ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the AB and BC.

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#### Solution

Given: ABCD is a parallelogram

To prove: BP × DQ = AB × BC

Proof: In ΔABP and ΔQDA

∠B = ∠D [Opposite angles of parallelogram]

∠BAP = ∠AQD [Alternate interior angles]

Then, ΔABP ~ ΔQDA [By AA similarity]

`therefore"AB"/"QD"="BP"/"DA"` [Corresponding parts of similar Δ are proportional]

But, DA = BC [Opposite sides of parallelogram]

Then, `therefore"AB"/"QD"="BP"/"BC"`

⇒ AB × BC = QD × BP

Concept: Criteria for Similarity of Triangles

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