https://file.aiquickdraw.com/mathbot/file/1728065771885n1ae7cin.jpg

The goal is to complete the proof that \( m \angle PRS + m \angle VUW = 180^\circ \).

Here's a breakdown of the reasoning leading to that conclusion, using the table in the image:

1. **\( \overleftrightarrow{GI} \parallel \overleftrightarrow{TV} \)** (Given)
2. **\( \overleftrightarrow{GI} \parallel \overleftrightarrow{QS} \)** (Given)
3. **\( \angle VUW \cong \angle IHW \)** (Corresponding Angles Theorem)
   - This step is justified because \( \overleftrightarrow{GI} \parallel \overleftrightarrow{TV} \), and the transversal \( \overleftrightarrow{HW} \) intersects them, forming corresponding angles \( \angle VUW \) and \( \angle IHW \).
4. **\( m \angle IHW + m \angle PRS = 180^\circ \)** (Same-Side Exterior Angles Theorem)
   - This follows from the fact that \( \overleftrightarrow{QS} \parallel \overleftrightarrow{GI} \), and the transversal \( \overleftrightarrow{PRS} \) creates supplementary angles.
5. **\( m \angle PRS + m \angle VUW = 180^\circ \)** (Substitution)
   - Since \( m \angle VUW = m \angle IHW \), we substitute \( m \angle VUW \) for \( m \angle IHW \) in step 4, completing the proof.

Would you like further clarification or details on any of these steps?

Here are five related questions to expand on this topic:
1. What is the Corresponding Angles Theorem, and why does it apply in this case?
2. Can the Same-Side Exterior Angles Theorem be used with any parallel lines and transversal?
3. Why are \( \angle VUW \) and \( \angle PRS \) supplementary in this configuration?
4. What other theorems could you use to prove angles are supplementary when dealing with parallel lines?
5. How would the proof change if the lines were not parallel?

**Tip:** Always check if lines are parallel before applying the Corresponding or Same-Side Angle Theorems.

Complete the proof that m∠PRS + m∠VUW = 180°.

This proof shows that m∠PRS + m∠VUW = 180° using the Corresponding Angles Theorem and the Same-Side Exterior Angles Theorem. Understand how to apply geometric properties and theorems with a detailed step-by-step solution, ideal for students in Grades 9-10.

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