Missing angle problems
So we're told that ray OL is perpendicular to ray ON. So OL is perpendicular to ON. So we know that this right over here is a right angle, or it's a 90-degree angle. Then they tell us that the measure of angle LOM is equal to 2x plus 46. So LOM-- so this angle right over here-- is equal to 2x plus 46. Then they tell us that the measure of angle MON is equal to 3x minus 6. So this angle right over here is 3x minus 6. And so they ask us find angle MON. So we need to find this angle right over here. And we'd be able to find it if we knew what x is. If we knew what x is, then it's going to be 3 times that minus 6. Now, the one thing that we know is that if we add up these two angles, these are adjacent angles. And their outside rays form a right angle. So these two angles are going to add up to 90 degrees. And I guess we can assume here that we're dealing in degrees. So let's say that 2x plus 46 plus 3x minus 6 is going to add up to 90 degrees. These two angles are complementary. So it's going to add up to 90 degrees, or it's going to add up to 90. And now, we just have to simplify. We have 2 x's. We have another 3 x's. So you add those two, and you're going to get 5 x's. And then you have 46. And you're going to subtract 6. So it's going to be plus 40 is equal to 90. Subtract 40 from both sides. So let's do that. Subtract 40 from both sides. And on the left-hand side, you're just left with a 5x. And on the right-hand side, you're left with a 50. Now, just divide both sides by 5. And we are left with x is equal to 10. Now, our answer is not 10. We've just figured out what x is. What they ask us for is what's the measure of angle MON? Well, we already know that the measure of angle MON is equal to 3x minus 6, which is equal to 3 times 10 minus 6-- I want to do that minus 6 in that green color-- which is equal to 24 degrees.