Question

determine if l || m based on the given information on the diagram. if yes, state the converse that proves the lines are parallel.

Explanation:

Step1: Recall parallel - line postulates

Use properties like corresponding angles, alternate - interior angles, same - side interior angles, etc.

Step2: Analyze diagram 1

In diagram 1, we have a pair of same - side interior angles. One angle is \(65^{\circ}\) and the other is \(115^{\circ}\). Since \(65^{\circ}+115^{\circ}=180^{\circ}\), by the same - side interior angles postulate (if same - side interior angles are supplementary, then the two lines are parallel), lines \(l\) and \(m\) are parallel.

Step3: Analyze diagram 2

In diagram 2, we have a pair of angles. One angle is \(52^{\circ}\) and the other is \(128^{\circ}\). These are same - side interior angles. Since \(52^{\circ}+128^{\circ}=180^{\circ}\), by the same - side interior angles postulate, lines \(l\) and \(m\) are parallel.

Step4: Analyze diagram 3

In diagram 3, we have two right - angled corners formed by the transversal and the two lines. If two lines are perpendicular to the same line, then they are parallel. Here, the transversal is perpendicular to both \(l\) and \(m\), so \(l\parallel m\).

Step5: Analyze diagram 4

In diagram 4, we do not have enough information (no angle measures or angle - relationship information) to prove that \(l\parallel m\).

Answer:

1. Yes, same - side interior angles are supplementary.
2. Yes, same - side interior angles are supplementary.
3. Yes, both lines are perpendicular to the transversal.
4. No, not enough information.