Question

find the value of x so that b || c.
(3x - 1)°
91°
x =

Explanation:

Step1: Identify angle relationship

For lines \( b \parallel c \) with transversal \( a \), the given angles \( (3x - 1)^\circ \) and \( 91^\circ \) are same - side interior angles? No, wait, actually, when two lines are parallel and cut by a transversal, same - side interior angles are supplementary? Wait, no, looking at the diagram, the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are actually same - side interior angles? Wait, no, if \( b\parallel c \), and \( a \) is the transversal, then the two angles \( (3x - 1)^\circ \) and \( 91^\circ \) should be supplementary? Wait, no, wait, actually, if we consider the linear pair or the fact that for parallel lines, consecutive interior angles are supplementary. Wait, let's re - examine. The angle \( (3x - 1)^\circ \) and \( 91^\circ \): if \( b\parallel c \), then these two angles should be supplementary? Wait, no, maybe they are same - side interior angles. Wait, no, actually, when two lines are parallel, same - side interior angles are supplementary. So \( (3x - 1)+91 = 180 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, looking at the diagram, the two angles \( (3x - 1)^\circ \) and \( 91^\circ \): if \( b\parallel c \), then the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are actually equal? Wait, no, that would be alternate interior angles. Wait, no, let's think again. Wait, the transversal \( a \) is a vertical line (assuming the horizontal lines are \( b \) and \( c \)). So the angle \( (3x - 1)^\circ \) and \( 91^\circ \): if \( b\parallel c \), then the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are same - side interior angles? Wait, no, maybe they are supplementary. Wait, let's check the sum. If \( (3x - 1)+91=180 \), then \( 3x+90 = 180 \), \( 3x = 90 \), \( x = 30 \). But wait, that doesn't seem right. Wait, maybe they are equal. Wait, if \( b\parallel c \), and the transversal is \( a \), then the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are corresponding angles? Wait, no, the angle \( (3x - 1)^\circ \) is above line \( c \) and to the right of transversal \( a \), and the angle \( 91^\circ \) is above line \( b \) and to the right of transversal \( a \). Wait, no, line \( b \) is below line \( c \). Wait, maybe the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are same - side interior angles. Wait, I think I messed up. Wait, let's start over.

Wait, the correct approach: For two parallel lines \( b \) and \( c \) cut by a transversal \( a \), the consecutive interior angles are supplementary. So the angle \( (3x - 1)^\circ \) and \( 91^\circ \) are consecutive interior angles. So their sum should be \( 180^\circ \).

So, \( (3x - 1)+91=180 \)

Step2: Solve the equation

Simplify the left - hand side of the equation: \( 3x-1 + 91=3x + 90 \)

So we have the equation \( 3x+90 = 180 \)

Subtract 90 from both sides: \( 3x=180 - 90=90 \)

Divide both sides by 3: \( x=\frac{90}{3}=30 \)

Wait, but let's check again. Wait, maybe the angles are equal. If \( (3x - 1)=91 \), then \( 3x=92 \), \( x=\frac{92}{3}\approx30.67 \), which doesn't seem right. Wait, maybe my initial assumption about the angle relationship is wrong. Wait, let's look at the diagram again. The line \( a \) is a vertical transversal (assuming the horizontal lines are \( b \) and \( c \)). The angle \( (3x - 1)^\circ \) is between line \( c \) and transversal \( a \), and the angle \( 91^\circ \) is between line \( b \) and transversal \( a \). If \( b\parallel c \), then these two angles should be supplementary because they are same - side interior angles. So \( (3x - 1)+91 = 180 \)

\( 3x-1+91 = 180 \)

\( 3x + 90=180 \)

\( 3x=180 - 90 \)

\( 3x = 90 \)

\( x = 30 \)

Answer:

\( x = 30 \)