Q. 133.9( 19 Votes )

# Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

Answer :

Let ABCD is a parallelogram.

Since, DC||AB and DA is transversal.

∠A + ∠D = 180

∠A + ∠D = 90

∠PAD + ∠PDA = 90

∠APD = 90

∠SPQ = 90

Similarly, ∠PQR = 90, ∠QRS = 90

And ∠PSR = 90

Thus, PQRS is a quadrilateral each of whose angles is 90.

Hence, PQRS is a rectangle.

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PREVIOUSE and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = (AB + CD).[Hint: Join BE and produce it to meet CD produced at G.]NEXTP and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.

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