Question

applying angle relationships to justify paralle
which lines are parallel? justify your answer.
lines p and q are parallel because same side interior angles are congruent.
lines p and q are parallel because alternate exterior angles are congruent
lines l and m are parallel because same side interior angles are supplementary
lines l and m are parallel because alternate interior angles are supplementary.

Explanation:

Step1: Recall angle - parallel line relationships

• Same - side interior angles: If two lines are cut by a transversal, and the sum of same - side interior angles is \(180^{\circ}\), the lines are parallel. If they are congruent (equal in measure) and not \(90^{\circ}\) each, the lines are not parallel.
• Alternate exterior angles: If two lines are cut by a transversal and alternate exterior angles are congruent, the lines are parallel.
• Alternate interior angles: If two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.

Step2: Analyze each option

• Option 1 (Lines \(p\) and \(q\) with same - side interior angles):
• For lines \(p\) and \(q\) cut by a transversal, if we assume the transversal, same - side interior angles are not a relevant pair for these lines in the given figure. Also, the rule for parallel lines using same - side interior angles is that they are supplementary (\(a + b=180^{\circ}\)), not congruent (unless \(a = b = 90^{\circ}\)).
• Option 2 (Lines \(p\) and \(q\) with alternate exterior angles):
• Consider the two \(130^{\circ}\) angles. When we look at lines \(p\) and \(q\) cut by a transversal (the non - parallel line among \(l\) and \(m\) that intersects \(p\) and \(q\)), the \(130^{\circ}\) angles are alternate exterior angles. By the alternate - exterior - angles theorem (if two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel), this is a valid reason for lines \(p\) and \(q\) to be parallel.
• Option 3 (Lines \(l\) and \(m\) with same - side interior angles):
• For lines \(l\) and \(m\), if we consider the angles \(130^{\circ}\) and \(80^{\circ}\), \(130+80 = 210

eq180\). So, the same - side interior angles (if we assume the correct transversal) are not supplementary.

• Option 4 (Lines \(l\) and \(m\) with alternate interior angles):
• Alternate interior angles should be congruent for lines to be parallel. The sum of angles (\(130^{\circ}\) and \(80^{\circ}\)) is not relevant for the alternate - interior - angles rule. The rule for alternate interior angles is \(a=b\) (congruent), not \(a + b=180^{\circ}\) (supplementary)

Answer:

Lines \(p\) and \(q\) are parallel because alternate exterior angles are congruent.