Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area

            Statement

           Reason

1. PS = QR.

1. Opposite sides of the parallelogram PQRS.

2. PN = QM.

2. Opposite sides of the parallelogram PQMN.

3. ∠SPT = ∠RQT.

3. Opposite sides PS and QR are parallel and TPQ is a transversal.

4. ∠NPT = ∠MQT.

4. Opposite sides PN and QM are parallel and TPQ is a transversal.

5. ∠NPS = ∠MQR.

5. Subtracting statements 3 and 4.

6. ∆PSN ≅ ∆RQM

6. By SAS axiom of congruency.

7. ar(∆PSN) ≅ ar(∆RQM).

7. By area axiom for congruent figures.

8. ar(∆PSN) + ar(quadrilateral PQRN) = ar(∆RQM) + ar(quadrilateral PQRN)

8. Adding the same area on both sides of the equality in statement 7.

9. ar(parallelogram PQRS) = ar(parallelogram PQMN). (Proved)

9. By addition axiom for area.