https://file.aiquickdraw.com/mathbot/file/17283408993449s9y2hu7.jpg

Let's work through the proof based on the image you uploaded.

### Problem
You are asked to complete the proof that \( m \angle TUZ + m \angle SXW = 180^\circ \). The diagram shows two pairs of parallel lines:
- \( \overleftrightarrow{TV} \parallel \overleftrightarrow{EG} \)
- \( \overleftrightarrow{EG} \parallel \overleftrightarrow{WY} \)

We also see that angles \( \angle TUZ \) and \( \angle SXW \) are involved. The table provides a step-by-step proof with missing justifications.

### Let's fill in the missing reasons:

1. **Statement 1**: \( \overleftrightarrow{TV} \parallel \overleftrightarrow{EG} \)  
   **Reason**: Given.

2. **Statement 2**: \( \overleftrightarrow{EG} \parallel \overleftrightarrow{WY} \)  
   **Reason**: Given.

3. **Statement 3**: \( \angle TUZ \cong \angle EFZ \)  
   **Reason**: Corresponding Angles Theorem.

   - Since \( \overleftrightarrow{TV} \parallel \overleftrightarrow{EG} \) and \( \overleftrightarrow{UZ} \) acts as a transversal, \( \angle TUZ \) and \( \angle EFZ \) are corresponding angles and thus congruent.

4. **Statement 4**: \( m \angle EFZ + m \angle SXW = 180^\circ \)  
   **Reason**: Linear Pair Postulate.

   - Angles \( \angle EFZ \) and \( \angle SXW \) form a linear pair because they are adjacent and on a straight line, meaning their measures sum to 180°.

5. **Statement 5**: \( m \angle TUZ + m \angle SXW = 180^\circ \)  
   **Reason**: Substitution.

   - Since \( m \angle TUZ = m \angle EFZ \) by congruence (from step 3), we can substitute \( m \angle TUZ \) for \( m \angle EFZ \) in the previous equation (step 4), yielding the desired result.

### Completed Proof:
We have shown that \( m \angle TUZ + m \angle SXW = 180^\circ \) by applying the corresponding angles theorem, the linear pair postulate, and substitution.

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Do you have any questions, or would you like more details on any of the steps?

Here are 5 related questions to expand understanding:
1. What are alternate interior angles, and how are they related to parallel lines?
2. Why is the Corresponding Angles Theorem valid for parallel lines?
3. What other types of angle relationships can form between parallel lines and a transversal?
4. Can two adjacent angles on a line ever sum to something other than 180°?
5. How can we use transversals to prove that lines are parallel?

### Tip:
When working with parallel lines, always look for corresponding, alternate interior, and consecutive interior angles, as these relationships often lead to important conclusions.

Complete the proof that m∠TUZ + m∠SXW = 180° given that lines TV and EG are parallel, and EG and WY are parallel.

Learn how to complete the proof that m∠TUZ + m∠SXW = 180° using the properties of parallel lines and angles. This step-by-step solution applies the Corresponding Angles Theorem and the Linear Pair Postulate, making it suitable for students in Grades 8-10.

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