Question

lines l, m, and n lie in a plane and are cut by a transversal, t. $angle 1$ is supplementary to $angle 2$, and $angle 2$ is supplementary to $angle 3$.
which lines, if any, are parallel?
what justifies your answer?

Explanation:

Step1: Define supplementary angles

If two angles are supplementary, their sum is $180^\circ$. So:
$\angle 1 + \angle 2 = 180^\circ$
$\angle 2 + \angle 3 = 180^\circ$

Step2: Set equations equal to each other

Since both equal $180^\circ$, equate the left sides:
$\angle 1 + \angle 2 = \angle 2 + \angle 3$

Step3: Simplify to find equal angles

Subtract $\angle 2$ from both sides:
$\angle 1 = \angle 3$

Step4: Relate to parallel lines

$\angle 1$ and $\angle 3$ are corresponding angles formed by transversal $t$ with lines $m$ and $n$. Corresponding angles being congruent proves the lines are parallel.

Answer:

Which lines, if any, are parallel?
Lines $m$ and $n$ are parallel.

What justifies your answer?
If two angles supplementary to the same angle are congruent, and those congruent angles are corresponding angles for two lines cut by a transversal, then the two lines are parallel (Congruent Corresponding Angles Theorem, derived from the Congruent Supplements Theorem).