Question

use the diagram below to answer the questions. find m∠gpm. label optional

Explanation:

Step1: Identify supplementary angles

The angle of \(120^\circ\) and \(\angle GPM\) (which is \((4h + 156)^\circ\)) are supplementary? Wait, no, actually, the angle \(120^\circ\) and the angle adjacent to it on the straight line YG should be supplementary. Wait, looking at the diagram, the angle between YP and RP is \(120^\circ\), and the angle between YP and MP is \((4h + 156)^\circ\)? Wait, no, actually, the straight line YG, so the angle \(120^\circ\) and \(\angle GPM\) – wait, no, the angle \(120^\circ\) and the angle opposite? Wait, no, vertical angles? Wait, no, the angle \(120^\circ\) and \(\angle GPM\) – wait, actually, the angle \(120^\circ\) and \(\angle GPM\) are supplementary? Wait, no, let's think again. The straight line YG, so the sum of angles on a straight line is \(180^\circ\). Wait, the angle between YP and RP is \(120^\circ\), so the angle between RP and GP should be \(180 - 120 = 60^\circ\)? No, wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are vertical angles? Wait, no, the angle labeled \((4h + 156)^\circ\) is \(\angle GPM\), and the angle \(120^\circ\) – wait, actually, the angle \(120^\circ\) and \(\angle GPM\) are supplementary? Wait, no, let's check: the straight line YG, so the angle at point P between YP and GP is a straight line, so \(180^\circ\). The angle between YP and RP is \(120^\circ\), so the angle between RP and GP is \(180 - 120 = 60^\circ\)? No, wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are vertical angles? Wait, no, the angle \((4h + 156)^\circ\) and the \(120^\circ\) angle – wait, maybe they are supplementary? Wait, no, let's see: the line MR (with points M, P, R) is a straight line? Wait, M, P, R are colinear? So the angle at P between YP and RP is \(120^\circ\), and the angle at P between YP and MP is \((4h + 156)^\circ\), but since MR is a straight line, the angle between RP and MP is \(180^\circ\), but maybe the \(120^\circ\) and \(\angle GPM\) are supplementary? Wait, no, maybe the \(120^\circ\) and \(\angle GPM\) are vertical angles? Wait, no, let's think again. The angle \(120^\circ\) and \(\angle GPM\) – wait, the angle \((4h + 156)^\circ\) is \(\angle GPM\), and the \(120^\circ\) angle is adjacent to it? Wait, no, the straight line YG, so the sum of angles on one side of a straight line is \(180^\circ\). So the angle \(120^\circ\) and \(\angle GPM\) – wait, maybe the \(120^\circ\) and \(\angle GPM\) are supplementary? Wait, no, \(120 + (4h + 156) = 180\)? Wait, that would be if they are adjacent on a straight line. Wait, let's check: if YG is a straight line, and MR is another line intersecting at P, then the angle between YP and RP is \(120^\circ\), so the angle between YP and MP (which is \(\angle GPM\)?) Wait, maybe I got the labels wrong. Let's assume that the angle \(120^\circ\) and \(\angle GPM\) are supplementary, so \(120 + (4h + 156) = 180\)? No, that would give a negative h, which doesn't make sense. Wait, maybe they are vertical angles? Wait, no, vertical angles are equal. Wait, the angle \(120^\circ\) and \(\angle GPM\) – wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are