this angle right over there. So this is 121 degrees plus this green angle, that has to be equal to 180 degrees. So this is going to be 180 minus 121. Let's see, that's the same thing as 80 minus 21. 80 minus 20 would be 60. So that's going to be 59 degrees. So let me write that down. That's going to be 59 degrees. Now we see that we have two angles of a triangle. If you have two angles of a triangle, you can figure out the third angle, because they need to add up to 180. Or you could say that this angle right over here-- so we'll call that question mark-- we know that 59 plus 29 plus question mark needs to be equal to 180 degrees. And if we subtract the 15 out of the 29 from both sides, we get question mark is equal to 180 minus 59 minus 29 degrees. So that is going to be 180 minus 59 minus 29, let's see, 180 minus 59, we already know, is 121. And then 121 minus 29. So if you subtract just 20, you get 101. You subtract another 9, you get 92. So that's going to be equal to 92 degrees. This is equal to 92 degrees. Well, this right here is equal to 92 degrees. This angle right here is vertical with that angle. So it is also going to be equal to 92 degrees. And now we're getting pretty close. We can zoom in on this triangle down here. And let me save some space here. So let me just say that that over there is also going to be 92 degrees. And at this triangle down here, we have two of the sides of the triangle. We just have to figure out the third. And actually, we don't even have to do much math here, because we have two of the angles of this triangle. We have to figure out the third angle. So over here, we have one angle that's 92, one angle that's 29. The other one will be 180 minus 92 minus 29. And we don't even have to do any math here, because essentially, this is the exact same angles that we have in this triangle right over here. We have a 92 degree angle, we have a 29 degree angle, and the other one is 59 degrees. So in this case, it has to be also 59 degrees, because over here they added up to 180. So over here, they'll also add up to 180. So that will also get us to 59 degrees. We could just get that by taking 180, subtracting 29, subtracting 92. And then if this is 59 degrees, then this angle is also going to be 59 degrees, because they are vertical angles. So we're done. x is equal to 59 degrees. Now there's multiple ways that you could have reasoned through this problem. You could have immediately said-- so let me start over, actually. Maybe a faster way, but you wouldn't have been able to do kind of this basic steps there, is you said, look, this is an exterior angle right over here. It is equal to the sum of the remote interior angles. So 121 is going to be 29 plus this thing right over here. And we ended up doing that when I did it step-by-step before. But here, we're just using kind of a few things that we know about triangles ahead of time to maybe skip a step or two. Although I like to do it the other way just so we make sure we don't do anything weird. So anyway, this is going to be 129 minus 29, which is going to be 92. And if this is 92, then this is also going to be 92. And then, if this is x, then this is also going to be x. And you could say x plus 92 plus 29 is equal to 180 degrees. And then you'd say x plus 92 plus 29 is going to be 121 degrees. We already knew that before. And so that is going to equal 180 degrees. And so x is equal to 59 degrees. So there's a ton of ways that you could have thought about this problem.