https://file.aiquickdraw.com/mathbot/file/1728916202186j7bjcf1f.jpg

Let's walk through the steps in this proof table and complete it based on the given angles and theorems.

### Given Information:
- **Given:** ∠RSU ≅ ∠VST 
- **Prove:** ∠RSV ≅ ∠UST

### Step-by-step explanation:

1. **Statement 1:**
   - **Statement:** ∠RSU ≅ ∠VST
   - **Reason:** Given

2. **Statement 2:**
   - **Statement:** \( m\angle RSU = m\angle VST \)
   - **Reason:** Definition of congruent angles (Congruent angles have equal measures)

3. **Statement 3:**
   - **Statement:** \( m\angle RSU + m\angle USV = m\angle VST + m\angle USV \)
   - **Reason:** Addition Property of Equality (add the same angle \( m\angle USV \) to both sides)

4. **Statement 4:**
   - **Statement:** \( m\angle RSV = m\angle UST \)
   - **Reason:** Angle Addition Postulate (The measure of a larger angle is the sum of its parts)

5. **Statement 5:**
   - **Statement:** \( \angle RSV \cong \angle UST \)
   - **Reason:** Definition of congruent angles (Angles with equal measures are congruent)

6. **Statement 6:**
   - **Statement:** \( m\angle RSV = m\angle UST \)
   - **Reason:** Definition of congruent angles (using the property that congruent angles have equal measures)

7. **Statement 7:**
   - **Statement:** ∠RSV ≅ ∠UST
   - **Reason:** Conclusion (as required to prove)

This completes the proof that ∠RSV ≅ ∠UST.

Would you like more details on any of these steps or have questions? Here are five related questions that can deepen understanding:

1. What is the significance of the Angle Addition Postulate in geometric proofs?
2. How can we apply the definition of congruent angles to solve other types of geometric problems?
3. What are the common reasons used in proving two angles congruent?
4. How does the Addition Property of Equality help in solving angle congruence problems?
5. What is the difference between "congruent" and "equal" in geometric terms?

**Tip:** In proofs, always ensure the reasoning is logically connected and each step flows clearly from the previous one to avoid gaps in the logic.

Given that ∠RSU ≅ ∠VST, prove that ∠RSV ≅ ∠UST.

This geometric proof demonstrates how to prove that ∠RSV ≅ ∠UST given that ∠RSU ≅ ∠VST, using the Angle Addition Postulate and congruence definitions. Step-by-step reasoning provided for Grades 8-10 students.

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