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The problem is asking to complete the proof showing that \(\triangle PST \cong \triangle QRT\).

### Let's break down the given information and complete the proof:

#### Given:
1. \( PS \perp PR \) (Line \( PS \) is perpendicular to line \( PR \)).
2. \( QS \perp QR \) (Line \( QS \) is perpendicular to line \( QR \)).
3. \( QT \cong PT \) (Line segment \( QT \) is congruent to line segment \( PT \)).

Now we can analyze the rest of the statements to complete the proof:

### Step-by-Step Reasoning:

1. \( \angle P \cong \angle Q \) because both angles are right angles (by the definition of perpendicular lines). Since \( PS \perp PR \) and \( QS \perp QR \), both angles \( \angle P \) and \( \angle Q \) are \( 90^\circ \).  
   **Reason:** Both are right angles.
   
2. \( \angle PTS \cong \angle QTR \). These are vertical angles, and vertical angles are congruent.
   **Reason:** Vertical angles are congruent.

3. \( \triangle PST \cong \triangle QRT \). This follows from the Angle-Side-Angle (ASA) Congruence Postulate. We have two congruent angles (right angles and vertical angles) and a congruent side \( QT \cong PT \).
   **Reason:** ASA Congruence Postulate.

### Final Completed Proof:

| Statement                     | Reason                            |
|--------------------------------|-----------------------------------|
| 1. \( PS \perp PR \)           | Given                             |
| 2. \( QS \perp QR \)           | Given                             |
| 3. \( QT \cong PT \)           | Given                             |
| 4. \( \angle P \cong \angle Q \)| Both are right angles            |
| 5. \( \angle PTS \cong \angle QTR \) | Vertical angles are congruent  |
| 6. \( \triangle PST \cong \triangle QRT \) | ASA Congruence Postulate  |

Would you like more details on any part of the proof? 

Here are five related questions you can explore:
1. How do perpendicular lines guarantee right angles?
2. What is the ASA Congruence Postulate?
3. Can vertical angles be used in other triangle congruence proofs?
4. What are some other postulates or theorems to prove triangle congruence?
5. How does the SSA (Side-Side-Angle) condition differ from ASA?

**Tip:** Always look for right angles and vertical angles in geometric proofs involving perpendicular lines and intersecting lines. These can be key to unlocking congruence between triangles.

Complete the proof that triangle PST is congruent to triangle QRT using given information such as perpendicular lines and congruent segments.

Learn how to complete a geometric proof that shows triangle PST is congruent to triangle QRT using the ASA Congruence Postulate. This problem involves concepts like vertical angles, right angles, and congruent segments. Suitable for students in Grades 8-10.

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