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The image presents a geometry proof where you are asked to show that \( m \angle RUT + m \angle QRU = 180^\circ \), utilizing properties of angles.

Here’s a breakdown of the reasoning steps based on the given figure:

1. **Given**: 
   - The diagram shows two parallel lines \( QS \parallel TV \) intersected by the transversal \( PU \).
   - This is written as the first statement in the proof: \( QS \parallel TV \).

2. **Corresponding Angles Theorem**: 
   - Because \( QS \parallel TV \), the angles \( \angle RUT \) and \( \angle PRQ \) are corresponding angles formed by the transversal \( PU \). Therefore, they are congruent, so:
   \[
   m \angle RUT = m \angle PRQ
   \]
   - This is the second step in the proof.

3. **Linear Pair**: 
   - The angles \( \angle PRQ \) and \( \angle QRU \) form a linear pair, as they are adjacent angles on a straight line. According to the linear pair postulate, their sum must be 180°:
   \[
   m \angle PRQ + m \angle QRU = 180^\circ
   \]
   - This is the third step in the proof.

4. **Conclusion**: 
   - Since \( m \angle RUT = m \angle PRQ \) (from step 2), you can substitute to conclude that:
   \[
   m \angle RUT + m \angle QRU = 180^\circ
   \]
   - This completes the proof.

Would you like more detailed explanations or clarifications on any of the steps? 

Here are 5 follow-up questions:
1. What is the Corresponding Angles Theorem?
2. Can you explain the Linear Pair Postulate in more detail?
3. How do we identify which angles are corresponding in this problem?
4. Why do parallel lines ensure that corresponding angles are congruent?
5. Can you think of another geometry theorem that might also apply to this diagram?

**Tip**: Always remember that corresponding angles are congruent when two parallel lines are cut by a transversal!

Complete the proof that m∠RUT + m∠QRU = 180°.

This proof demonstrates how to use the Corresponding Angles Theorem and the Linear Pair Postulate to show that the sum of m∠RUT and m∠QRU equals 180°. The problem involves two parallel lines cut by a transversal and requires understanding of geometry concepts. Suitable for students in Grades 9-10.

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