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we're told that the measure of angle QPR so 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 RPS so that's this angle right over here is 2x plus 22 degrees and they asked 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 big big idea here the thing that pops out here is that the outside rays for both of these angle form align these two angles form a line you could you could say that they are supplementary these are those of these angles are supplementary 2x plus 22 plus another 2x plus 122 is going to add up to 180 we know that this isn't 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 1 22 plus the green angle plus our angle RPS so plus 2x plus 22 is going to be equal to 180 degrees is going to be equal to 180 and now we can add we can start simplifying this we have 2 X's we have another 2 X's so those are going to add up to be 4x and then we have 1 22 plus 22 so that's going to be 144 and that's the sum of those two are going to be equal to 180 degrees we can subtract 144 from both sides 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