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The problem asks you to complete a proof showing that \( SW \cong ST \), using a two-column proof format with statements and reasons.

### Here's how the reasoning follows:

1. **\( \angle SWV = \angle SVT \)** — *Given*  
   This means that angles \( SWV \) and \( SVT \) are congruent, which is part of the information provided.

2. **\( \angle TSU = \angle USV \)** — *Given*  
   Here, angles \( TSU \) and \( USV \) are also congruent, as given.

3. **\( SV = SU \)** — *Given*  
   Segment \( SV \) is congruent to segment \( SU \), another given fact.

4. **\( m\angle TSV = m\angle USV + m\angle LSV \)** — *Additive Property of Angle Measure*  
   Using the additive property of angles, we can express \( m\angle TSV \) as the sum of two other angles: \( m\angle USV \) and \( m\angle LSV \).

5. **\( m\angle USW = m\angle LSW + m\angle TSV \)** — *Additive Property of Angle Measure*  
   Similarly, \( m\angle USW \) can be expressed as the sum of two angles: \( m\angle LSW \) and \( m\angle TSV \).

6. **\( m\angle LTSV = m\angle LUSW \)** — *Substitution*  
   By substituting equivalent angle measures, we establish the equality of the two angle measures.

7. **\( \triangle SWV \cong \triangle SVT \)** — *Transitive Property of Equality*  
   By the transitive property, the triangles are congruent because their corresponding parts are congruent.

8. **\( SW \cong ST \)** — *CPCTC (Corresponding Parts of Congruent Triangles are Congruent)*  
   Since the triangles are congruent, their corresponding sides \( SW \) and \( ST \) are congruent.

This proof completes the process showing that \( SW \cong ST \).

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Would you like a more detailed explanation on any specific step? Here are five related questions to deepen your understanding:

1. Why do we use the additive property of angle measures in the proof?
2. How does the transitive property help in proving triangle congruence?
3. What is the role of CPCTC in triangle congruence proofs?
4. Could you apply a similar proof strategy to other geometric figures?
5. How does substitution of angle measures ensure equality in proofs?

**Tip:** In geometry, always ensure that corresponding parts of congruent triangles are clearly identified before concluding that sides or angles are congruent using CPCTC.

This problem demonstrates a proof that SW is congruent to ST using properties like angle addition and theorems such as CPCTC. The steps involve identifying congruent triangles and applying substitution and the transitive property. Suitable for high school geometry students in grades 9-11.

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