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? linear pair postulate corresponding angles theorem same side interior angle theorem converse of same side interior angle theorem

Explanation:

Step1: Analyze angle - angle relationships

Given $\angle1$ is supplementary to $\angle2$ and $\angle2$ is supplementary to $\angle3$. So, $\angle1+\angle2 = 180^{\circ}$ and $\angle2+\angle3=180^{\circ}$, which implies $\angle1=\angle3$.

Step2: Apply parallel - line theorem

$\angle1$ and $\angle3$ are same - side interior angles for lines $m$ and $n$ with transversal $t$. Since $\angle1=\angle3$, by the converse of the same - side interior angle theorem, if same - side interior angles are congruent, the lines are parallel.

Answer:

Lines $m$ and $n$ are parallel; converse of same side interior angle theorem