Question

select all the pairs of supplementary angles.
∠tsv and ∠rsv ∠rsq and ∠tsq
∠tsq and ∠tsv ∠rsv and ∠rsq

Explanation:

Step1: Recall Supplementary Angles Definition

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). We analyze each pair:

Step2: Analyze \(\angle TSV\) and \(\angle RSV\)

\(RT\) is a straight line, so \(\angle TSV + \angle RSV = 180^\circ\) (linear pair). Thus, they are supplementary.

Step3: Analyze \(\angle RSQ\) and \(\angle TSQ\)

\(QX\) is a straight line, so \(\angle RSQ + \angle TSQ = 180^\circ\) (linear pair). Thus, they are supplementary.

Step4: Analyze \(\angle TSQ\) and \(\angle TSV\)

These are vertical angles (formed by intersecting lines \(RT\) and \(QX\))? Wait, no—wait, \(\angle TSQ\) and \(\angle RSV\) are vertical? Wait, no, let's check again. Wait, \(\angle TSQ\) and \(\angle RSV\) are vertical? Wait, no, \(\angle TSQ\) and \(\angle RSV\) are equal (vertical angles), but \(\angle TSQ\) and \(\angle TSV\): Wait, \(RT\) and \(UW\) are parallel? Wait, no, the diagram: \(RT\) and \(UW\) are parallel, cut by transversal \(QX\). But \(\angle TSQ\) and \(\angle TSV\): Wait, no, \(\angle TSQ\) and \(\angle TSV\) – wait, \(S\) is on \(RT\), \(V\) is on \(UW\). Wait, maybe I made a mistake. Wait, no, the key is linear pairs. Wait, \(\angle TSQ\) and \(\angle TSV\): are they adjacent? No, \(\angle TSQ\) is at \(S\) between \(Q\) and \(T\), \(\angle TSV\) is at \(S\) between \(T\) and \(V\)? Wait, no, the points: \(Q - S - X\) is a line, \(R - S - T\) is a line. So \(\angle RSQ\) and \(\angle TSQ\) are linear pair (supplementary). \(\angle TSV\) and \(\angle RSV\) are linear pair (supplementary). Now, \(\angle TSQ\) and \(\angle TSV\): are they supplementary? Wait, no, because \(\angle TSQ\) and \(\angle RSV\) are vertical angles (equal), and \(\angle RSV\) and \(\angle TSV\) are supplementary, so \(\angle TSQ\) and \(\angle TSV\) would also be supplementary? Wait, no, wait: \(\angle TSQ = \angle RSV\) (vertical angles), and \(\angle RSV + \angle TSV = 180^\circ\), so \(\angle TSQ + \angle TSV = 180^\circ\). Wait, so maybe I was wrong earlier. Wait, let's re-express:

Wait, \(RT\) is straight, so \(\angle RSV + \angle TSV = 180^\circ\) (linear pair). \(QX\) is straight, so \(\angle RSQ + \angle TSQ = 180^\circ\) (linear pair). Now, \(\angle RSQ\) and \(\angle TSV\): are they equal? Wait, \(RT \parallel UW\) (since \(RT\) and \(UW\) are both horizontal, parallel), cut by transversal \(QX\), so \(\angle RSQ = \angle UXV\) (corresponding angles), but \(\angle TSV\) and \(\angle UXV\) are vertical angles? Wait, maybe the diagram has \(RT \parallel UW\), so \(\angle TSQ\) (at \(S\), between \(Q\) and \(T\)) and \(\angle TSV\) (at \(S\), between \(T\) and \(V\)): wait, no, \(V\) is on \(UW\). Wait, maybe the correct pairs are:

- \(\angle TSV\) and \(\angle RSV\): linear pair, supplementary.
- \(\angle RSQ\) and \(\angle TSQ\): linear pair, supplementary.
- \(\angle TSQ\) and \(\angle TSV\): since \(\angle RSQ = \angle TSV\) (alternate interior angles, as \(RT \parallel UW\), transversal \(QX\)), and \(\angle RSQ + \angle TSQ = 180^\circ\), so \(\angle TSV + \angle TSQ = 180^\circ\), so they are supplementary.
- \(\angle RSV\) and \(\angle RSQ\): since \(\angle RSV = \angle TSQ\) (vertical angles? Wait, no, \(\angle RSV\) and \(\angle TSQ\) are equal (alternate interior angles), and \(\angle RSQ + \angle TSQ = 180^\circ\), so \(\angle RSQ + \angle RSV = 180^\circ\), so they are supplementary.

