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  • In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (v) APCQ is a parallelogram

Q : 9  In parallelogram ABCD, two points P and Q are taken on diagonal BD such that  \small DP=BQ (see Fig. \small 8.20). Show that:

 APCQ is a parallelogram

    

Answers (1)

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Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ.

To prove: APCQ is a parallelogram

Proof :

In \small \Delta APD\, \, and\, \, \Delta CQB,

 DP=BQ       (Given )

 \angleADP=\angleCBQ (alternate angles)

 AD=BC       (opposite sides of a parallelogram)

\small \Delta APD\cong \Delta CQB    (By SAS)

 \small AP=CQ       (CPCT)...............................................................1

Also,

In \small \Delta AQB\, \, and\, \, \Delta CPD,

 DP=BQ       (Given )

 \angleABQ=\angleCDP  (alternate angles)

 AB=CD       (opposite sides of a parallelogram)

\small \Delta AQB\cong \Delta CPD      (By SAS)          

   \small AQ=CP           (CPCT)........................................2

From equation 1 and 2, we get 

\small AP=CQ

\small AQ=CP

Thus, opposite sides of quadrilateral APCQ are equal so APCQ is a parallelogram.

 

 

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