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To complete the proof that \( \triangle RSU \cong \triangle RTU \), let's examine the given information and fill in the missing reasons.

### Proof:

1. **Statement**: \( U \) is the midpoint of \( \overline{ST} \)  
   **Reason**: Given

2. **Statement**: \( \overline{RU} \perp \overline{ST} \)  
   **Reason**: Given

3. **Statement**: \( \angle RUS \cong \angle RUT \)  
   **Reason**: **Definition of perpendicular lines** (since \( \overline{RU} \perp \overline{ST} \), the angles \( \angle RUS \) and \( \angle RUT \) are right angles and thus congruent)

4. **Statement**: \( \overline{SU} \cong \overline{TU} \)  
   **Reason**: **Definition of midpoint** (since \( U \) is the midpoint of \( \overline{ST} \), it divides \( \overline{ST} \) into two congruent segments)

5. **Statement**: \( \overline{RU} \cong \overline{RU} \)  
   **Reason**: Reflexive Property of Congruence (any segment is congruent to itself)

6. **Statement**: \( \triangle RSU \cong \triangle RTU \)  
   **Reason**: **SAS** (Side-Angle-Side congruence postulate)

### Summary:
The triangles \( \triangle RSU \) and \( \triangle RTU \) are congruent by the SAS postulate, as they have a pair of congruent sides (\( \overline{SU} \cong \overline{TU} \)), a pair of congruent angles (\( \angle RUS \cong \angle RUT \)), and another pair of congruent sides (\( \overline{RU} \cong \overline{RU} \)).

Would you like more details on any specific step, or do you have any questions about this proof?

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Here are five related questions to deepen your understanding:

1. What other methods (besides SAS) can prove triangles congruent?
2. Why is the Reflexive Property necessary in triangle congruence proofs?
3. How does the definition of perpendicular lines help in proving angle congruence?
4. What role does the definition of a midpoint play in congruence proofs?
5. Can you use this proof method in other shapes besides triangles?

**Tip**: Always check the requirements of congruence postulates (like SAS, SSS, or ASA) when proving triangles are congruent.

U is the midpoint of ST. Complete the proof that ΔRSU ≅ ΔRTU.

Learn how to prove that triangles ΔRSU and ΔRTU are congruent using the SAS postulate. This geometry problem covers key concepts such as midpoints, perpendicular lines, and the reflexive property of congruence. Ideal for students in Grades 8-10.

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