Here we will discuss about the diagonals of a parallelogram bisect each other.

In a parallelogram, diagonals bisect each other and each diagonal bisects the parallelogram into two congruent triangles.

**Given:** PQRS is a parallelogram in which PQ ∥ SR and PS ∥ QR.
Its diagonals PR and QS cut each other at O.

**To prove:** (i) ∆PQR ≅ ∆RSP, ∆PQS ≅ ∆RSQ.

(ii) OP = OR, OQ = OS.

**Proof:**

1. ∠QPR = ∠PRS 2. ∠QRP = ∠RPS 3. PR = PR 4. ∆PQR ≅ ∆RSP. Similarly, ∆PQS ≅ ∆RSQ. (Proved) |
1. PQ ∥ SR and PR is a transversal. 2. PS ∥ QR and PR is a transversal. 3. Common side. 4. By ASA criterion of congruency. |

5. PQ = RS 6. ∠QPO = ∠ORS 7. ∠PQO = ∠RSO 8. ∆OPQ ≅ ∆ORS. Therefore, OP = OR, QO = OS (Proved). |
5. CPCTC from statement 4. 6. PQ ∥ SR and PR is a transversal. 7. PQ ∥ SR and QS is a transversal. 8. By SAS criterion of congruency. CPCTC. |

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