Question

find the value of x.
(9x - 5)°
(3x)°
163°
x =

Explanation:

Step1: Identify angle sum

Angles around a point sum to \(360^\circ\), but here, since \(MQ\) is a straight line (supplementary to \(163^\circ\))? Wait, no—wait, the angles on one side of a straight line sum to \(180^\circ\)? Wait, no, looking at the diagram: the angles \(163^\circ\), \(3x\), \(9x - 5\), and the angle opposite? Wait, no, actually, the angles around point \(S\) on the straight line? Wait, no, let's re-examine. Wait, the straight line is \(MQ\) (horizontal), and the other angles: \(163^\circ\) is adjacent to the angles \(3x\), \(9x - 5\), and the angle between \(R\) and \(M\)? Wait, no, maybe the sum of angles on a straight line (since \(MQ\) is a straight line, the angles on one side of \(MQ\) should sum to \(180^\circ\)? Wait, no, the \(163^\circ\) is given, and the other angles: \(3x\), \(9x - 5\), and the angle between \(R\) and \(M\)? Wait, maybe I made a mistake. Wait, actually, the angles \(3x\), \(9x - 5\), and the angle adjacent to \(163^\circ\) (which is \(180 - 163 = 17^\circ\))? Wait, no, let's look again. Wait, the diagram: points \(M\), \(S\), \(Q\) are colinear? Wait, \(M\) and \(Q\) are on a straight line, so the sum of angles around \(S\) on that line? Wait, no, the angles given: \(163^\circ\), \(3x\), \(9x - 5\), and the angle between \(R\) and \(M\) (but \(R\) and \(M\) are both leftward, so maybe \(R\) and \(M\) are a straight line? No, the arrows: \(M\) and \(R\) are both left, but \(N\) is up-left, \(P\) is up-right, \(Q\) is right. Wait, maybe the sum of angles \(3x\), \(9x - 5\), and the angle that is supplementary to \(163^\circ\) (since \(163^\circ\) and that angle are adjacent to a straight line) is \(180^\circ\)? Wait, \(180 - 163 = 17^\circ\)? No, that doesn't make sense. Wait, no—wait, the correct approach: the angles \(3x\), \(9x - 5\), and the angle between \(N\) and \(P\), and the \(163^\circ\) angle. Wait, maybe the sum of angles \(3x + (9x - 5) + 3x\)? No, I'm confused. Wait, let's start over.

Wait, the key is that the sum of angles around a point is \(360^\circ\), but if \(M\) and \(Q\) are colinear (straight line), then the angles on one side of \(MQ\) (the left side: \(R\), \(M\), \(N\), \(P\)) should sum to \(180^\circ\)? Wait, no, the \(163^\circ\) is on the right side of \(MQ\), so the left side (angles \(3x\), \(9x - 5\), and the angle between \(R\) and \(M\)) should sum to \(180^\circ\)? Wait, no, the \(163^\circ\) is given, and the other angles: \(3x\), \(9x - 5\), and the angle adjacent to \(163^\circ\) (which is \(180 - 163 = 17^\circ\))? Wait, no, maybe the angles \(3x\), \(9x - 5\), and \(17^\circ\) sum to \(180^\circ\)? Wait, no, let's look at the diagram again. The angles at point \(S\): \(163^\circ\) is below the straight line \(MQ\), and above \(MQ\), we have angles \(3x\) (between \(M\) and \(N\)), \(9x - 5\) (between \(N\) and \(P\)), and the angle between \(P\) and \(Q\) (but \(P\) is up-right, \(Q\) is right, so that angle is small). Wait, no, maybe the sum of angles above \(MQ\) (i.e., \(3x\), \(9x - 5\), and the angle between \(P\) and \(Q\)) plus the angle below \(MQ\) (163°) should equal \(360^\circ\), but that's not right. Wait, I think I made a mistake. Let's check the correct method.

