Question

two parallel lines are crossed by a transversal. what is the value of b? m p b° 128° q 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, but here \(b\) and \(128^\circ\) are same - side exterior angles? Wait, no. Actually, when two parallel lines are cut by a transversal, alternate interior angles are equal, and also, if we look at the angles, \(b\) and the \(128^\circ\) angle are same - side interior angles? Wait, no, let's re - examine. The lines \(p\) and \(q\) are parallel, and \(m\) is the transversal. The angle \(b\) and the \(128^\circ\) angle are same - side interior angles? Wait, no, actually, if we consider the linear pair or the alternate interior angles. Wait, no, the correct relationship here is that \(b\) and \(128^\circ\) are same - side interior angles? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, no, in this case, the angle \(b\) and the \(128^\circ\) angle are actually same - side interior angles? Wait, no, let's think again. The lines \(p\) and \(q\) are parallel. The transversal \(m\) intersects them. The angle \(b\) and the \(128^\circ\) angle: if we look at the positions, \(b\) and the angle adjacent to \(128^\circ\) (linear pair) would be alternate interior angles. Wait, no, the \(128^\circ\) angle and \(b\) are same - side interior angles? Wait, no, the sum of same - side interior angles is \(180^\circ\)? Wait, no, I made a mistake. Actually, when two parallel lines are cut by a transversal, alternate interior angles are equal, and consecutive interior angles (same - side interior angles) are supplementary. But in this case, the angle \(b\) and the \(128^\circ\) angle are actually same - side interior angles? Wait, no, let's count the positions. The line \(p\) and \(q\) are parallel. The transversal \(m\) crosses them. The angle \(b\) is on line \(p\), and the \(128^\circ\) is on line \(q\). They are on the same side of the transversal. So, same - side interior angles are supplementary. Wait, but that would mean \(b + 128=180\), so \(b = 52\)? Wait, no, maybe I got the angle positions wrong. Wait, no, let's look at the diagram again. The angle \(b\) and the \(128^\circ\) angle: if we consider the vertical angles or alternate interior angles. Wait, no, the correct approach is: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but in the diagram, the angle \(b\) and the \(128^\circ\) angle are same - side interior angles. So \(b+128 = 180\), so \(b=180 - 128=52\)? Wait, no, that can't be. Wait, maybe I mixed up the angle types. Wait, no, another approach: the angle \(b\) and the angle adjacent to \(128^\circ\) (which is \(180 - 128 = 52^\circ\)) are alternate interior angles? No, that doesn't make sense. Wait, no, let's start over.

Two parallel lines \(p\) and \(q\), transversal \(m\). The angle \(128^\circ\) and angle \(b\): are they same - side interior angles? Let's recall the definition: same - side interior angles are two angles that lie between the two parallel lines and on the same side of the transversal. So, if \(p\) and \(q\) are parallel, and \(m\) is the transversal, then the angle \(b\) (on line \(p\)) and the \(128^\circ\) angle (on line \(q\)) are same - side interior angles. So, by the same - side interior angles theorem, they are supplementary. So, \(b + 128=180\).

Step2: Solve for \(b\)

We have the equation \(b + 128=180\). To solve for \(b\), we subtract \(128\) from both sides of the equation.

\(b=180 - 128\)

\(b = 52\)

Answer:

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