The diagonals of a parallelogram are equal. (True Or False)

Answer

To determine whether the statement "The diagonals of a parallelogram are equal" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of a Parallelogram**:
- A parallelogram is a quadrilateral with opposite sides that are equal and parallel. The opposite angles are also equal.

2. **Drawing a Parallelogram**:
- Let's draw a parallelogram ABCD. Label the vertices as A, B, C, and D.

3. **Identifying the Diagonals**:
- The diagonals of the parallelogram are AC and BD. We need to analyze these diagonals.

4. **Using Congruent Triangles**:
- To determine if the diagonals are equal, we can use the concept of congruent triangles. We will consider triangles ADC and BCD.

5. **Identifying Sides and Angles**:
- In triangles ADC and BCD:
- AD = BC (opposite sides of a parallelogram are equal).
- CD is common to both triangles.
- We need to check if angles ACD and BCD are equal.

6. **Analyzing Angles**:
- The angles ACD and BCD may not necessarily be equal unless the parallelogram is a special type (like a rectangle or square). In a general parallelogram, these angles can be different.

7. **Conclusion**:
- Since we cannot guarantee that angles ACD and BCD are equal in a general parallelogram, we cannot conclude that the triangles ADC and BCD are congruent. Therefore, the diagonals AC and BD are not necessarily equal.

8. **Final Statement**:
- The statement "The diagonals of a parallelogram are equal" is **False**.