If ABCD is a parallelogram with diagonals intersecting at `O`, then the number of distinct pairs of congruent triangles formed is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the number of distinct pairs of congruent triangles formed when the diagonals of a parallelogram intersect. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Parallelogram**:
- Let ABCD be a parallelogram with diagonals AC and BD intersecting at point O.

2. **Properties of Diagonals in a Parallelogram**:
- The diagonals of a parallelogram bisect each other. Therefore, AO = OC and BO = OD.

3. **Identifying Congruent Triangles**:
- We can identify triangles formed by the diagonals:
- Triangle AOD and Triangle BOC.
- Triangle AOB and Triangle COD.

4. **Using the Congruence Criteria**:
- For Triangle AOD and Triangle BOC:
- AO = OC (as diagonals bisect each other)
- BO = OD (as diagonals bisect each other)
- Angle AOD = Angle BOC (vertically opposite angles)
- By the Side-Angle-Side (SAS) congruence criterion, Triangle AOD ≅ Triangle BOC.

5. **Identifying Another Pair of Congruent Triangles**:
- For Triangle AOB and Triangle COD:
- AO = OC (as diagonals bisect each other)
- BO = OD (as diagonals bisect each other)
- Angle AOB = Angle COD (vertically opposite angles)
- Again, by the SAS congruence criterion, Triangle AOB ≅ Triangle COD.

6. **Considering the Whole Parallelogram**:
- The diagonal AC divides the parallelogram into two congruent triangles:
- Triangle ABC and Triangle CDA.
- By the Side-Side-Side (SSS) congruence criterion, Triangle ABC ≅ Triangle CDA.

7. **Another Set of Congruent Triangles**:
- The diagonal BD divides the parallelogram into two congruent triangles:
- Triangle BCD and Triangle DAB.
- By the SSS congruence criterion, Triangle BCD ≅ Triangle DAB.

8. **Counting Distinct Pairs of Congruent Triangles**:
- We have identified the following pairs of congruent triangles:
1. Triangle AOD ≅ Triangle BOC
2. Triangle AOB ≅ Triangle COD
3. Triangle ABC ≅ Triangle CDA
4. Triangle BCD ≅ Triangle DAB

9. **Conclusion**:
- Therefore, the total number of distinct pairs of congruent triangles formed is **4**.

### Final Answer:
The number of distinct pairs of congruent triangles formed is **4**.