Missing angle problems
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So we have two intersecting lines here, and then we have this other purple-looking line. And then they've given us some angles. They tell us that the measure of this angle right over here is 7x. The measure of this angle right over here is 60 degrees. And the measure of this angle right over here is x. So let's try to figure out what all of these angles are. So to do that, we have to figure out what x is. And there's a big clue here, because the 60-degree angle plus the x angle, they're adjacent. And if you add these two angles together, their outer rays, our vertical angle with this 7x angle. So we could say-- and just to visualize that a little better, let me color it in. Actually, let me do it this way. You see that this angle out here-- let me do it in a color I haven't used yet. This entire angle over here, which is going to be 60 degrees plus x, that's a vertical angle with this angle, the one that has measure 7x. So we could say that 60 degrees plus x is equal to 7x because vertical angles are equal. So let's write that down. We get 60. And we'll assume that everything is in degrees. 60 plus-- let me do that in this other color. 60 plus x is going to be equal to 7x. And now we just have to solve for x. So the simplest thing to do would be to get all over x's on one side of the equation. I've already gotten seven x's on this right-hand side, so let's get rid of all of the x's on the left-hand side. And the easiest way to get rid of this x is to subtract x from the left-hand side. But of course, in order to keep it an equation, we can't just do something to one side. Otherwise, it won't be equal anymore. We have to do it to both sides. So let's subtract x from both sides. And on the left-hand side, we are left with just the 60. So we're left with just a 60. And then that is going to be equal to 7x minus x. If I have 7 of something and I get rid of 1 of them, I'm going to have 6 of that something left. So that's going to be equal to 6x. So we have 6 times something is equal to 60. You could probably figure that out in your head. But I will do it a little systematically. We can divide both sides by 6 to solve for x. So let's do that. And we would be left with x is equal to. 60 divided by 6 is 10. And we reminded ourselves that everything was in degrees. And we could even do that here. This was in degrees. This is in degrees. And so this is in degrees right over here. So the measure of this angle right over here is 10 degrees. So this one right over there is 10 degrees. This is, of course, 60 degrees. You add them together, 60 degrees plus 10 degrees is 70 degrees. So this bigger angle right over here is 70 degrees. And of course, this one over here, it's a vertical angle. It's going to have to be the same. And we see that, 7 times x. 7 times 10 degrees is 70 degrees as well.