https://file.aiquickdraw.com/mathbot/file/1727470193251vzvak971.jpg

Let's analyze the image and fill in the missing reasoning for the proof that \(\angle V \cong \angle TUW\).

### Proof Steps:

1. **Given:** \( SV \parallel TU \) (lines are parallel)
2. **Given:** \(\angle VSW \cong \angle UTW\) (corresponding angles based on the parallel lines and transversal \(SW\))
3. **Missing Reason for Step 3:** 
   - **Reason:** Vertical Angles Theorem (since \(\angle VSW \) and \(\angle T UW\) are vertically opposite angles)
4. **Reason for Step 4:** Alternate Interior Angles Theorem (since \(SV \parallel TU\), \(\angle UTW \cong \angle V\))
5. **Reason for Step 5:** Transitive Property of Congruence (combining the earlier congruences, \(\angle VSW \cong \angle V \) and \(\angle UTW \cong \angle VSW\), we get \(\angle UTW \cong \angle V\))
6. **Reason for Step 6:** Transitive Property of Congruence (final conclusion of congruent angles \(\angle V \cong \angle TUW\)).

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Would you like further details on the vertical angles theorem or alternate interior angles? Here are some follow-up questions to consider:

1. What is the Vertical Angles Theorem, and how does it apply here?
2. How do alternate interior angles relate to parallel lines and a transversal?
3. Can the Transitive Property of Congruence be used in other geometric proofs?
4. How does the Corresponding Angles Postulate differ from the Alternate Interior Angles Theorem?
5. What other theorems are commonly used in congruence proofs?

**Tip:** Always look for parallel lines and transversals when working with angle congruence proofs—they often lead to important angle relationships!

Learn how to complete the proof that ∠V is congruent to ∠TUW using properties of parallel lines, transversals, and congruent angles. This problem involves key geometric concepts such as the Vertical Angles Theorem, Alternate Interior Angles Theorem, and the Transitive Property of Congruence. Suitable for students in Grades 9-10.

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