Question

which angles are supplementary angles?
∠tsp and ∠rsu ∠qps and ∠opn
∠qps and ∠rsp ∠opn and ∠qpn

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). They can be adjacent (forming a linear pair) or non - adjacent.

Step2: Analyze Each Option

- Option 1: \(\angle TSP\) and \(\angle RSU\)
\(\angle RSU\) and \(\angle TSP\) are vertical angles (opposite angles formed by the intersection of two lines). Vertical angles are equal, not supplementary (unless each is \(90^\circ\), which is not indicated here).
- Option 2: \(\angle QPS\) and \(\angle OPN\)
\(\angle QPS\) and \(\angle OPN\) are vertical angles (formed by the intersection of lines \(OQ\) and \(NP\)). Vertical angles are equal, not supplementary (unless each is \(90^\circ\), which is not indicated here).
- Option 3: \(\angle QPS\) and \(\angle RSP\)
Lines \(OQ\) and \(RT\) are parallel (since \(PQ\) and \(ST\) are parallel as they are both arrows in the same direction). Line \(NU\) is a transversal. \(\angle QPS\) and \(\angle RSP\) are same - side interior angles? No, wait. Let's look at the straight line. Wait, actually, \(\angle QPS\) and \(\angle RSP\): Wait, no, let's check the fourth option first. Wait, no, let's re - examine. Wait, \(\angle OPN\) and \(\angle QPN\): \(\angle OPN\) and \(\angle QPN\) form a linear pair (they are adjacent and their non - common sides form a straight line \(OQ\)). So \(\angle OPN+\angle QPN = 180^\circ\), so they are supplementary. Wait, no, let's check the third option again. Wait, \(\angle QPS\) and \(\angle RSP\): If \(OQ\parallel RT\) and \(NU\) is a transversal, then \(\angle QPS\) and \(\angle RSP\) are same - side interior angles? No, maybe I made a mistake. Wait, let's check the fourth option: \(\angle OPN\) and \(\angle QPN\). \(\angle OPN\) and \(\angle QPN\) are adjacent angles with a common side \(PN\) and their non - common sides \(OP\) and \(QP\) form a straight line \(OQ\). So by the definition of a linear pair, their sum is \(180^\circ\), so they are supplementary. Wait, no, let's check the third option: \(\angle QPS\) and \(\angle RSP\). Wait, maybe I messed up. Wait, the correct approach: Supplementary angles sum to \(180^\circ\). Let's check each pair:
- \(\angle TSP\) and \(\angle RSU\): Vertical angles, equal, not supplementary.
- \(\angle QPS\) and \(\angle OPN\): Vertical angles, equal, not supplementary.
- \(\angle QPS\) and \(\angle RSP\): Let's see, if \(OQ\parallel RT\), then \(\angle QPS\) and \(\angle RSP\) are same - side interior angles? No, actually, if we look at the lines, \(\angle QPS\) and \(\angle RSP\): Wait, maybe not. Wait, the fourth option: \(\angle OPN\) and \(\angle QPN\). These two angles are adjacent and form a linear pair (they lie on a straight line \(OQ\)). So \(\angle OPN+\angle QPN = 180^\circ\), so they are supplementary. Wait, but let's check the third option again. Wait, maybe the correct answer is \(\angle QPS\) and \(\angle RSP\)? No, I think I made a mistake. Wait, let's re - define: A linear pair of angles is supplementary. So angles that form a linear pair are supplementary. \(\angle OPN\) and \(\angle QPN\) form a linear pair (they are adjacent, share a common vertex \(P\), share a common side \(PN\), and their other sides \(OP\) and \(QP\) are opposite rays, forming a straight line). So \(\angle OPN+\angle QPN = 180^\circ\), so they are supplementary. Wait, but the third option: \(\angle QPS\) and \(\angle RSP\). If \(OQ\parallel RT\), then \(\angle QPS\) and \(\angle RSP\) are same - side interior angles, and same - side interior angles are supplementary when the lines are parallel. Since \(PQ\) and \(ST\) are parallel (as they are both in the same direction, so \(OQ\parallel RT\)), then \(\angle QPS\) and \(\angle RSP\) are same - side interior angles and are supplementary. Wait, now I'm confused. Wait, let's look at the diagram again. The lines \(OQ\) and \(RT\) are parallel (since \(PQ\) and \(ST\) are parallel, as they are both arrows going in the same direction). The transversal is \(NU\). So \(\angle QPS\) and \(\angle RSP\) are same - side interior angles. Same - side interior angles are supplementary when the lines are parallel. So \(\angle QPS+\angle RSP = 180^\circ\), so they are supplementary. And \(\angle OPN\) and \(\angle QPN\) form a linear pair, so they are also supplementary? Wait, no, maybe the options are labeled differently. Wait, the options are:
1. \(\angle TSP\) and \(\angle RSU\)
2. \(\angle QPS\) and \(\angle OPN\)
3. \(\angle QPS\) and \(\angle RSP\)
4. \(\angle OPN\) and \(\angle QPN\)

Wait, \(\angle OPN\) and \(\angle QPN\): \(\angle OPN\) and \(\angle QPN\) are adjacent and their non - common sides form a straight line, so they are a linear pair, hence supplementary. But also, \(\angle QPS\) and \(\angle RSP\): if \(OQ\parallel RT\), then same - side interior angles are supplementary. Wait, maybe the correct answer is \(\angle QPS\) and \(\angle RSP\)? Wait, no, let's check the angles. \(\angle QPS\) and \(\angle RSP\): Let's assume \(OQ\parallel RT\). Then, by the same - side interior angles theorem, same - side interior angles are supplementary. So \(\angle QPS\) and \(\angle RSP\) are same - side interior angles, so they are supplementary. And \(\angle OPN\) and \(\angle QPN\) are a linear pair, so they are supplementary. Wait, but maybe the diagram shows that \(\angle QPS\) and \(\angle RSP\) are supplementary. Wait, I think I made a mistake earlier. Let's go back to the definition. Supplementary angles sum to \(180^\circ\). A linear pair is a type of supplementary angles (adjacent). Same - side interior angles (when lines are parallel) are also supplementary (non - adjacent). So let's check each option:

- \(\angle TSP\) and \(\angle RSU\): Vertical angles, equal, not supplementary.
- \(\angle QPS\) and \(\angle OPN\): Vertical angles, equal, not supplementary.
- \(\angle QPS\) and \(\angle RSP\): Same - side interior angles (since \(OQ\parallel RT\) and \(NU\) is transversal), so supplementary.
- \(\angle OPN\) and \(\angle QPN\): Linear pair, supplementary.

Wait, but maybe the diagram has \(OQ\) and \(RT\) parallel, so \(\angle QPS\) and \(\angle RSP\) are supplementary. So the correct answer is \(\angle QPS\) and \(\angle RSP\).

Answer:

\(\angle QPS\) and \(\angle RSP\)