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In this problem, we are dealing with parallel lines and angles formed by a transversal. Let's analyze the situation step by step.

### Given Information:
- \( \overrightarrow{RT} \parallel \overrightarrow{UW} \) (These lines are parallel).
- \( m\angle TSV = 126^\circ \).
- We are asked to find \( m\angle WVS \).

### Observations:
1. \( \overrightarrow{RT} \) and \( \overrightarrow{UW} \) are parallel lines, and \( \overrightarrow{SQ} \) is a transversal cutting across them.
2. \( \angle TSV = 126^\circ \) is an exterior angle at point \( S \).
3. The angle \( \angle WVS \) that we are asked to find is the corresponding interior angle at point \( V \).

### Reasoning:
Since \( \overrightarrow{RT} \parallel \overrightarrow{UW} \), the corresponding angles formed by the transversal \( \overrightarrow{SQ} \) must be equal.

- \( \angle TSV \) is supplementary to the interior angle at point \( S \) on the same side of the transversal, so:
  \[
  m\angle TSV + m\angle WVS = 180^\circ
  \]
  Substituting the value of \( m\angle TSV = 126^\circ \):
  \[
  126^\circ + m\angle WVS = 180^\circ
  \]
  \[
  m\angle WVS = 180^\circ - 126^\circ = 54^\circ
  \]

### Final Answer:
\( m\angle WVS = 54^\circ \).

Would you like more details or have any further questions?

Here are five related questions to extend your understanding:
1. What is the relationship between alternate interior angles formed by parallel lines?
2. How do you calculate the supplement of an angle?
3. Can you identify other pairs of corresponding angles in this diagram?
4. What are alternate exterior angles in the context of parallel lines?
5. How would the angle measures change if the transversal were perpendicular to the parallel lines?

**Tip:** Always check for parallel line relationships in geometry problems, as they provide key angle congruencies or supplementary relationships.

If RT and UW are parallel lines and m∠TSV = 126°, what is m∠WVS?

This problem involves finding the measure of angle WVS using supplementary angles formed by a transversal cutting parallel lines. Given that m∠TSV = 126°, we can calculate m∠WVS by using angle relationships. Ideal for students in Grades 8-10 studying geometry and parallel lines.

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