https://file.aiquickdraw.com/mathbot/file/17292070625012ui5ur4t.jpg

Let's work through this proof based on the provided image. We are trying to prove that \( SX \cong VW \), using the given statements and applying properties of geometry.

Here’s the breakdown of the reasoning:

### Given Information:
- \( WX \cong TU \) (Statement 1: Given)
- \( \angle VUW \cong \angle STX \) (Statement 2: Given)
- \( \angle SXT \cong \angle UWV \) (Statement 3: Given)

### Next Steps:
Looking at the statement-reason table, we can now fill in the missing reasons.

1. **Statement 4: \( TX = WX + TW \)** (Reason: Additive Property of Length)
   - This expresses the total length \( TX \) as the sum of the lengths \( WX \) and \( TW \).

2. **Statement 5: \( UW = TU + TW \)** (Reason: Additive Property of Length)
   - Similarly, the total length \( UW \) is the sum of \( TU \) and \( TW \).

3. **Statement 6: \( TX = TU + TW \)** (Reason: Substitution)
   - This follows from the fact that \( WX \cong TU \) (Given in Statement 1), so \( TX = TU + TW \).

4. **Statement 7: \( UW = TX \)** (Reason: Substitution)
   - Again by substitution, we equate \( UW \) and \( TX \), based on prior expressions for \( UW \) and \( TX \).

5. **Statement 8: \( \triangle STX \cong \triangle VUW \)** (Reason: Transitive Property of Equality or Congruence)
   - The two triangles \( \triangle STX \) and \( \triangle VUW \) are congruent by the given angles and sides.

6. **Statement 9: \( SX \cong VW \)** (Reason: Corresponding Parts of Congruent Triangles (CPCTC))
   - Since the triangles are congruent, corresponding sides are congruent, so \( SX \cong VW \).

Thus, we have completed the proof that \( SX \cong VW \).

Would you like more details on any part of this proof or clarification on certain steps?

### Related Questions:
1. How does the Additive Property of Length work in geometric proofs?
2. Why is triangle congruence important in proving side lengths are equal?
3. What role do corresponding angles play in triangle congruence?
4. Can this type of proof be applied to non-congruent triangles in some cases?
5. How does the CPCTC property apply to other shapes, such as quadrilaterals?

### Tip:
In geometry, always keep track of congruent triangles. Once triangles are proven congruent, many properties (angles, sides) follow automatically, simplifying complex problems.

WX ≅ TU, ∠VUW ≅ ∠STX, and ∠SXT ≅ ∠UWV. Complete the proof that SX ≅ VW.

This proof demonstrates how to prove SX ≅ VW using triangle congruence principles, the additive property of length, and CPCTC. Suitable for high school students learning geometry.

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