Question

solve for x.
(4x - 3)°
123°
x =
submit

Explanation:

Step1: Identify Parallel Lines and Corresponding Angles

Lines \( RS \) and \( TU \) are parallel, and \( PQ \) is a transversal. So, \( \angle RV P \) (which is \( (4x - 3)^\circ \)) and the angle corresponding to \( 123^\circ \) are equal (corresponding angles postulate).

Step2: Set Up the Equation

Since \( (4x - 3)^\circ = 123^\circ \), we solve for \( x \).
Add 3 to both sides: \( 4x = 123 + 3 \)
\( 4x = 126 \)
Divide both sides by 4: \( x=\frac{126}{4}= 31.5 \)? Wait, no, wait. Wait, actually, the angle at \( W \), the adjacent angle to \( 123^\circ \) on the straight line would be \( 180 - 123 = 57 \)? No, wait, no. Wait, looking at the diagram again: \( RS \parallel TU \), and \( PQ \) is the transversal. So \( \angle RV P \) and the angle at \( W \) that is equal to it. Wait, maybe I made a mistake. Wait, the angle \( (4x - 3)^\circ \) and the angle supplementary to \( 123^\circ \)? No, wait, no. Wait, \( RS \) and \( TU \) are parallel, so corresponding angles: \( \angle RV P = \angle TW Q \)? Wait, no, the angle at \( W \), the one adjacent to \( 123^\circ \) is \( 180 - 123 = 57 \)? No, that's not right. Wait, no, the angle \( (4x - 3)^\circ \) and the angle \( 123^\circ \) are equal because they are corresponding angles? Wait, no, maybe they are alternate exterior angles or something. Wait, let's re-examine.

Wait, \( RS \) is parallel to \( TU \), and \( PQ \) is the transversal. So \( \angle RV P \) (which is \( (4x - 3)^\circ \)) and the angle at \( W \) that is \( 123^\circ \) are equal? Wait, no, that can't be. Wait, maybe the angle \( (4x - 3)^\circ \) and the angle supplementary to \( 123^\circ \)? Wait, no, let's think again.

Wait, the line \( TU \) has a straight angle at \( W \), so the angle adjacent to \( 123^\circ \) is \( 180 - 123 = 57^\circ \)? No, that's not. Wait, no, the angle \( (4x - 3)^\circ \) and the angle \( 123^\circ \) are equal because \( RS \parallel TU \), so corresponding angles. Wait, maybe I messed up the diagram. Let me look again.

The diagram: \( RS \) is a horizontal line (left to right: \( R \) to \( V \) to \( S \)), \( TU \) is another horizontal line (left to right: \( T \) to \( W \) to \( U \)). The transversal \( PQ \) goes through \( V \) and \( W \), with \( P \) above \( V \) and \( Q \) below \( W \). The angle at \( V \) on \( RS \) is \( (4x - 3)^\circ \) (between \( RV \) and \( PV \)), and at \( W \) on \( TU \), the angle below \( TU \) (between \( WQ \) and \( TW \)) is \( 123^\circ \). So since \( RS \parallel TU \), the corresponding angles: \( \angle RV P \) (above \( RS \), between \( RV \) and \( PV \)) and the angle above \( TU \), between \( TW \) and \( PW \) (but \( PW \) is the same as \( PQ \)) would be equal. Wait, the angle above \( TU \) at \( W \) would be equal to \( (4x - 3)^\circ \), and the angle below \( TU \) is \( 123^\circ \), so those two angles (above and below \( TU \)) are supplementary? Wait, no, they are adjacent angles on a straight line, so they should add up to \( 180^\circ \). Wait, no, that's not. Wait, \( \angle RV P \) and the angle above \( TU \) at \( W \) are equal (corresponding angles), and that angle and \( 123^\circ \) are supplementary (since they form a linear pair). So \( (4x - 3) + 123 = 180 \)? Wait, that makes sense. Let's check:

If \( RS \parallel TU \), then \( \angle RV P \) (above \( RS \)) and the angle above \( TU \) at \( W \) are equal (corresponding angles). Then, that angle and \( 123^\circ \) (below \( TU \)) are adjacent and form a linear pair, so they add up to \( 180^\circ \). So:

\( (4x - 3) + 123 = 180 \)

Simplify:

\( 4x + 120 = 180 \)

\( 4x = 60 \)

\( x = 15 \)? Wait, that can't be. Wait, no, maybe I got the angles wrong.

