Question

lines l, m, and n lie in a plane and are cut by a transversal, t. ∠1 is supplementary to ∠2, and ∠2 is supplementary to ∠3. which lines, if any, are parallel? m and n m and l none all three lines

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles sum to \(180^\circ\). So, \(\angle1 + \angle2 = 180^\circ\) (given \(\angle1\) supplementary to \(\angle2\)) and \(\angle2 + \angle3 = 180^\circ\) (given \(\angle2\) supplementary to \(\angle3\)).

Step2: Transitive Property of Equality

From \(\angle1 + \angle2 = 180^\circ\) and \(\angle2 + \angle3 = 180^\circ\), we can set them equal: \(\angle1 + \angle2=\angle2 + \angle3\). Subtract \(\angle2\) from both sides: \(\angle1=\angle3\).

Step3: Corresponding Angles and Parallel Lines

\(\angle1\) and \(\angle3\) are corresponding angles (if we consider lines \(m\) and \(n\) cut by transversal \(t\)). When corresponding angles are equal, the lines are parallel. So, \(m \parallel n\). Wait, but let's check again. Wait, \(\angle1\) and \(\angle2\) are adjacent supplementary angles (linear pair), so that's standard. Then \(\angle2\) and \(\angle3\) being supplementary implies \(\angle1=\angle3\) (since both are supplementary to \(\angle2\)). So \(\angle1\) and \(\angle3\) are corresponding angles for lines \(m\) and \(n\) with transversal \(t\), so \(m \parallel n\). Wait, but the options have "m and n". Wait, maybe I misread. Wait, the lines are \(l\), \(m\), \(n\). Let's see the diagram: \(m\) and \(n\) have angles \(\angle1\) (with \(m\)) and \(\angle3\) (with \(n\)) related. So if \(\angle1=\angle3\) (corresponding angles), then \(m \parallel n\).

Wait, but let's re-express:

Given \(\angle1\) supplementary to \(\angle2\): \(\angle1 + \angle2 = 180^\circ\)

\(\angle2\) supplementary to \(\angle3\): \(\angle2 + \angle3 = 180^\circ\)

So, \(\angle1 = 180^\circ - \angle2\) and \(\angle3 = 180^\circ - \angle2\), so \(\angle1 = \angle3\). These are corresponding angles for lines \(m\) and \(n\) (since \(t\) is the transversal), so by Corresponding Angles Postulate, \(m \parallel n\).

Wait, but the options include "m and n". So the correct answer should be "m and n".

Wait, maybe I made a mistake earlier. Let's check the diagram again. The lines \(m\), \(l\), \(n\) with transversal \(t\). \(\angle1\) is between \(m\) and \(t\), \(\angle2\) between \(m\) and \(l\)? No, wait, the diagram: \(t\) is the horizontal transversal. \(m\) is the left line, \(l\) is the middle, \(n\) is the right. \(\angle1\) is between \(m\) and \(t\) (below \(t\)? Wait, no, the arrows: \(m\) and \(n\) are slanting, \(l\) is vertical? Wait, maybe the angles: \(\angle1\) and \(\angle2\) are adjacent (linear pair), so \(\angle1 + \angle2 = 180^\circ\). Then \(\angle2\) and \(\angle3\) are supplementary, so \(\angle1 = \angle3\) (since both are supplementary to \(\angle2\)). So \(\angle1\) and \(\angle3\) are corresponding angles for lines \(m\) and \(n\) (since they are in the same position relative to transversal \(t\) and lines \(m\) and \(n\)), so \(m \parallel n\).

So the correct option is "m and n".

Answer:

m and n