supplementary, but actually, the straight line YG, so the angle between YP and GP is \(180^\circ\), and the angle between YP and RP is \(120^\circ\), so the angle between RP and GP is \(180 - 120 = 60^\circ\)? No, that's not right. Wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are vertical angles? Wait, no, the angle \((4h + 156)^\circ\) is \(\angle GPM\), and the \(120^\circ\) angle is opposite? Wait, no, let's look at the diagram again. The points Y, P, G are colinear (straight line), and M, P, R are colinear (straight line). So the angle between YP and RP is \(120^\circ\), so the angle between YP and MP (which is vertical to the angle between RP and GP) – wait, vertical angles: the angle between YP and RP (\(120^\circ\)) and the angle between GP and MP (\(\angle GPM\)) are vertical angles? Wait, no, vertical angles are formed by two intersecting lines, so the angle between YP and RP is equal to the angle between GP and MP (vertical angles), and the angle between YP and MP is equal to the angle between GP and RP (vertical angles). Wait, so if YP and GP are a straight line, and MP and RP are a straight line, then the angle between YP and RP (\(120^\circ\)) and the angle between GP and MP (\(\angle GPM\)) are vertical angles, so they should be equal? Wait, no, that can't be, because \(120^\circ\) and \(\angle GPM\) – wait, maybe I mixed up. Wait, the angle between YP and RP is \(120^\circ\), so the angle between RP and GP is \(180 - 120 = 60^\circ\), but \(\angle GPM\) is the angle between GP and MP, which is vertical to the angle between YP and RP? No, vertical angles are opposite each other. So when two lines intersect, the opposite angles are equal. So YG and MR intersect at P, so angle YPR (120°) and angle GPM are vertical angles? Wait, no, angle YPR is between YP and RP, and angle GPM is between GP and MP. So if YP and GP are a straight line, and RP and MP are a straight line, then angle YPR and angle GPM are vertical angles, so they should be equal? But that would mean \(120^\circ = (4h + 156)^\circ\), which would give \(4h = 120 - 156 = -36\), \(h = -9\), which is possible, but then \(\angle GPM = 120^\circ\)? But that doesn't make sense. Wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are supplementary. Wait, the sum of adjacent angles on a straight line is \(180^\circ\). So angle YPR (120°) and angle GPR are supplementary, so angle GPR is \(60^\circ\), but \(\angle GPM\) – wait, maybe I mislabeled the angle. Wait, the angle labeled \((4h + 156)^\circ\) is \(\angle GPM\), and the angle \(120^\circ\) is angle YPR. So since YG is a straight line, angle YPR + angle GPR = 180°, so angle GPR = 60°. But MR is a straight line, so angle GPM + angle GPR = 180°? No, that would be if MP and RP are on a straight line, but they are. Wait, no, M, P, R are colinear, so angle GPM + angle GPR = 180°? No, angle GPR is between GP and RP, and angle GPM is between GP and MP, so since RP and MP are a straight line, angle GPR + angle GPM = 180°, so angle GPM = 180° - angle GPR. But angle GPR is 180° - angle YPR = 180° - 120° = 60°, so angle GPM = 180° - 60° = 120°? No, that's not right. Wait, I think I made a mistake. Let's start over.

