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? what justifies your answer?

Explanation:

Step1: Use the property of supplementary angles

If \(\angle1\) is supplementary to \(\angle2\), then \(\angle1+\angle2 = 180^{\circ}\). If \(\angle2\) is supplementary to \(\angle3\), then \(\angle2+\angle3=180^{\circ}\).

Step2: Apply the transitive property

From \(\angle1+\angle2 = 180^{\circ}\) and \(\angle2+\angle3 = 180^{\circ}\), we can get \(\angle1=\angle3\) (by subtracting \(\angle2\) from both equations: \(\angle1 = 180^{\circ}-\angle2\) and \(\angle3=180^{\circ}-\angle2\)).

Step3: Use the converse of the corresponding angles postulate

When two lines (\(m\) and \(n\)) are cut by a transversal (\(t\)) and the corresponding angles (\(\angle1\) and \(\angle3\)) are equal, then the two lines (\(m\) and \(n\)) are parallel.

Answer:

Lines \(m\) and \(n\) are parallel.