Question

Prove that the area of a quadrilateral between a base and parallel lines is equal.​

Answer 1

Answer:

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.

Step-by-step explanation:

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