Wait, actually, the straight line is \(MQ\), so the sum of angles on one side of \(MQ\) (the upper side: \(M\) to \(N\) to \(P\) to \(Q\)) should sum to \(180^\circ\)? No, \(M\) and \(Q\) are colinear, so the angles above \(MQ\) (between \(M\), \(N\), \(P\), \(Q\)) should sum to \(180^\circ\), and the angle below \(MQ\) is \(163^\circ\), so the upper angles sum to \(180^\circ\). Wait, the upper angles: \(3x\) (between \(M\) and \(N\)), \(9x - 5\) (between \(N\) and \(P\)), and the angle between \(P\) and \(Q\) (let's call it \(y\)). But we don't know \(y\). Wait, no, maybe \(M\), \(S\), \(Q\) are colinear, so the sum of angles around \(S\) on the line \(MQ\) is \(180^\circ\) on each side. Wait, the angle below \(MQ\) is \(163^\circ\), so the angle above \(MQ\) (adjacent to \(163^\circ\)) is \(180 - 163 = 17^\circ\)? No, that's not. Wait, maybe the angles \(3x\), \(9x - 5\), and \(3x\)? No, I'm stuck. Wait, let's look at the problem again. The user provided the diagram with angles \(3x\), \(9x - 5\), and \(163^\circ\). Wait, maybe the sum of \(3x + (9x - 5) + 3x = 163\)? No, that doesn't make sense. Wait, no—wait, the correct equation: since \(M\), \(S\), \(Q\) are colinear, the sum of angles on one side (the left side: \(R\), \(M\), \(N\), \(P\)) and the right side (\(Q\))? Wait, I think I made a mistake. Let's try again.

Wait, the key is that the angles \(3x\), \(9x - 5\), and the angle that is supplementary to \(163^\circ\) (because \(163^\circ\) and that angle are adjacent to a straight line) sum to \(180^\circ\). Wait, \(180 - 163 = 17^\circ\), so \(3x + (9x - 5) + 17 = 180\)? No, that's not. Wait, no—wait, the diagram: the angles \(3x\), \(9x - 5\), and the angle between \(R\) and \(M\) (but \(R\) and \(M\) are both left, so maybe \(R\) and \(M\) are a straight line, so the angle between \(R\) and \(M\) is \(0\)? No, that can't be. Wait, maybe the sum of \(3x\), \(9x - 5\), and the angle adjacent to \(163^\circ\) (which is \(180 - 163 = 17^\circ\)) is \(180^\circ\)? Wait, \(3x + (9x - 5) + 17 = 180\)? Let's compute that: \(12x + 12 = 180\) → \(12x = 168\) → \(x = 14\). Wait, but let's check. If \(x = 14\), then \(3x = 42\), \(9x - 5 = 121\), and \(42 + 121 + 17 = 180\). Yes, that works. So the correct equation is \(3x + (9x - 5) + (180 - 163) = 180\). Wait, because the angle adjacent to \(163^\circ\) is \(180 - 163 = 17^\circ\) (since \(M\) and \(Q\) are colinear, so those two angles are supplementary). Then, the sum of \(3x\), \(9x - 5\), and \(17^\circ\) is \(180^\circ\) (since they are on the straight line \(MQ\), above it). So:

Step1: Calculate supplementary angle

The angle supplementary to \(163^\circ\) is \(180 - 163 = 17^\circ\).

Step2: Set up equation

The sum of angles \(3x\), \(9x - 5\), and \(17^\circ\) is \(180^\circ\) (since they lie on a straight line \(MQ\) above it). So:
\[ 3x + (9x - 5) + 17 = 180 \]

Step3: Simplify equation

Combine like terms:
\[ 12x + 12 = 180 \]

Step4: Solve for \(x\)

Subtract 12 from both sides:
\[ 12x = 168 \]
Divide by 12:
\[ x = 14 \]

Answer:

\(x = 14\)