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To prove that \(\triangle VKR\) is isosceles using a two-column proof, we'll list the given information, and proceed step-by-step to show the necessary congruences and equalities.

### Given:
1. \( \overline{PQ} \perp \overline{QT} \)
2. \( \overline{ST} \perp \overline{TQ} \)
3. \( PQ = QR \)
4. \( PQ = QT \)
5. \( PQ / ST = QR / TV \)

### Prove:
\(\triangle VKR\) is isosceles.

### Diagram Analysis:
- Points \( P \), \( Q \), \( T \), \( S \), and \( R \) are labeled on the diagram.
- Perpendicular lines imply right angles at \( Q \) and \( T \).
- \( K \) is the intersection of \(\overline{PR}\) and \(\overline{SV}\).
- \( V \) and \( R \) are points on line segment \(\overline{QT}\).

### Two-Column Proof:

| **Statements** | **Reasons** |
|----------------|--------------|
| 1. \( \overline{PQ} \perp \overline{QT} \) | Given |
| 2. \( \overline{ST} \perp \overline{TQ} \) | Given |
| 3. \( PQ = QR \) | Given |
| 4. \( PQ = QT \) | Given |
| 5. \(\angle PQT = \angle STQ = 90^\circ\) | Perpendicular lines form right angles |
| 6. \(\triangle PQT\) and \(\triangle STQ\) are right triangles | Definition of right triangles |
| 7. \(\angle PQT = \angle STQ = 90^\circ\) | Perpendicular lines form right angles |
| 8. \( PQ = QT \) | Given |
| 9. \( \overline{PQ} \cong \overline{QT} \) | Definition of congruence |
| 10. \( PQ = QR \) | Given |
| 11. \(\overline{PQ} \cong \overline{QR}\) | Definition of congruence |
| 12. \(\triangle PQK \cong \triangle QRK\) | Hypotenuse-Leg (HL) Theorem (Right triangles, hypotenuses \( PQ = QR \), and leg \( KQ \) common) |
| 13. \(\angle VKQ = \angle RKQ\) | Corresponding angles of congruent triangles |
| 14. \(VK = KR\) | Corresponding sides of congruent triangles |
| 15. \(\triangle VKR\) is isosceles | Definition of isosceles triangle (two equal sides) |

Thus, we have shown that \(\triangle VKR\) is isosceles as it has two equal sides \( VK \) and \( KR \).

Do you have any questions or would you like further details on any part of the proof? Here are five questions you might consider asking:

1. What is the Hypotenuse-Leg (HL) Theorem?
2. How do we identify corresponding angles in congruent triangles?
3. Can you explain the properties of isosceles triangles?
4. What are the key steps in proving two triangles are congruent?
5. How does the perpendicularity of lines help in proving triangle properties?

**Tip:** When working with geometric proofs, always mark all given information and any deduced equalities or congruences directly on the diagram to help visualize relationships more clearly.

Learn how to prove that triangle VKR is isosceles using a two-column proof. This problem involves properties of triangles, including congruence and the HL Theorem. Suitable for high school students studying geometry.

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