According to the Lines Perpendicular to a Transversal Theorem, lines that are perpendicular to the same line are parallel to each other. In this case, both n and p are perpendicular to the transversal R. Therefore, we can say with certainty that n and p are parallel.
Regarding m and R, it sure looks like they are parallel. However looks can be deceiving. The fact is, we just dont have enough information.