Question

lines m and l are parallel. what is the measure of ∠x? 80 100 110 70

Explanation:

Step1: Identify the angle relationship

Since lines \( m \) and \( l \) are parallel, and we have a transversal, we can use the property of supplementary angles (if consecutive interior angles or corresponding/alternate angles with a linear pair). The given angle is \( 110^\circ \), and \( \angle x \) and this \( 110^\circ \) angle are supplementary (they form a linear pair or are same - side interior angles? Wait, actually, if we look at the parallel lines and transversal, the angle adjacent to \( 110^\circ \) (vertical or linear pair) and \( \angle x \) would be related. Wait, more accurately, if two parallel lines are cut by a transversal, same - side interior angles are supplementary. But also, a linear pair of angles sums to \( 180^\circ \). Let's assume the given \( 110^\circ \) angle and the angle supplementary to \( \angle x \) are equal (corresponding angles). So, if one angle is \( 110^\circ \), then \( \angle x + 110^\circ= 180^\circ \)? Wait, no, maybe the angle adjacent to \( 110^\circ \) (vertical angle) is \( 110^\circ \), and \( \angle x \) and that angle are same - side interior angles? Wait, maybe a better approach: the angle given as \( 110^\circ \) and \( \angle x \) are supplementary? Wait, no, let's think again. If lines are parallel, alternate interior angles are equal, corresponding angles are equal, and same - side interior angles are supplementary. Let's suppose that the angle with measure \( 110^\circ \) and \( \angle x \) are same - side interior angles? No, wait, maybe the angle vertical to the \( 110^\circ \) angle is \( 110^\circ \), and \( \angle x \) and that angle are supplementary. So \( \angle x=180 - 110 = 70^\circ \)? Wait, no, the options include 70, 80, 100, 110. Wait, maybe I made a mistake. Wait, the given angle is \( 110^\circ \), and \( \angle x \) and the angle that is supplementary to \( 110^\circ \) (i.e., \( 70^\circ \)) are equal? No, wait, maybe the angle adjacent to \( 110^\circ \) (linear pair) is \( 70^\circ \), and \( \angle x \) is equal to that \( 70^\circ \) angle (alternate interior angles). So \( \angle x = 70^\circ \)? Wait, no, the options have 70 as one of them. Wait, let's recast: if two parallel lines are cut by a transversal, and we have an angle of \( 110^\circ \), then the angle supplementary to \( 110^\circ \) is \( 180 - 110=70^\circ \). And if \( \angle x \) is equal to that supplementary angle (by alternate interior angles or corresponding angles), then \( \angle x = 70^\circ \). But wait, the options are 80, 100, 110, 70. Wait, maybe the given angle is not \( 110^\circ \) but another angle. Wait, the image shows a \( 110^\circ \) angle, and the options are 80, 100, 110, 70. Wait, maybe the angle is \( 110^\circ \), and \( \angle x \) is equal to \( 70^\circ \) because \( 180 - 110 = 70 \).

Step2: Calculate \( \angle x \)

Using the linear pair or supplementary angle property: If two angles form a linear pair (sum to \( 180^\circ \)) and one angle is \( 110^\circ \), then the other angle (which is related to \( \angle x \) by parallel lines) is \( 180 - 110=70^\circ \). So \( \angle x = 70^\circ \). Wait, but let's check the options. The options are 80, 100, 110, 70. So 70 is an option.

Answer:

70