Question

which lines are parallel? justify your answer.
lines e and f are parallel because their alternate exterior angles are congruent.
lines a and b are parallel because their same side exterior angles are supplementary.
lines e and f are parallel because their same side exterior angles are congruent.
lines a and b are parallel because their alternate exterior angles are congruent.

Explanation:

Step1: Recall parallel line theorems

For two lines to be parallel, certain angle - related conditions must be met. Alternate exterior angles are formed when a transversal crosses two lines. If alternate exterior angles are congruent, then the two lines are parallel.

Step2: Analyze the angles for lines \(a\) and \(b\)

For lines \(a\) and \(b\), we check the alternate exterior angles. The angle of \(75^{\circ}\) for line \(a\) and the angle that should be compared for line \(b\) (if we consider the transversal). But for lines \(a\) and \(b\), their alternate exterior angles are not congruent (\(75^{\circ}\) and \(115^{\circ}\) - related angles).

Step3: Analyze the angles for lines \(e\) and \(f\)

For lines \(e\) and \(f\), the alternate exterior angles are both \(75^{\circ}\). According to the alternate - exterior - angles theorem, if two lines are cut by a transversal and the alternate exterior angles are congruent, then the two lines are parallel.

Answer:

Lines \(e\) and \(f\) are parallel because their alternate exterior angles are congruent.