Question

classify the angle pair \\(\angle mnq\\) and \\(\angle qno\\).
options:

• adjacent and complementary
• adjacent and neither complementary nor supplementary
• a linear pair
• nonadjacent and complementary

Explanation:

Step1: Recall the definition of a linear pair

A linear pair of angles is formed when two adjacent angles form a straight line (sum to \(180^\circ\)) and share a common side and a common vertex.

Step2: Analyze \(\angle MNQ\) and \(\angle QNO\)

• First, check if they are adjacent: \(\angle MNQ\) and \(\angle QNO\) share the common vertex \(N\) and the common side \(NQ\), so they are adjacent.
• Then, check if their sum is \(180^\circ\): From the diagram, \(\angle MNQ = 80^\circ\) (wait, actually, looking at the right angle at \(N\) (since \(ON\) and \(PM\) are perpendicular? Wait, no, let's re - examine. Wait, \(ON\) is a straight line? Wait, no, \(O - N - M\)? Wait, no, the diagram shows \(ON\) and \(PM\) are perpendicular (right angle at \(N\) between \(ON\) and \(PN\)? Wait, maybe I misread. Wait, actually, \(\angle MNQ\) and \(\angle QNO\): Let's see, the straight line here is \(O - N - M\)? No, wait, \(O\) and \(M\) are on a straight line? Wait, no, \(ON\) is one ray, \(MN\) is another, and \(QN\) is between them. Wait, actually, \(\angle MNQ+\angle QNO\): If we look at the straight line \(OM\) (assuming \(O\) and \(M\) are on a straight line), then \(\angle MNQ+\angle QNO = 80^\circ + 100^\circ\)? Wait, no, maybe I made a mistake. Wait, the right angle is at \(N\) between \(ON\) and \(PN\), so \(ON\perp PN\), meaning \(\angle ONP = 90^\circ\). Then \(\angle MNQ = 80^\circ\), \(\angle QNP=40^\circ\), so \(\angle MNP=\angle MNQ+\angle QNP = 80 + 40=120^\circ\)? No, that's not right. Wait, maybe the straight line is \(O - N - M\), so \(\angle ONM = 180^\circ\) (since \(O\) and \(M\) are on a straight line). Then \(\angle MNQ+\angle QNO=\angle ONM = 180^\circ\), and they are adjacent (share \(NQ\) and vertex \(N\)). So they form a linear pair.

Answer:

A linear pair