Wait, let's start over. The key is that \( RS \parallel TU \), so the corresponding angles are equal. The angle \( (4x - 3)^\circ \) and the angle at \( W \) that is equal to it. Wait, the angle at \( W \), the one that is vertical to the angle adjacent to \( 123^\circ \)? No, this is confusing. Wait, maybe the angle \( (4x - 3)^\circ \) is equal to \( 123^\circ \) because they are corresponding angles. Let's try that:

\( 4x - 3 = 123 \)

\( 4x = 126 \)

\( x = 31.5 \). But that seems odd. Wait, maybe the diagram is such that \( (4x - 3)^\circ \) and \( 123^\circ \) are alternate interior angles? No, alternate interior angles would be between the two lines. Wait, \( RS \) and \( TU \) are parallel, so alternate interior angles: \( \angle RV W \) and \( \angle TW Q \)? No, maybe I need to look at the positions.

Wait, the angle \( (4x - 3)^\circ \) is at \( V \), between \( RV \) and \( PV \). The angle \( 123^\circ \) is at \( W \), between \( TW \) and \( QW \). Since \( RS \parallel TU \) and \( PQ \) is the transversal, these two angles are corresponding angles, so they should be equal. So:

\( 4x - 3 = 123 \)

\( 4x = 123 + 3 = 126 \)

\( x = 126 / 4 = 31.5 \). But that's a decimal. Wait, maybe the problem is that the angle \( (4x - 3)^\circ \) and the angle supplementary to \( 123^\circ \) are equal. Let's check:

Supplementary angle to \( 123^\circ \) is \( 180 - 123 = 57^\circ \). So:

\( 4x - 3 = 57 \)

\( 4x = 60 \)

\( x = 15 \). Ah, that makes sense. So why? Because the angle at \( W \), the one above \( TU \) (between \( TW \) and \( PW \)) is supplementary to \( 123^\circ \), and since \( RS \parallel TU \), that angle is equal to \( (4x - 3)^\circ \) (corresponding angles). So:

\( 4x - 3 = 180 - 123 \)

\( 4x - 3 = 57 \)

\( 4x = 60 \)

\( x = 15 \). Yes, that makes sense. So I made a mistake earlier in identifying the corresponding angle. The angle \( (4x - 3)^\circ \) is above \( RS \), and the angle above \( TU \) (adjacent to \( 123^\circ \)) is equal to it, and that angle is supplementary to \( 123^\circ \) because they form a linear pair. So:

Step1: Determine the Relationship

Since \( RS \parallel TU \) and \( PQ \) is a transversal, \( \angle RV P = \angle TW W' \) (where \( W' \) is the point above \( W \) on \( PQ \)), and \( \angle TW W' + 123^\circ = 180^\circ \) (linear pair). Thus, \( \angle RV P = 180^\circ - 123^\circ \).

Step2: Set Up the Equation

\( 4x - 3 = 180 - 123 \)

\( 4x - 3 = 57 \)

Step3: Solve for \( x \)

Add 3 to both sides: \( 4x = 57 + 3 = 60 \)

Divide by 4: \( x = \frac{60}{4} = 15 \)

Yes, that's correct. So the mistake earlier was not realizing that the angle \( (4x - 3)^\circ \) is equal to the supplementary angle of \( 123^\circ \) because of the parallel lines and transversal.

Answer:

\( x = 15 \)