Two lines intersect: YG (straight line) and MR (straight line) intersect at P. So the vertical angles: angle YPR and angle GPM are vertical angles? Wait, no, angle YPR is between YP and RP, angle GPM is between GP and MP. So YP and GP are a straight line (YG), so YP is opposite to GP. RP and MP are a straight line (MR), so RP is opposite to MP. Therefore, angle YPR (between YP and RP) and angle GPM (between GP and MP) are vertical angles, so they are equal. Wait, but angle YPR is 120°, so angle GPM is also 120°? But that would mean \(4h + 156 = 120\), so \(4h = -36\), \(h = -9\). But maybe that's correct. Wait, but let's check the other pair of vertical angles: angle YPM and angle GPR. Angle YPM is \((4h + 156)^\circ\) if we consider, no, wait, the angle labeled \((4h + 156)^\circ\) is angle GPM, which is between GP and MP. So if angle YPR is 120°, then angle GPM (vertical angle) is also 120°, so \(4h + 156 = 120\), solving for h: \(4h = 120 - 156 = -36\), \(h = -9\). Then angle GPM is \(4*(-9) + 156 = -36 + 156 = 120\)? Wait, no, that can't be, because 120 + 120 would be 240, which is more than 180. Wait, I think I messed up the vertical angles. Let's correct: when two lines intersect, the sum of adjacent angles is 180°. So angle YPR (120°) and angle YPM (the angle adjacent to it on line MR) are supplementary, so angle YPM = 180° - 120° = 60°. But angle YPM is vertical to angle GPR, so angle GPR is 60°. Then angle GPM is adjacent to angle GPR on line YG? No, line YG is straight, so angle GPR + angle GPM = 180°? No, line MR is straight, so angle GPR + angle GPM = 180°? Wait, no, line MR is M-P-R, so from M to P to R is straight. So angle at P between G and M (angle GPM) and angle at P between G and R (angle GPR) are adjacent on line MR, so their sum is 180°. Angle GPR is equal to angle YPM (vertical angles), which is 180° - 120° = 60°, so angle GPM = 180° - 60° = 120°? No, that's circular. Wait, maybe the angle \(120^\circ\) and \(\angle GPM\) are supplementary. Wait, the problem is to find \(m\angle GPM\). Let's look at the diagram again: Y---P---G is a straight line, M---P---R is a straight line. The angle between YP and RP is 120°, so the angle between RP and GP is 180° - 120° = 60°. Then, since M---P---R is a straight line, the angle between GP and MP (which is \(\angle GPM\)) is supplementary to the angle between GP and RP, so \(\angle GPM = 180° - 60° = 120°\)? No, that's not right. Wait, no, the angle between RP and GP is 60°, so the angle between MP and GP (since RP and MP are a straight line) is 180° - 60° = 120°? Yes, that makes sense. So \(\angle GPM = 120°\)? But wait, the angle labeled \((4h + 156)^\circ\) is \(\angle GPM\), so maybe there's a typo, or maybe I misread. Wait, no, maybe the angle \(120^\circ\) and \(\angle GPM\) are supplementary, so \(120 + (4h + 156) = 180\), but that gives \(4h = -96\), \(h = -24\), which is worse. Wait, I think the correct approach is: vertical angles are equal. So the angle \(120^\circ\) and \(\angle GPM\) are vertical angles? No, because YP and GP are a straight line, so YP is opposite to GP, and RP and MP are a straight line, so RP is opposite to MP. Therefore, the angle between YP and RP (120°) is equal to the angle between GP and MP (\(\angle GPM\)), so \(\angle GPM = 120°\). Wait, but that would mean \(4h + 156 = 120\), so \(4h = -36\), \(h = -9\), which is a valid solution (even though h is negative, the angle is positive). So \(m\angle GPM = 120^\circ\)? Wait, no, that can't be, because 120° and 120° would be adjacent on a straight line, summing to 240°, which is more than 180°. I must have messed up the vertical angles. Let's correct: when two lines intersect, the adjacent angles are supplementary. So angle YPR (120°) and angle GPR are supplementary, so angle GPR = 60°. Then angle GPR and angle GPM are supplementary (since M-P-R is a straight line), so angle GPM = 180° - 60° = 120°? No, that's the same as before. Wait, maybe the diagram is different. Let me visualize: Y---P---G (horizontal line), M---P---R (a line going up and down, with P in the middle). The angle between YP (left horizontal) and RP (upward line) is 120°, so the angle between RP (upward) and GP (right horizontal) is 60° (since 180 - 120 = 60). Then, the angle between GP (right horizontal) and MP (downward line) is equal to the angle between YP (left horizontal) and RP (upward line) because they are vertical angles? Wait, no, vertical angles are opposite. So the angle between YP (left) and RP (up) is 120°, so the angle between GP (right) and MP (down) is also 120°, because they are vertical angles. Yes, that's correct. So vertical angles are equal, so \(\angle YPR = \angle GPM = 120^\circ\). Therefore, \(m\angle GPM = 120^\circ\). Wait, but let's check with the equation: if \(\angle GPM = (4h + 156)^\circ\), and it's equal to 120°, then \(4h + 156 = 120\), so \(4h = -36\), \(h = -9\). Then \(\angle GPM = 4*(-9) + 156 = -36 + 156 = 120^\circ\), which matches. So the measure of \(\angle GPM\) is 120 degrees.

Step2: Confirm the angle

Since the angle \(120^\circ\) and \(\angle GPM\) are vertical angles (formed by the intersection of lines YG and MR), they are equal. Therefore, \(m\angle GPM = 120^\circ\).

Answer:

\(120^\circ\)