Question

1. look at the diagram, diagram omitted which equation can be used to solve for x? 5x + 72 = 180 5x + 38 = 72 5x = 110 5x + 110 = 180 solve for x. x =

Explanation:

Part 1: Determine the equation to solve for \( x \)

Step1: Identify vertical angles or linear pairs

Angles \( \angle SWQ \) and \( \angle RW T \) are vertical angles? Wait, no. Wait, the straight line: Let's see, the angles on a straight line sum to \( 180^\circ \). Wait, the angles around point \( W \): Wait, the angle between \( QW \) and \( UW \) is \( 72^\circ \), between \( UW \) and \( TW \) is \( 5x^\circ \), and between \( TW \) and \( RW \) is \( 38^\circ \). Wait, actually, the angle opposite to \( \angle SWQ \) would be \( \angle RW T \)? Wait, no, let's look at the straight line. Wait, \( QW \) and \( RW \) are not a straight line. Wait, maybe \( SW \) and \( TW \) are a straight line? No, the diagram: \( S, W, T \) – wait, no, the points: \( S \) is on a line, \( Q \) is on a vertical line? Wait, maybe the angles \( 72^\circ \), \( 5x^\circ \), and \( 38^\circ \) are related to a straight line? Wait, no, the sum of angles on a straight line is \( 180^\circ \). Wait, actually, the angle \( 72^\circ + 5x^\circ + 38^\circ = 180^\circ \)? Wait, no, \( 72 + 38 = 110 \), so \( 5x + 110 = 180 \). Wait, let's check:

Wait, the angles adjacent to a straight line: If we consider the line that \( QW \) and \( RW \) are not on, but the line that \( UW \) and... Wait, maybe the angle \( \angle QWU = 72^\circ \), \( \angle UWT = 5x^\circ \), and \( \angle TWR = 38^\circ \), and \( \angle QWR \) is a straight line? Wait, no, \( QW \) and \( RW \) are opposite rays? Wait, the diagram: \( Q \) is up, \( R \) is down, so \( QW \) and \( RW \) are a straight line (vertical line). Then the angles on that straight line: \( \angle QWU = 72^\circ \), \( \angle UWT = 5x^\circ \), \( \angle TWR = 38^\circ \). So their sum should be \( 180^\circ \) (since they are on a straight line). So \( 72 + 5x + 38 = 180 \). Simplify \( 72 + 38 = 110 \), so \( 5x + 110 = 180 \). So the equation is \( 5x + 110 = 180 \).

Step2: Confirm the equation

So the correct equation is \( 5x + 110 = 180 \).

Part 2: Solve for \( x \)

Step1: Subtract 110 from both sides

Starting with \( 5x + 110 = 180 \), subtract 110 from both sides:
\( 5x + 110 - 110 = 180 - 110 \)
\( 5x = 70 \) Wait, no, wait: 180 - 110 is 70? Wait, no, 180 - 110 is 70? Wait, 110 + 70 = 180. Wait, but earlier I thought 72 + 38 is 110. Wait, 72 + 38 is 110, yes. So 5x + 110 = 180. Then 5x = 180 - 110 = 70? Wait, no, 180 - 110 is 70? Wait, 110 + 70 = 180. Then 5x = 70? Wait, no, wait, maybe I made a mistake. Wait, let's recalculate: 72 + 38 = 110. Then 5x + 110 = 180. So 5x = 180 - 110 = 70? Wait, 180 - 110 is 70? Yes. Then x = 70 / 5 = 14? Wait, but let's check again.

Wait, maybe the straight line is \( SW \) and \( TW \)? No, the diagram: \( S \) is on a line, \( T \) is on another. Wait, maybe the vertical angles: \( \angle SWQ \) and \( \angle RW T \) are vertical angles? Wait, \( \angle SWQ \) would be equal to \( \angle RW T \)? Wait, no, \( \angle RW T \) is 38°, so \( \angle SWQ \) is 38°? Then the angle between \( QW \) and \( UW \) is 72°, between \( UW \) and \( TW \) is 5x°, so the sum of \( 72 + 5x + 38 = 180 \)? Wait, no, if \( QW \) and \( RW \) are a straight line, then the angles on that line are \( \angle QWU = 72^\circ \), \( \angle UWT = 5x^\circ \), \( \angle TWR = 38^\circ \), so their sum is 180. So 72 + 5x + 38 = 180 → 5x + 110 = 180. Then solving:

Step1: Subtract 110 from both sides

\( 5x + 110 - 110 = 180 - 110 \)
\( 5x = 70 \)

Step2: Divide by 5

\( x = \frac{70}{5} = 14 \)

Answer:

(Equation):
\( 5x + 110 = 180 \)