Question

two parallel lines are crossed by a transversal. what is the value of x? diagram: two parallel lines (s, r) with transversal t; 115° angle above line s, x° angle below line r. options: x = 45, x = 65, x = 95, x = 115

Explanation:

Step1: Identify angle relationship

When two parallel lines are cut by a transversal, corresponding angles are equal, and also, consecutive interior angles are supplementary, but here we can see that the \(115^\circ\) angle and \(x^\circ\) angle are actually equal because they are alternate interior angles (or corresponding angles in this case as the lines are parallel). Wait, no, actually, let's check the positions. Wait, the two lines \(s\) and \(r\) are parallel, and transversal \(t\) cuts them. The angle of \(115^\circ\) and \(x\) are same - side? No, wait, actually, when two parallel lines are cut by a transversal, alternate interior angles are equal, and also, if we look at the vertical angles or the corresponding angles. Wait, actually, the \(115^\circ\) angle and \(x\) are equal because they are corresponding angles (since lines \(s\) and \(r\) are parallel, and transversal \(t\) creates corresponding angles which are equal). Wait, no, wait, maybe I made a mistake. Wait, actually, the angle \(115^\circ\) and \(x\) are same - side? No, let's think again. The sum of consecutive interior angles is \(180^\circ\)? Wait, no, the \(115^\circ\) angle and \(x\) are actually equal because they are alternate interior angles. Wait, no, let's look at the diagram. The two parallel lines \(s\) and \(r\), transversal \(t\). The angle above line \(s\) is \(115^\circ\), and the angle below line \(r\) (wait, no, the angle at line \(r\) is \(x\). Wait, actually, the \(115^\circ\) angle and \(x\) are equal because they are corresponding angles. Wait, no, maybe the \(115^\circ\) and \(x\) are same - side exterior or interior? Wait, no, let's recall the properties. When two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, alternate exterior angles are equal, and consecutive interior angles are supplementary. In this case, the \(115^\circ\) angle and \(x\) are corresponding angles (or alternate interior angles), so they should be equal? Wait, no, that can't be. Wait, maybe the \(115^\circ\) and \(x\) are supplementary? Wait, no, let's calculate. If we consider that the angle adjacent to \(115^\circ\) on line \(s\) is \(180 - 115=65^\circ\), but that's not the case. Wait, no, the correct property here: when two parallel lines are cut by a transversal, the alternate interior angles are equal. Wait, the \(115^\circ\) angle and \(x\) are alternate interior angles? Wait, no, maybe the \(115^\circ\) and \(x\) are same - side interior angles? No, same - side interior angles sum to \(180^\circ\). Wait, I think I made a mistake. Let's look at the diagram again. The two parallel lines \(s\) and \(r\), transversal \(t\). The angle marked \(115^\circ\) is above line \(s\), and the angle marked \(x\) is below line \(r\)? No, wait, the arrows on the lines \(s\) and \(r\) are in the same direction, so they are parallel. The transversal \(t\) intersects them. The angle of \(115^\circ\) and \(x\) are actually equal because they are corresponding angles. Wait, no, that would mean \(x = 115\), but let's check the options. One of the options is \(x = 115\). Wait, but let's confirm. When two parallel lines are cut by a transversal, corresponding angles are equal. So if the angle on line \(s\) is \(115^\circ\), the corresponding angle on line \(r\) (which is \(x\)) should also be \(115^\circ\). So \(x=115\).

Step2: Confirm the angle relationship

Since lines \(s\) and \(r\) are parallel and cut by transversal \(t\), the angle \(x\) and the \(115^\circ\) angle are corresponding angles (or alternate interior angles), so they are equal. Thus, \(x = 115\).

Answer:

\(x = 115\) (the option: \(x = 115\))