Question

find the value of x so that b || c.
(5x + 3)°
32°
x =

Explanation:

Step1: Identify the angle relationship

For lines \( b \parallel c \), the corresponding angles should be equal. So, \( 5x + 3 = 32 + 180 \)? Wait, no, actually, when lines are parallel, the alternate interior angles or corresponding angles. Wait, looking at the diagram, the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—wait, no, actually, if \( b \parallel c \), then the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle should be equal? Wait, no, maybe it's a corresponding angle or alternate interior. Wait, actually, when two lines are parallel, the corresponding angles are equal. Wait, maybe the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—wait, no, let's re-examine. Wait, the transversal cuts through \( b \) and \( c \). So, if \( b \parallel c \), then the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle should be equal? Wait, no, maybe it's a supplementary angle? Wait, no, let's think again. Wait, the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—if \( b \parallel c \), then the angle \( (5x + 3)^\circ \) should be equal to \( 32^\circ + 180^\circ \)? No, that can't be. Wait, maybe I made a mistake. Wait, actually, when two lines are parallel, the alternate interior angles are equal. Wait, maybe the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—wait, no, let's look at the diagram. The line \( a \) is a transversal, cutting \( b \) and \( c \). So, if \( b \parallel c \), then the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle should be equal? Wait, no, maybe it's a corresponding angle. Wait, actually, the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—if \( b \parallel c \), then \( 5x + 3 = 32 + 180 \)? No, that's not right. Wait, maybe the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle are equal. Wait, no, let's check the diagram again. Wait, the line \( b \) is parallel to \( c \), so the transversal (line \( a \)) creates corresponding angles. So, the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—wait, maybe the angle \( (5x + 3)^\circ \) is equal to \( 32^\circ \)? No, that would give \( 5x + 3 = 32 \), \( 5x = 29 \), \( x = 5.8 \), which doesn't seem right. Wait, maybe it's a supplementary angle. Wait, if \( b \parallel c \), then the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle are supplementary? Wait, no, that would be \( 5x + 3 + 32 = 180 \), so \( 5x + 35 = 180 \), \( 5x = 145 \), \( x = 29 \). Wait, that makes more sense. Wait, why? Because if \( b \parallel c \), then the consecutive interior angles are supplementary. Wait, let's see: the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle—if they are consecutive interior angles, then they should add up to \( 180^\circ \). Wait, no, maybe the angle \( (5x + 3)^\circ \) and the \( 32^\circ \) angle are same-side interior angles, so they are supplementary. So, \( 5x + 3 + 32 = 180 \)? Wait, no, \( 5x + 3 + 32 = 180 \)? Wait, \( 5x + 35 = 180 \), \( 5x = 145 \), \( x = 29 \). Wait, but let's check again. Wait, the diagram: line \( b \) and \( c \) are parallel, transversal is line \( a \). The angle \( (5x + 3)^\circ \) is on line \( b \), and the \( 32^\circ \) is on line \( c \). So, if they are same-side interior angles, then they are supplementary. So, \( 5x + 3 + 32 = 180 \)? Wait, no, \( 5x + 3 = 180 - 32 \)? Wait, \( 180 - 32 = 148 \), so \( 5x + 3 = 148 \), \( 5x = 145 \), \( x = 29 \). Yes, that's correct. So, the equation is \( 5x + 3 = 180 - 32 \)? Wait, no, same-side interior angles are supplementary, so \( (5x + 3) + 32 = 180 \)? Wait, no, \( 5x + 3 = 180 - 32 \), because \( 32 + (5x + 3) = 180 \). So, \( 5x + 3 = 148 \), \( 5x = 145 \), \( x = 29 \).

Step2: Solve the equation

So, we have the equation \( 5x + 3 = 180 - 32 \). Wait, \( 180 - 32 = 148 \), so \( 5x + 3 = 148 \). Subtract 3 from both sides: \( 5x = 148 - 3 = 145 \). Then divide both sides by 5: \( x = \frac{145}{5} = 29 \).

Answer:

\( x = 29 \)