Question

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

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \), the angle \( x^\circ \) and the \( 109^\circ \) angle are same - side interior angles? No, wait, actually, looking at the diagram, the angle \( x \) and the \( 109^\circ \) angle are supplementary? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, vertical angles and corresponding angles. Wait, actually, the angle \( x \) and the \( 109^\circ \) angle are same - side interior angles? No, let's think again. If we consider the transversal that cuts \( m \) and \( n \), the angle \( x \) and the \( 109^\circ \) angle: actually, \( x \) and \( 109^\circ \) are supplementary? Wait, no, when two parallel lines are cut by a transversal, consecutive interior angles (same - side interior angles) are supplementary. But also, if we look at the vertical angles or corresponding angles. Wait, actually, the angle \( x \) and the \( 109^\circ \) angle are same - side interior angles? Wait, no, let's see: the two parallel lines \( m \) and \( n \), and the transversal is the horizontal line. Then the angle \( x \) and the \( 109^\circ \) angle: since \( m\parallel n \), the angle \( x \) and \( 109^\circ \) are supplementary? Wait, no, actually, \( x \) and \( 109^\circ \) are same - side interior angles, so they should be supplementary? Wait, no, wait, maybe I made a mistake. Wait, the angle \( x \) and the \( 109^\circ \) angle: if we consider the linear pair or the parallel lines. Wait, actually, the angle \( x \) and \( 109^\circ \) are supplementary because they are same - side interior angles. Wait, no, same - side interior angles add up to \( 180^\circ \). Wait, but let's check: if \( m\parallel n \), then the angle adjacent to \( x \) (vertical angle or corresponding) and \( 109^\circ \) would be equal? Wait, no, maybe the angle \( x \) and \( 109^\circ \) are supplementary. Wait, let's calculate: \( x + 109=180 \), so \( x = 180 - 109 \).

Step2: Calculate the value of \( x \)

We know that for two parallel lines cut by a transversal, same - side interior angles are supplementary. So \( x+109 = 180 \).
To find \( x \), we solve the equation: \( x=180 - 109 \)
\( x = 71 \)

Answer:

\( 71 \)