Wait, but let's check the options:

1. \(\angle TSV\) and \(\angle RSV\): supplementary (linear pair) – correct.
2. \(\angle RSQ\) and \(\angle TSQ\): supplementary (linear pair) – correct.
3. \(\angle TSQ\) and \(\angle TSV\): supplementary (as \(\angle TSQ + \angle TSV = 180^\circ\)) – correct.
4. \(\angle RSV\) and \(\angle RSQ\): supplementary (as \(\angle RSV + \angle RSQ = 180^\circ\)) – correct.

Wait, but maybe the diagram shows \(RT\) and \(UW\) parallel, so all these pairs are supplementary? Wait, the original options:

- \(\angle TSV\) and \(\angle RSV\): linear pair (supplementary) – yes.
- \(\angle RSQ\) and \(\angle TSQ\): linear pair (supplementary) – yes.
- \(\angle TSQ\) and \(\angle TSV\): since \(\angle TSQ = \angle RSV\) (alternate interior angles), and \(\angle RSV + \angle TSV = 180^\circ\), so \(\angle TSQ + \angle TSV = 180^\circ\) – supplementary.
- \(\angle RSV\) and \(\angle RSQ\): since \(\angle RSV = \angle TSQ\) (alternate interior angles), and \(\angle RSQ + \angle TSQ = 180^\circ\), so \(\angle RSQ + \angle RSV = 180^\circ\) – supplementary.

But maybe the intended answer is the linear pairs first, but according to the definition, all four pairs? Wait, no, maybe I misread. Wait, the problem says "select all pairs". Let's re-express:

Supplementary angles sum to \(180^\circ\). Linear pairs (adjacent angles forming a straight line) are supplementary. Also, if two angles are same-side interior angles (for parallel lines) or corresponding, but here:

- \(\angle TSV\) and \(\angle RSV\): linear pair (supplementary) – correct.
- \(\angle RSQ\) and \(\angle TSQ\): linear pair (supplementary) – correct.
- \(\angle TSQ\) and \(\angle TSV\): since \(RT \parallel UW\), \(\angle TSQ = \angle RSV\) (alternate interior), and \(\angle RSV + \angle TSV = 180^\circ\), so \(\angle TSQ + \angle TSV = 180^\circ\) – supplementary.
- \(\angle RSV\) and \(\angle RSQ\): since \(\angle RSV = \angle TSQ\) (alternate interior), and \(\angle RSQ + \angle TSQ = 180^\circ\), so \(\angle RSQ + \angle RSV = 180^\circ\) – supplementary.

But maybe the options are all correct? Wait, the user's diagram shows four options, all with checkmarks? Wait, the user's image has four options, each with a checkmark? Wait, the original problem: the options are:

- \(\angle TSV\) and \(\angle RSV\)
- \(\angle RSQ\) and \(\angle TSQ\)
- \(\angle TSQ\) and \(\angle TSV\)
- \(\angle RSV\) and \(\angle RSQ\)

So all four pairs are supplementary? Wait, let's confirm:

1. \(\angle TSV + \angle RSV = 180^\circ\) (linear pair, \(RT\) is straight) – yes.
2. \(\angle RSQ + \angle TSQ = 180^\circ\) (linear pair, \(QX\) is straight) – yes.
3. \(\angle TSQ + \angle TSV\): since \(RT \parallel UW\), \(\angle TSQ = \angle RSV\) (alternate interior angles), and \(\angle RSV + \angle TSV = 180^\circ\), so \(\angle TSQ + \angle TSV = 180^\circ\) – yes.
4. \(\angle RSV + \angle RSQ\): since \(\angle RSV = \angle TSQ\) (alternate interior angles), and \(\angle RSQ + \angle TSQ = 180^\circ\), so \(\angle RSQ + \angle RSV = 180^\circ\) – yes.

Thus, all four pairs are supplementary.

Answer:

All pairs: \(\angle TSV\) and \(\angle RSV\), \(\angle RSQ\) and \(\angle TSQ\), \(\angle TSQ\) and \(\angle TSV\), \(\angle RSV\) and \(\angle RSQ\) are supplementary. (If the options are these four, all are correct.)