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 ≅ ∆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.





9th Grade Math

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