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

Here we will prove that parallelogram on the same base and between the same parallel lines are equal in area.

Given: PQRS and PQMN are two parallelograms on the same base PQ and between same parallel lines PQ and SM.

To prove: ar(parallelogram PQRS) = ar(parallelogram PQMN).

Construction: Produce QP to T.

Proof:

 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.

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