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In this proof, we are asked to show that triangles \( \triangle QRT \) and \( \triangle QST \) are congruent. Here’s how to complete the missing reason for Statement 3.

**Proof Structure**

### Given Information:
1. \( T \) is the midpoint of \( \overline{RS} \).
2. \( \overline{RS} \perp \overline{QT} \).

### Statements and Reasons:
1. **Statement:** \( T \) is the midpoint of \( \overline{RS} \)  
   **Reason:** Given.

2. **Statement:** \( \overline{RS} \perp \overline{QT} \)  
   **Reason:** Given.

3. **Statement:** \( \angle QRT \cong \angle QTS \)  
   **Reason:** _Perpendicular lines form congruent right angles._

4. **Statement:** \( \overline{RT} \cong \overline{ST} \)  
   **Reason:** Since \( T \) is the midpoint of \( \overline{RS} \).

5. **Statement:** \( \overline{QT} \cong \overline{QT} \)  
   **Reason:** Reflexive Property of Congruence.

6. **Statement:** \( \triangle QRT \cong \triangle QST \)  
   **Reason:** Side-Angle-Side (SAS) congruence.

With this reasoning, we complete Statement 3 by stating:
- **Reason:** "Perpendicular lines form congruent right angles."

Would you like further clarification on any step?

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### Related Questions
1. Why is the midpoint property important for this proof?
2. How does the Side-Angle-Side (SAS) criterion work in proving triangle congruence?
3. Can triangles still be congruent without perpendicular angles?
4. How does the Reflexive Property of Congruence help in proofs?
5. What other congruence criteria (like ASA or SSS) could apply here?

### Tip
Remember, perpendicular lines always form right angles, which are congruent. This is often crucial in triangle congruence proofs.

T is the midpoint of RS and RS ⊥ QT. Complete the proof that ΔQRT ≅ ΔQST.

Learn how to complete a proof to show that triangles ΔQRT and ΔQST are congruent using the Side-Angle-Side (SAS) theorem. The problem explores key concepts like midpoints, perpendicular lines, and the properties of congruent triangles. Suitable for Grade 9 students.

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