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Prove the statement using a two-column proof or paragraph proof.
M
L
Given: JKLM is a parallelogram; KL≈ LM; JL MK
Prove: JKLM is a rhombus.
Complete your proof in the box below. You will be awarded 5 points for your
statements and 5 points for your reasons.

To prove that \( JKLM \) is a rhombus, we can use a two-column proof. Here’s how the proof works based on the given information:

### **Given:**
1. \( JKLM \) is a parallelogram.
2. \( KL \cong LM \) (congruent segments).
3. \( JL \perp MK \) (perpendicular diagonals).

### **To Prove:**
\( JKLM \) is a rhombus.

### **Proof:**

| **Statements**                                             | **Reasons**                                                   |
|------------------------------------------------------------|---------------------------------------------------------------|
| 1. \( JKLM \) is a parallelogram.                          | Given                                                         |
| 2. \( KL \cong LM \)                                       | Given                                                         |
| 3. \( JL \perp MK \)                                       | Given                                                         |
| 4. Opposite sides of a parallelogram are congruent.        | Definition of parallelogram                                   |
| 5. \( JK \cong ML \) and \( JM \cong KL \)                 | Opposite sides of a parallelogram are congruent (Statement 4) |
| 6. \( KL \cong ML \)                                       | Transitive property (Statement 2 and Statement 5)             |
| 7. \( JK \cong KL \cong LM \cong MJ \)                     | All sides are congruent (Statement 2, 4, and 6)               |
| 8. \( JL \perp MK \)                                        | Given (perpendicular diagonals in a rhombus)                  |
| 9. \( JKLM \) is a rhombus.                                | A parallelogram with all sides congruent and perpendicular diagonals is a rhombus.  |

### Explanation:

1. **Statement 1:** \( JKLM \) is a parallelogram. This is given.
2. **Statement 2:** \( KL \cong LM \), also given, implies two adjacent sides of the parallelogram are congruent.
3. **Statement 3:** \( JL \perp MK \), meaning the diagonals are perpendicular, which is a property of rhombuses.
4. **Statement 4:** Opposite sides of a parallelogram are congruent. Therefore, \( JK \cong ML \) and \( JM \cong KL \).
5. **Statement 6:** Given that \( KL \cong LM \) and \( ML \cong JK \), by transitive property, all sides are congruent.
6. **Statement 8:** Perpendicular diagonals (given) confirm that \( JKLM \) must be a rhombus, as the diagonals of a rhombus are perpendicular to each other.

### **Conclusion:**
Since all sides of \( JKLM \) are congruent and the diagonals are perpendicular, \( JKLM \) is a rhombus.

Would you like more details or have any further questions?

### Related Questions:

1. What is the difference between a rhombus and a square?
2. Can a parallelogram be a rhombus without having perpendicular diagonals?
3. How do the properties of a rhombus differ from those of a general parallelogram?
4. What are the conditions for a parallelogram to be classified as a rectangle?
5. Can a parallelogram be a square? If so, under what conditions?
6. How do you prove that a quadrilateral is a parallelogram?
7. What properties of diagonals can be used to determine if a parallelogram is a rhombus?
8. If one angle of a parallelogram is a right angle, what can you conclude about the parallelogram?

### Tip:
When proving a geometric figure is a specific type of quadrilateral, always consider the properties unique to that figure, such as congruent sides, perpendicular diagonals, or equal angles.

Learn how to prove that JKLM is a rhombus using properties of parallelograms and rhombuses. Detailed two-column proof with step-by-step reasoning and congruence principles. Suitable for high school geometry students.

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