In the given image, we are asked to find which angles are supplementary. The diagram shows two parallel lines $\overline{SU}$ and $\overline{VX}$, intersected by a transversal line $\overline{RXY}$, and possibly another transversal through point $W$.

Supplementary angles are two angles whose measures add up to 180 degrees. In the case of parallel lines cut by a transversal, supplementary angles are often a pair of consecutive (or same-side interior) angles along the transversal.

Let’s review the options and check for supplementary angle pairs:

1. $\angle VWY$ and $\angle STW$: These two angles are on opposite sides of the transversal, making them alternate interior angles, not supplementary.

2. $\angle STR$ and $\angle VWT$: These angles do not add up to 180° since they are not consecutive or on the same side of the transversal. This option is incorrect.

3. $\angle XWT$ and $\angle UTR$: This pair seems to be adjacent angles at point $T$, forming a straight line. Since adjacent angles on a straight line are supplementary, this pair is correct.

4. $\angle XWY$ and $\angle VWY$: These two angles share a common side but are not consecutive interior or exterior angles, so they are not supplementary.

Therefore, the correct answer is $\angle XWT$ and $\angle UTR$.

Do you want more details on how supplementary angles work in parallel lines, or do you have other questions?

Here are 5 related questions:

1. What are the properties of alternate interior angles in parallel lines?
2. How can you identify consecutive interior angles in a diagram like this?
3. What is the difference between supplementary and complementary angles?
4. Can exterior angles also be supplementary in parallel lines cut by a transversal?
5. How do you calculate the measure of an unknown angle when given a pair of supplementary angles?

Tip: When dealing with parallel lines and transversals, look for straight lines and adjacent angles to identify supplementary pairs easily!