Missing angle problems
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We're told that the measure of angle QPR-- so that's this angle right over here-- is 2x plus 122. And I'll assume that these are in degrees. So it's 2x plus 122 degrees. And the measure of angle RPS-- so that's this angle right over here-- is 2x plus 22 degrees. And they ask us to find the measure of angle RPS. So we need to figure out this right over here. So we would be able to figure that out if we just knew what x is. And lucky for us, we can use the information given to solve for x and then figure out what 2 times x plus 22 is. And the main big idea here, the thing that pops out here, is that the outside rays for both of these angle form a line. These two angles form a line. You could say that they are supplementary. Both of these angles are supplementary. 2x plus 22 plus another 2x plus 122 is going to add up to 180. We know that this entire angle right over here is 180 degrees. So we can say that the measure of angle QPR, this angle right over here, 2x plus 122, plus the green angle, plus angle RPS-- so plus 2x plus 22-- is going to be equal to 180 degrees. And now we can start simplifying this. We have two x's. We have another two x's. So those are going to add up to be 4x. And then we have 122 plus 22. So that's going to be 144. And the sum of those two are going to be equal to 180 degrees. We can subtract 144 from both sides. On the left-hand side, we're just going to be left with a 4x, this 4x right here. And on the right-hand side, we're going to have-- let's see, if we were subtracting 140, we would have 40 left. And then we have to subtract another 4, so it's going to be 36. Divide both sides by 4, and we get x is equal to 9. Now remember, we're not done yet. They didn't say solve for x. They said find the measure of angle RPS, which is 2 times x plus 22 or 2 times 9 plus 22, which is 18 plus 22, which is equal to 40. So the measure of angle RPS is 40 degrees.