The diagonals of parallelogram bisect each other ( True/ False)

Answer

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

### Step-by-Step Solution:

1. **Draw a Parallelogram**:
- Begin by sketching a parallelogram and label its vertices as A, B, C, and D.

2. **Draw the Diagonals**:
- Draw the diagonals AC and BD. Let the point where the diagonals intersect be labeled as O.

3. **Identify the Properties of a Parallelogram**:
- Recall that in a parallelogram, opposite sides are equal in length. Thus, we have:
- \( AD = BC \) (opposite sides)
- \( AB = CD \) (also opposite sides)

4. **Consider Triangles**:
- Focus on triangles AOD and BOC. We will prove that these two triangles are congruent.

5. **Use the Properties of Parallel Lines**:
- Since AD is parallel to BC, we can use the properties of alternate interior angles:
- \( \angle ADO = \angle CBO \) (alternate interior angles)
- \( \angle DAO = \angle BCO \) (alternate interior angles)

6. **Apply the ASA Congruence Criterion**:
- We have:
- \( AD = BC \) (sides)
- \( \angle ADO = \angle CBO \) (angles)
- \( \angle DAO = \angle BCO \) (angles)
- Therefore, by the Angle-Side-Angle (ASA) congruence criterion, we can conclude that:
- \( \triangle AOD \cong \triangle BOC \)

7. **Conclude with CPCT**:
- From the congruence of triangles AOD and BOC, we can state that:
- \( AO = CO \) (corresponding parts of congruent triangles)
- \( BO = DO \) (corresponding parts of congruent triangles)

8. **Final Conclusion**:
- Since AO = CO and BO = DO, we can conclude that the diagonals of the parallelogram bisect each other.

### Final Answer:
Thus, the statement "The diagonals of a parallelogram bisect each other" is **True**.

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