# Diagonals of a Parallelogram Bisect each Other

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:

 Statement(i) In ∆PQR ≅ ∆RSP1. ∠QPR = ∠PRS2. ∠QRP = ∠RPS3. PR = PR4. ∆PQR ≅ ∆RSP. Similarly, ∆PQS ≅ ∆RSQ. (Proved) Reason1. PQ ∥ SR and PR is a transversal.2. PS ∥ QR and PR is a transversal.3. Common side.4. By ASA criterion of congruency.
 (ii). In ∆OPQ ≅ ∆ORS5. PQ = RS6. ∠QPO = ∠ORS7. ∠PQO = ∠RSO8. ∆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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