Question

select all the pairs of supplementary angles.

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze \(\angle NMP\) and \(\angle NMK\)

These two angles form a linear - pair (they are adjacent and their non - common sides form a straight line). By the linear - pair postulate, if two angles form a linear pair, then they are supplementary. So \(\angle NMP+\angle NMK = 180^{\circ}\).

Step3: Analyze \(\angle OPM\) and \(\angle LMK\)

Since \(LN\parallel OQ\) (assumed from the parallel - line appearance in the figure), and \(KR\) is a transversal. \(\angle OPM\) and \(\angle LMP\) are supplementary (by the consecutive - interior - angles theorem for parallel lines \(LN\) and \(OQ\) with transversal \(KR\)). Also, \(\angle LMP=\angle LMK\) (vertical angles are equal). So \(\angle OPM+\angle LMK = 180^{\circ}\).

Step4: Analyze \(\angle QPM\) and \(\angle LMP\)

Since \(LN\parallel OQ\) and \(KR\) is a transversal. \(\angle QPM\) and \(\angle LMP\) are supplementary (by the consecutive - interior - angles theorem: for parallel lines \(LN\) and \(OQ\) cut by transversal \(KR\), consecutive interior angles are supplementary).

Step5: Analyze \(\angle OPR\) and \(\angle QPR\)

These two angles form a linear - pair (they are adjacent and their non - common sides form a straight line). By the linear - pair postulate, \(\angle OPR+\angle QPR=180^{\circ}\).

Answer:

\(\angle NMP\) and \(\angle NMK\), \(\angle OPM\) and \(\angle LMK\), \(\angle QPM\) and \(\angle LMP\), \(\angle OPR\) and \(\angle QPR\)