To determine which angles are adjacent, let's first clarify what adjacent angles are: Adjacent angles are two angles that share a common side and a common vertex, but do not overlap.

In this diagram, the question is asking for adjacent angles involving $\angle XWY$.

Analyzing each option:

1. $\angle XWY$ and $\angle UTR$: These angles do not share a common side or vertex, so they are not adjacent.

2. $\angle XWY$ and $\angle VWT$: These two angles share the vertex $W$ and the side $\overline{VW}$, making them adjacent.

3. $\angle XWY$ and $\angle VWY$: Although they share the vertex $W$, they do not have a common side as $\angle VWY$ does not exist in this configuration.

4. $\angle XWY$ and $\angle STW$: These two angles do not share a common side or vertex, so they are not adjacent.

The correct answer is: $\angle XWY$ and $\angle VWT$.

Would you like further details on how adjacent angles are identified?

Here are some related questions for further exploration:

1. What are other pairs of adjacent angles in this diagram?
2. How do parallel lines affect angle relationships?
3. What are vertically opposite angles, and can they be found here?
4. Can we identify any corresponding angles in this diagram?
5. How does the presence of parallel lines help identify angle relationships?

Tip: When working with parallel lines and a transversal, keep in mind the specific angle relationships, such as alternate interior, corresponding, and consecutive interior angles.