Question

two parallel lines are crossed by a transversal. what is the value of b?

b = 32

b = 52

b = 118

b = 128

Explanation:

Step1: Identify angle relationship

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary (sum to \(180^\circ\))? Wait, no, actually, here \(b\) and the \(128^\circ\) angle—wait, no, looking at the diagram, lines \(p\) and \(q\) are parallel, transversal \(m\). The angle \(b\) and the \(128^\circ\) angle—wait, no, actually, they are same - side interior? Wait, no, maybe alternate exterior? Wait, no, let's see: if we look at the angles, \(b\) and the angle adjacent to \(128^\circ\) (the linear pair) would be equal? Wait, no, actually, \(b\) and \(128^\circ\) are same - side interior? Wait, no, let's recall: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, no, in this case, \(b\) and \(128^\circ\) are same - side interior? Wait, no, maybe \(b\) and \(128^\circ\) are supplementary? Wait, no, wait the diagram: line \(p\) and \(q\) are parallel, transversal \(m\). The angle \(b\) is on line \(p\), and the \(128^\circ\) is on line \(q\). Wait, actually, \(b\) and \(128^\circ\) are same - side interior angles? No, wait, no—wait, the angle \(b\) and the angle that is supplementary to \(128^\circ\)? Wait, no, let's calculate: if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Wait, \(b + 128^\circ=180^\circ\)? No, that would give \(b = 52^\circ\)? Wait, no, wait maybe I got the relationship wrong. Wait, no, looking at the diagram again: the angle \(b\) and the \(128^\circ\) angle—wait, maybe they are same - side interior? Wait, no, let's think again. The sum of same - side interior angles is \(180^\circ\). So \(b+128 = 180\)? Then \(b=180 - 128=52\)? Wait, no, that can't be. Wait, no, maybe the angle \(b\) and the \(128^\circ\) angle are alternate exterior? No, wait, maybe I made a mistake. Wait, no, let's check the options. The options are 32, 52, 118, 128. Wait, if \(b\) and \(128^\circ\) are supplementary, then \(b = 180 - 128 = 52\)? Wait, no, wait the diagram: line \(p\) and \(q\) are parallel, transversal \(m\). The angle \(b\) is on the upper parallel line, and the \(128^\circ\) is on the lower parallel line. Wait, actually, \(b\) and \(128^\circ\) are same - side interior angles, so they should be supplementary. So \(b=180 - 128 = 52\)? Wait, but let's confirm. Consecutive interior angles (same - side interior angles) are supplementary when lines are parallel. So yes, \(b + 128^\circ=180^\circ\), so \(b = 180 - 128=52\).

Step2: Calculate \(b\)

We know that for two parallel lines cut by a transversal, consecutive interior angles are supplementary. So we have the equation:
\(b + 128^\circ=180^\circ\)
To find \(b\), we solve for \(b\):
\(b=180^\circ - 128^\circ\)
\(b = 52^\circ\)

Answer:

\(b = 52\) (corresponding to the option \(b = 52\))