Question

two parallel lines are crossed by a transversal. what is the value of x? 115° x° options: x = 45, x = 65, x = 95, x = 115

Explanation:

Step1: Identify Angle Relationship

The two lines \( s \) and \( r \) are parallel, and \( t \) is a transversal. The \( 115^\circ \) angle and \( x^\circ \) angle are same - side interior angles? No, wait, actually, the \( 115^\circ \) angle and the angle adjacent to \( x \) (on the same transversal) are supplementary? Wait, no, let's look again. Wait, the \( 115^\circ \) angle and \( x \) are actually same - side interior angles? No, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, no, wait the \( 115^\circ \) angle and \( x \): Wait, no, maybe I made a mistake. Wait, actually, the \( 115^\circ \) angle and \( x \) are same - side interior angles? Wait, no, let's think about the properties of parallel lines and transversals. The consecutive interior angles (same - side interior angles) are supplementary. But wait, in the diagram, the \( 115^\circ \) angle and \( x \): Wait, no, maybe the \( 115^\circ \) angle and \( x \) are actually equal? Wait, no, that can't be. Wait, no, wait the angle of \( 115^\circ \) and \( x \): Wait, maybe they are same - side interior angles? Wait, no, let's check the diagram again. The two parallel lines are \( s \) and \( r \), transversal \( t \). The \( 115^\circ \) angle is above line \( s \), and \( x \) is below line \( r \)? No, wait, the \( 115^\circ \) and \( x \) are on the same side of the transversal. Wait, no, actually, the \( 115^\circ \) angle and \( x \) are same - side interior angles, so they should be supplementary? Wait, no, that would mean \( x + 115=180 \), so \( x = 65 \)? Wait, no, that's not one of the options? Wait, no, wait I think I messed up the angle relationship. Wait, maybe the \( 115^\circ \) angle and \( x \) are alternate interior angles? No, alternate interior angles are equal. Wait, no, the \( 115^\circ \) angle and \( x \): Wait, maybe the \( 115^\circ \) angle and \( x \) are same - side interior angles? Wait, no, let's calculate. If two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \( x+115 = 180\), then \( x=180 - 115=65 \)? But wait, the options have \( x = 65 \) as one of the options (the second option: \( x = 65 \)). Wait, but let's confirm.

Wait, maybe I made a mistake in the angle relationship. Wait, the \( 115^\circ \) angle and \( x \) are same - side interior angles, so they are supplementary. So \( x=180 - 115 = 65 \).

Step2: Verify with Options

We calculated \( x = 65 \), and looking at the options, the second option is \( x = 65 \).

Answer:

\( x = 65 \) (corresponding to the option "x = 65")