Which of the following statements is true ?

Options

A. In a parallelogram, the diagonals are equal.
B. In a parallelogram, the diagonals bisect each other
C. In a parallelogram, the diagonals intersect each other at right angles.
D. In any quadrilateral, if a pair of opposite sides are equal, it is parallelogram.

Answer

To determine which statement about parallelograms and quadrilaterals is true, let's analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding the Properties of a Parallelogram**:
- A parallelogram is defined as a quadrilateral with opposite sides that are equal in length and parallel.

2. **Evaluating Statement A**:
- **Statement A**: "In a parallelogram, the diagonals are equal."
- This statement is **false**. In a general parallelogram, the diagonals are not necessarily equal. They are equal only in the special case of a rectangle.

3. **Evaluating Statement B**:
- **Statement B**: "In a parallelogram, the diagonals bisect each other."
- This statement is **true**. In any parallelogram, the diagonals do bisect each other, meaning they cut each other into two equal halves.

4. **Evaluating Statement C**:
- **Statement C**: "In a parallelogram, the diagonals intersect each other."
- This statement is somewhat misleading. While it is true that the diagonals intersect, the statement does not specify that they bisect each other, which is a more precise property. However, since they do intersect, this statement is technically true but less informative than statement B.

5. **Evaluating Statement D**:
- **Statement D**: "In a quadrilateral, if a pair of opposite sides are equal."
- This statement is **false**. A quadrilateral can have a pair of opposite sides equal without being a parallelogram. For example, a kite has two pairs of adjacent sides equal but does not satisfy the properties of a parallelogram.

6. **Conclusion**:
- The only statement that is definitively true regarding the properties of a parallelogram is **Statement B**: "In a parallelogram, the diagonals bisect each other."

### Final Answer:
The true statement is **B**: "In a parallelogram, the diagonals bisect each other."