Question

1. diagram: two parallel horizontal lines cut by a transversal (a slanted line). on the upper line, an angle formed with the transversal is labeled ( x + 66 ). on the lower line, an angle formed with the transversal is labeled ( 120^circ ).

Explanation:

Step1: Identify angle relationship

The two angles \( x + 66 \) and \( 120^\circ \) are same - side interior angles? No, wait, actually, since the lines are parallel (indicated by the arrows) and cut by a transversal, the angle \( x + 66 \) and \( 120^\circ \) are supplementary? Wait, no, let's check again. Wait, actually, if we look at the diagram, the angle \( x + 66 \) and \( 120^\circ \) are same - side interior angles? No, maybe they are equal? Wait, no, let's think about parallel lines and transversals. If two parallel lines are cut by a transversal, same - side interior angles are supplementary, alternate interior angles are equal, corresponding angles are equal. Wait, in this case, the angle \( x + 66 \) and \( 120^\circ \): let's see, if we consider the adjacent angles, maybe \( x + 66=120\)? Wait, no, maybe I made a mistake. Wait, actually, when two parallel lines are cut by a transversal, the sum of same - side interior angles is \( 180^\circ \)? No, wait, no, let's look at the diagram again. Wait, the angle \( x + 66 \) and \( 120^\circ \): maybe they are equal? Wait, no, let's set up the equation. Wait, if the two lines are parallel, then the angle \( x + 66 \) and \( 120^\circ \) are equal? Wait, no, maybe \( x + 66+120 = 180\)? No, that would be supplementary. Wait, let's check: if two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \( (x + 66)+120=180 \)? Wait, no, that would be if they are same - side interior. Wait, maybe the angle \( x + 66 \) and \( 120^\circ \) are equal? Wait, no, let's solve for \( x \). Wait, maybe the correct relationship is \( x + 66 = 120 \)? No, that would give \( x = 54 \), but let's check. Wait, no, maybe the angle \( x + 66 \) and \( 120^\circ \) are supplementary? Wait, \( x+66 + 120=180\), then \( x+186 = 180\), which would give \( x=-6\), which doesn't make sense. Wait, maybe I got the angle relationship wrong. Wait, actually, the angle \( x + 66 \) and the angle adjacent to \( 120^\circ \) (the vertical angle or alternate interior angle) are equal. Wait, the angle adjacent to \( 120^\circ \) is \( 60^\circ \) (since \( 180 - 120 = 60 \))? No, that's not right. Wait, no, let's start over.

Wait, the two lines are parallel, cut by a transversal. The angle \( x + 66 \) and \( 120^\circ \): if we look at the diagram, maybe \( x + 66=120 \), so we can solve for \( x \).

Step2: Solve for \( x \)

If \( x + 66=120 \), then subtract 66 from both sides:
\( x=120 - 66 \)
\( x = 54 \)

Wait, but let's check the angle relationship again. If the two lines are parallel, and the transversal cuts them, then the angle \( x + 66 \) and \( 120^\circ \) are corresponding angles or alternate interior angles, so they are equal. So the equation \( x + 66 = 120 \) is correct.

Answer:

\( x = 54 \)