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.
(i) In ∆PQR ≅ ∆RSP
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.
(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.