Question

question 7. in the figure, line q is parallel to line r, and both lines are intersected by line s. if y = 2x + 8, what is the value of x? note: figure not drawn to scale. a. 57° b. 58° c. 122° d. 12°

Explanation:

Step1: Identify angle relationship

Since line \( q \parallel r \) and intersected by transversal \( s \), \( y \) and the \( 58^\circ \) angle are same - side interior angles? Wait, no, actually, the \( 58^\circ \) angle and the angle adjacent to \( y \) (vertical angle or corresponding angle) - wait, looking at the diagram, \( y \) and the \( 58^\circ \) angle are supplementary? Wait, no, let's correct. The \( 58^\circ \) angle and the angle that is supplementary to \( y \) (since \( q\parallel r \), consecutive interior angles are supplementary). Wait, actually, the \( 58^\circ \) angle and \( y \) are same - side interior angles? Wait, no, the \( 58^\circ \) angle and the angle adjacent to \( y \) (the one that forms a linear pair with \( y \)): Wait, no, let's think again. When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. The \( 58^\circ \) angle and \( y \) are same - side interior angles? Wait, no, the \( 58^\circ \) angle and the angle that is vertical to the angle adjacent to \( y \). Wait, maybe a better approach: The \( 58^\circ \) angle and \( y \) are supplementary? Wait, no, let's see the diagram. The \( 58^\circ \) angle is on line \( q \), and \( y \) is on line \( r \). Since \( q\parallel r \), the \( 58^\circ \) angle and \( y \) are same - side interior angles, so they should be supplementary? Wait, no, same - side interior angles are supplementary. Wait, but let's check the linear pair. Wait, actually, the \( 58^\circ \) angle and the angle that is vertical to the angle adjacent to \( y \): Wait, maybe I made a mistake. Let's look at the equation \( y = 2x + 8 \). Also, since \( q\parallel r \), the \( 58^\circ \) angle and \( y \) are supplementary? Wait, no, the \( 58^\circ \) angle and \( y \) are same - side interior angles, so \( 58 + y=180 \)? Wait, no, that can't be, because if \( y = 2x + 8 \), and if \( 58 + y = 180 \), then \( y=122 \), and then \( 122 = 2x + 8 \), \( 2x=114 \), \( x = 57 \). Wait, that makes sense. Let's verify:

Step1: Determine the relationship between \( y \) and \( 58^\circ \)

Since \( q\parallel r \) and cut by transversal \( s \), \( y \) and \( 58^\circ \) are same - side interior angles, so they are supplementary. So \( y+58 = 180 \), so \( y=180 - 58=122 \).

Step2: Solve for \( x \) using \( y = 2x + 8 \)

We know \( y = 122 \), so substitute into \( y = 2x+8 \):
\( 122=2x + 8 \)
Subtract 8 from both sides: \( 122 - 8=2x \)
\( 114 = 2x \)
Divide both sides by 2: \( x=\frac{114}{2}=57 \)

Answer:

\( 57^\circ \) (Option A)