Question

if m∠nsm = 72°, then what is m∠psm?

Explanation:

Step1: Find vertical angle

Since \(m\angle1 = 72^{\circ}\) (vertical angles are equal, \(\angle1\) and \(\angle NSM\) are vertical angles), and \(\angle1\) and \(\angle2\) are supplementary (\(\angle1+\angle2 = 180^{\circ}\), linear - pair of angles).

Step2: Calculate \(\angle2\)

\(m\angle2=180^{\circ}-m\angle1 = 180^{\circ}- 72^{\circ}=108^{\circ}\)

Step3: Use vertical - angle property again

\(\angle2\) and \(\angle4\) are vertical angles (\(JK\parallel NP\), corresponding angles or vertical - angle relationship). \(\angle PSM\) and \(\angle4\) are supplementary. Wait, no, \(\angle2\) and \(\angle4\) are vertical angles (if we consider the parallel lines \(JK\) and \(NP\) cut by transversal \(LM\), but actually, \(\angle NSM\) and \(\angle1\) are vertical angles (\(m\angle1 = 72^{\circ}\)), \(\angle1\) and \(\angle2\) are supplementary (\(m\angle2=108^{\circ}\)), and \(\angle2\) and \(\angle PSM\) are vertical angles. So \(m\angle PSM=m\angle2\)

Answer:

\(108^{\circ}\)