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 ≅ ∆RSP 1. ∠QPR = ∠PRS 2. ∠QRP = ∠RPS 3. PR = PR 4. ∆PQR ≅ ∆RSP. Similarly, ∆PQS ≅ ∆RSQ. (Proved) |
Reason 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. |
(ii). In ∆OPQ ≅ ∆ORS 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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