Question

using vertical angles to find unknown values
what is ( mangle srw )?
54
108
126
18

Explanation:

Step1: Identify vertical angles

The angles \((2x + 18)^\circ\) and \((3x)^\circ\) are vertical angles? Wait, no, actually, looking at the diagram, \(\angle TRS\) (which is \((2x + 18)^\circ\)) and \(\angle VRW\) (which is \((3x)^\circ\))? Wait, no, maybe \(\angle TRS\) and \(\angle VRW\) are not vertical. Wait, actually, the angle \((2x + 18)^\circ\) and \((3x)^\circ\) are adjacent to a straight line? Wait, no, let's correct. The angles \((2x + 18)^\circ\) and \((3x)^\circ\) are supplementary? Wait, no, vertical angles are equal. Wait, maybe \(\angle TRS\) (which is \((2x + 18)^\circ\)) and \(\angle VRW\) (which is \((3x)^\circ\)) are vertical angles? Wait, no, looking at the diagram, the lines \(TS\) and \(VW\) intersect at \(R\), so \(\angle TRS\) and \(\angle VRW\) are vertical angles? Wait, no, \(\angle TRS\) is \((2x + 18)^\circ\) and \(\angle VRW\) is \((3x)^\circ\)? Wait, no, maybe \(\angle TRS\) and \(\angle VRW\) are vertical angles, so they should be equal? Wait, no, that can't be. Wait, maybe \(\angle TRS\) and \(\angle SRW\) are supplementary? Wait, no, let's re-examine.

Wait, the angle \((2x + 18)^\circ\) and \((3x)^\circ\) are adjacent to form a linear pair? Wait, no, the lines \(TS\) and \(VW\) intersect at \(R\), so \(\angle TRS\) and \(\angle VRW\) are vertical angles, so they are equal? Wait, no, that would mean \(2x + 18 = 3x\), so \(x = 18\). Then \(\angle VRW = 3x = 54^\circ\), but \(\angle SRW\) is supplementary to \(\angle TRS\)? Wait, no, \(\angle TRS\) is \((2x + 18)^\circ = 2*18 + 18 = 54^\circ\), so \(\angle SRW\) is \(180^\circ - 54^\circ = 126^\circ\). Wait, let's do it step by step.

Step1: Set vertical angles equal (wait, no, maybe \((2x + 18)^\circ\) and \((3x)^\circ\) are vertical angles? Wait, no, looking at the diagram, the angle \((2x + 18)^\circ\) is at \(R\) between \(T\) and \(S\), and \((3x)^\circ\) is between \(V\) and \(W\). Wait, actually, the lines \(TV\) and \(SW\) intersect? No, the lines are \(TS\) (with points \(T\), \(R\), \(S\)) and \(VW\) (with points \(V\), \(R\), \(W\)). So \(\angle TRV\) and \(\angle SRW\) are vertical angles, and \(\angle TRS\) (which is \((2x + 18)^\circ\)) and \(\angle VRW\) (which is \((3x)^\circ\)) are vertical angles. Wait, no, vertical angles are opposite each other when two lines intersect. So when two lines intersect, the opposite angles are vertical angles and are equal. So if lines \(TS\) and \(VW\) intersect at \(R\), then \(\angle TRS\) and \(\angle VRW\) are vertical angles, so they are equal. Therefore:

\(2x + 18 = 3x\)

Step2: Solve for \(x\)

Subtract \(2x\) from both sides:

\(18 = x\)

So \(x = 18\)

Step3: Find \(m\angle SRW\)

Now, \(\angle SRW\) is supplementary to \(\angle TRS\) because they form a linear pair (they are adjacent and form a straight line). So:

\(m\angle SRW + m\angle TRS = 180^\circ\)

We know \(m\angle TRS = 2x + 18 = 2*18 + 18 = 54^\circ\)

Therefore:

\(m\angle SRW = 180^\circ - 54^\circ = 126^\circ\)

Answer:

126