https://file.aiquickdraw.com/mathbot/file/1729094874269bfdppccr.jpg

Based on the given image, which depicts two parallel lines \( \overleftrightarrow{TV} \) and \( \overleftrightarrow{WY} \), and a transversal line \( \overleftrightarrow{ZV} \) intersecting these lines, we need to identify the **adjacent angles**. 

### Explanation of Adjacent Angles:
Adjacent angles share a common vertex and a common side but do not overlap. They are "next to each other."

### Identifying Adjacent Angles in the Diagram:
1. **Angle pairs at point \( X \)**:
   - \( \angle WXY \) and \( \angle ZXY \) share vertex \( X \) and side \( \overrightarrow{XW} \).
   - \( \angle ZXY \) and \( \angle ZXV \) share vertex \( X \) and side \( \overrightarrow{XZ} \).
   
2. **Angle pairs at point \( U \)**:
   - \( \angle VUS \) and \( \angle ZUS \) share vertex \( U \) and side \( \overrightarrow{US} \).
   - \( \angle ZUS \) and \( \angle ZUW \) share vertex \( U \) and side \( \overrightarrow{UZ} \).
   
3. **Angle pairs at point \( V \)**:
   - \( \angle TVU \) and \( \angle SVU \) share vertex \( V \) and side \( \overrightarrow{VU} \).
   - \( \angle SVU \) and \( \angle YVU \) share vertex \( V \) and side \( \overrightarrow{VS} \).

### Answer:
The adjacent angles in this diagram are:
1. \( \angle WXY \) and \( \angle ZXY \)
2. \( \angle ZXY \) and \( \angle ZXV \)
3. \( \angle VUS \) and \( \angle ZUS \)
4. \( \angle ZUS \) and \( \angle ZUW \)
5. \( \angle TVU \) and \( \angle SVU \)
6. \( \angle SVU \) and \( \angle YVU \)

These angle pairs share both a common vertex and a common side.

### Would you like to further analyze any specific angle relationships or explore properties like alternate interior or corresponding angles in the context of parallel lines?

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#### Related Questions:
1. Which angles in this figure are **corresponding angles**?
2. Can you identify any **alternate interior angles** based on the transversal line?
3. How do the angles formed by the transversal relate to each other if the lines \( TV \) and \( WY \) are parallel?
4. What is the sum of the adjacent angles at each intersection point?
5. How do you prove that \( TV \parallel WY \) using angle properties?

**Tip**: When two lines are parallel and a transversal cuts across them, corresponding angles are congruent, and alternate interior angles are equal.

Which angles are adjacent angles in the given diagram, where lines TV and WY are parallel?

This problem involves identifying adjacent angles in a diagram where two parallel lines are intersected by a transversal. Learn how to find pairs of angles that share a common vertex and side. Suitable for students in Grades 6-8.

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