Question

two parallel lines are crossed by a transversal. what is the value of g? \\( g = 75 \\) \\( g = 80 \\) \\( g = 100 \\) \\( g = 105 \\) (diagram shows angle 105° and lines t, u, w with arrows)

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, the \(105^\circ\) angle and the angle adjacent to \(g\) (let's see the diagram: the \(105^\circ\) and the angle that would be same - side with \(g\)? Wait, no, actually, looking at the diagram, the \(105^\circ\) angle and \(g\) are same - side interior angles? Wait, no, maybe alternate interior or corresponding? Wait, no, let's think again. Wait, the two parallel lines are \(u\) and \(w\), cut by transversal \(t\)? Wait, no, the transversal is the line with the \(105^\circ\) and the other line. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, vertical angles or corresponding angles. Wait, no, the \(105^\circ\) angle and \(g\): wait, maybe the \(105^\circ\) and \(g\) are same - side interior angles? No, wait, actually, the angle adjacent to \(105^\circ\) (let's say the angle that is supplementary to \(105^\circ\)) and \(g\) are corresponding angles. Wait, the angle supplementary to \(105^\circ\) is \(180 - 105=75^\circ\)? No, that's not right. Wait, no, let's look at the diagram again. The two parallel lines ( \(u\) and \(w\)) are cut by a transversal (the line with the \(105^\circ\) angle and the other transversal? Wait, no, the problem says two parallel lines are crossed by a transversal. So the two parallel lines are, say, the two lines with the arrows ( \(u\) and \(w\)), and the transversal is the line \(t\) and the other line. Wait, the \(105^\circ\) angle and \(g\): actually, \(g\) and the \(105^\circ\) angle are same - side interior angles? No, wait, no. Wait, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, no, the \(105^\circ\) angle and \(g\): wait, maybe the \(105^\circ\) angle and \(g\) are supplementary? No, that would be if they are same - side. Wait, no, let's calculate. Wait, the angle that is vertical to the angle supplementary to \(105^\circ\)? No, maybe I made a mistake. Wait, the correct approach: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, the \(105^\circ\) angle and the angle that is adjacent to \(g\) (let's call it \(x\)): if \(u\parallel w\), then \(105^\circ + x=180^\circ\), so \(x = 75^\circ\)? No, that's not. Wait, no, actually, \(g\) and the \(105^\circ\) angle: wait, maybe \(g\) is equal to \(105^\circ\)? No, that can't be. Wait, no, let's think again. Wait, the two parallel lines are \(u\) and \(w\), and the transversal is the line that makes the \(105^\circ\) angle and the line with \(g\). Wait, the \(105^\circ\) angle and \(g\) are same - side interior angles? No, same - side interior angles sum to \(180^\circ\), so \(g + 105^\circ=180^\circ\)? No, that would give \(g = 75^\circ\), but that's not one of the options? Wait, no, the options are \(g = 75\), \(80\), \(100\), \(105\). Wait, maybe I misidentified the angle. Wait, maybe the \(105^\circ\) angle and \(g\) are corresponding angles? No, corresponding angles are equal. Wait, no, maybe the angle that is vertical to the \(105^\circ\) angle and \(g\) are supplementary. Wait, the vertical angle of \(105^\circ\) is also \(105^\circ\), and if that vertical angle and \(g\) are same - side interior angles, then \(105^\circ+g = 180^\circ\), so \(g = 75^\circ\)? But \(g = 75\) is an option. Wait, but let's check the diagram again. Wait, the user's diagram: two parallel lines ( \(u\) and \(w\)) with arrows, and a transversal (line \(t\)) and another transversal. The \(105^\circ\) is on line \(t\), and \(g\) is on the other transversal, between the two parallel lines. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So the angle of \(105^\circ\) and \(g\) are same - side interior angles? No, wait, maybe the \(105^\circ\) angle and \(g\) are supplementary? Wait, \(180 - 105=75\), so \(g = 75\)? But let's check the options. The first option is \(g = 75\). Wait, maybe I was wrong earlier. Let's do it step by step.

Step1: Recall same - side interior angles theorem

When two parallel lines are cut by a transversal, same - side interior angles are supplementary (their sum is \(180^\circ\)).

Step2: Identify the angles

Let the two parallel lines be \(u\) and \(w\), and the transversal be the line that creates the \(105^\circ\) angle and the angle \(g\). The \(105^\circ\) angle and \(g\) are same - side interior angles. So we have the equation:

\(105^\circ+g = 180^\circ\)

Step3: Solve for \(g\)

Subtract \(105^\circ\) from both sides of the equation:

\(g=180^\circ - 105^\circ\)

\(g = 75^\circ\)

Answer:

\(g = 75\) (corresponding to the option \(g = 75\))