Question

1. given ( m parallel n ), find the value of ( x ).

Explanation:

Step1: Identify the relationship

Since \( m \parallel n \), the angle of \( 118^\circ \) and \( x^\circ \) are same - side interior angles? Wait, no, actually, the angle adjacent to \( 118^\circ \) and \( x \) should be supplementary? Wait, no, let's look at the diagram again. The \( 118^\circ \) angle and the angle that is vertical or corresponding? Wait, actually, the \( 118^\circ \) and \( x \) are same - side interior angles? No, wait, the correct relationship: when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, the angle supplementary to \( 118^\circ \) and \( x \) have a relationship. Wait, the angle adjacent to \( 118^\circ \) (linear pair) is \( 180 - 118=62^\circ \)? No, wait, no. Wait, the \( 118^\circ \) and \( x \) are actually same - side interior angles? Wait, no, let's think again. The two parallel lines \( m \) and \( n \), cut by a transversal. The \( 118^\circ \) angle and \( x \) angle: since they are on the same side of the transversal and inside the two parallel lines? No, wait, the \( 118^\circ \) and \( x \) are actually supplementary? Wait, no, the correct approach: the angle of \( 118^\circ \) and \( x \) are same - side interior angles? Wait, no, the angle that is equal to \( x \) should be the supplementary angle of \( 118^\circ \)? Wait, no, let's recall: when two parallel lines are cut by a transversal, consecutive interior angles (same - side interior angles) are supplementary. But also, the angle \( 118^\circ \) and the angle adjacent to \( x \) (vertical angle or corresponding) – wait, actually, the \( 118^\circ \) and \( x \) are same - side interior angles? Wait, no, the sum of same - side interior angles is \( 180^\circ \)? No, wait, no: same - side interior angles are supplementary, meaning their sum is \( 180^\circ \). Wait, but in the diagram, the \( 118^\circ \) and \( x \) – wait, maybe I made a mistake. Wait, the correct relationship: the angle of \( 118^\circ \) and \( x \) are same - side interior angles, so \( 118 + x=180 \)? No, that would be if they are same - side interior. Wait, no, let's look at the diagram again. The two parallel lines \( m \) and \( n \), the transversal cuts them. The \( 118^\circ \) angle and \( x \) angle: actually, the \( 118^\circ \) and \( x \) are supplementary? Wait, no, the angle that is vertical to the angle supplementary to \( 118^\circ \). Wait, the linear pair of \( 118^\circ \) is \( 180 - 118 = 62^\circ \), but that's not right. Wait, no, the correct way: since \( m\parallel n \), the angle \( x \) and the angle supplementary to \( 118^\circ \) are equal? No, wait, let's use the consecutive interior angles theorem. Consecutive interior angles are supplementary. The \( 118^\circ \) angle and \( x \) are consecutive interior angles? Wait, no, the \( 118^\circ \) angle and \( x \) – let's see, the transversal cuts \( m \) and \( n \). The \( 118^\circ \) is above \( m \), and \( x \) is below \( n \) on the same side of the transversal. So they are same - side interior angles, so their sum is \( 180^\circ \)? Wait, no, that would mean \( x = 180 - 118=62 \)? No, wait, no, maybe I got the diagram wrong. Wait, actually, the \( 118^\circ \) and \( x \) are alternate interior angles? No, alternate interior angles are equal. Wait, maybe the \( 118^\circ \) and \( x \) are same - side interior angles, so \( x = 180 - 118 = 62 \)? Wait, no, let's check again. The two parallel lines, transversal. The angle of \( 118^\circ \) and \( x \): if we look at the diagram, the \( 118^\circ \) angle and \( x \) are same - side interior angles, so they should be supplementary. So \( x=180 - 118 \).

Step2: Calculate \( x \)

\( x = 180-118 \)
\( x = 62 \)

Answer:

\( x = 62 \)