Question

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

Explanation:

Step1: Identify angle relationship

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary (sum to \(180^\circ\)), and corresponding angles or alternate interior angles have specific relationships. Here, the \(105^\circ\) angle and the angle adjacent to \(g^\circ\) (let's call it \(x\)) are supplementary? Wait, no—actually, looking at the diagram, the \(105^\circ\) angle and the angle that would be same - side with \(g\) (but actually, the angle vertical to the one adjacent to \(105^\circ\) or maybe the consecutive interior? Wait, no, let's re - examine. The two parallel lines (u and w) are cut by transversal t? Wait, no, the transversal is the line with \(g^\circ\) and the other transversal? Wait, no, the two parallel lines are u and w, and the transversals are t and the line with \(g^\circ\). Wait, actually, the \(105^\circ\) angle and the angle that is supplementary to \(g\) (if we consider consecutive interior angles) or maybe the angle \(105^\circ\) and \(g\) are same - side exterior or something? Wait, no, let's think again. 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 say the angle between the transversal with \(g\) and line u) – no, actually, the \(105^\circ\) angle and \(g\) should be supplementary? Wait, no, \(180 - 105=75\)? No, that's not right. Wait, maybe the \(105^\circ\) angle and \(g\) are corresponding angles? No, wait, the correct relationship: if we have two parallel lines, and a transversal, then consecutive interior angles are supplementary. Wait, the \(105^\circ\) angle and the angle that is equal to \(g\) – no, wait, let's look at the diagram. The \(105^\circ\) angle and \(g\) are same - side exterior angles? No, actually, the angle adjacent to \(105^\circ\) (let's call it \(y\)) is \(180 - 105 = 75^\circ\), but that's not \(g\). Wait, no, maybe I made a mistake. Wait, the two parallel lines are u and w, and the transversal is the line with \(g\) and the other transversal is t. Wait, the \(105^\circ\) angle and \(g\) are actually same - side interior angles? No, same - side interior angles sum to \(180^\circ\). Wait, no, the correct approach: the angle \(105^\circ\) and the angle that is vertical to the angle adjacent to \(g\) – no, let's start over.

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Let's assume that the line with \(g^\circ\) and the line t are transversals cutting the parallel lines u and w. The \(105^\circ\) angle and \(g^\circ\) are same - side interior angles? Wait, no, if we look at the diagram, the \(105^\circ\) angle and \(g\) should be supplementary? Wait, \(180-105 = 75\)? No, that's not one of the options. Wait, no, maybe the \(105^\circ\) angle and \(g\) are corresponding angles? No, that would make \(g = 105\), but that's not supplementary. Wait, no, I think I messed up the diagram. Wait, the two parallel lines are the ones with the arrows (u and w), and the transversal is the line with \(g\) and the other transversal is t. The angle of \(105^\circ\) and \(g\) – actually, the angle adjacent to \(105^\circ\) (let's say the angle formed by transversal t and line u) and \(g\) are corresponding angles? No, wait, the correct relationship is that when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, the \(105^\circ\) angle and \(g\) are same - side exterior angles? No, same - side exterior angles also sum to \(180^\circ\). Wait, no, the answer options include \(g = 105\)? Wait, no, the options are \(g = 75\), \(g = 80\), \(g = 100\), \(g = 105\). Wait, maybe the \(105^\circ\) angle and \(g\) are vertical angles or corresponding angles. Wait, no, let's think about the diagram again. If we have two parallel lines, and a transversal, then alternate interior angles are equal, corresponding angles are equal, consecutive interior angles are supplementary.

Wait, maybe the \(105^\circ\) angle and \(g\) are same - side interior angles? No, that would mean \(105+g = 180\), so \(g = 75\), but that's option A. Wait, but maybe I got the diagram wrong. Wait, the user's diagram: two parallel lines (u and w) with arrows, transversal t and another transversal with \(g^\circ\). The angle of \(105^\circ\) is on transversal t and line t. Wait, maybe the \(105^\circ\) angle and \(g\) are supplementary? Wait, \(180 - 105=75\), so \(g = 75\)? But let's check the options. Option A is \(g = 75\). Wait, but maybe I made a mistake. Wait, no, let's recall: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. So if one angle is \(105^\circ\), the consecutive interior angle is \(180 - 105 = 75^\circ\), and if \(g\) is equal to that angle (by corresponding angles or alternate interior angles), then \(g = 75\).

Step2: Calculate \(g\)

Since consecutive interior angles are supplementary, and the angle supplementary to \(105^\circ\) is \(180 - 105=\boldsymbol{75}\), and \(g\) is equal to that angle (by the properties of parallel lines and transversals), so \(g = 75\).

Answer:

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