If the diagonals of a parallelogram are equal, then it is a

Options

A. Rectangle
B. Rhombus
C. Square
D. Kite

Explanation

To prove that if the diagonals of a parallelogram are equal, then it is a rectangle, we can follow these steps:

### Step-by-step Solution:

1. **Draw a Parallelogram**:
- Let’s consider a parallelogram PQRS where the diagonals PR and QS are drawn.

2. **Given Information**:
- We are given that the diagonals PR and QS are equal, i.e., PR = QS.

3. **Identify Triangles**:
- We will analyze triangles PQR and SRQ.

4. **Use Properties of Parallelograms**:
- In parallelogram PQRS, opposite sides are equal:
- PQ = SR (opposite sides)
- QR = QR (common side)

5. **Apply the Given Condition**:
- We know that PR = QS (given).

6. **Congruence of Triangles**:
- By the Side-Side-Side (SSS) congruence criterion, we can conclude that:
- Triangle PQR ≅ Triangle SRQ.

7. **Corresponding Angles**:
- Since the triangles are congruent, the corresponding angles are equal:
- ∠PQR = ∠SRQ.

8. **Use the Property of Parallelograms**:
- In a parallelogram, the sum of adjacent angles is 180 degrees:
- ∠PQR + ∠SRQ = 180°.

9. **Substituting the Angles**:
- Since ∠PQR = ∠SRQ, we can substitute:
- ∠PQR + ∠PQR = 180°,
- 2∠PQR = 180°.

10. **Solving for Angles**:
- Dividing both sides by 2 gives:
- ∠PQR = 90°.

11. **Conclusion About Other Angles**:
- Since ∠PQR = 90°, and ∠SRQ = 90°, we can conclude that:
- All angles in parallelogram PQRS are 90°.

12. **Final Conclusion**:
- A parallelogram with all angles equal to 90° is a rectangle. Therefore, PQRS is a rectangle.

### Final Statement:
If the diagonals of a parallelogram are equal, then it is a rectangle.

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