Question

if ( mangle nsm = 72^circ ), then what is ( mangle psm )?

Explanation:

Step1: Find the measure of ∠3

Since ∠1 and ∠3 are vertical angles, and vertical angles are equal. Given \(m\angle1 = 72^{\circ}\), so \(m\angle3=72^{\circ}\).

Step2: Use the linear - pair relationship

∠3 and ∠2 form a linear pair (\(\angle3+\angle2 = 180^{\circ}\)), but we also know that ∠2 and ∠4 are vertical angles (so \(m\angle2 = m\angle4\)). Another way is to note that ∠NSM and ∠PSM are supplementary (they form a linear pair).
If two angles form a linear pair, then the sum of their measures is \(180^{\circ}\). Let \(x=m\angle PSM\).
We know that \(m\angle NSM + m\angle PSM=180^{\circ}\).
Substitute \(m\angle NSM = 72^{\circ}\) into the equation: \(72^{\circ}+x = 180^{\circ}\).
Solve for \(x\): \(x=180^{\circ}-72^{\circ}\).

Answer:

\(108\)