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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:

              (v) APCQ is a parallelogram

                

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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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