ABCD-is-a-parallelogram-P-and-Q-are-two-points-on-the-diagonal-BD-such-that-DP-QB-Prove-that-APCQ-is-a-parallelogram?

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ABCD-is-a-parallelogram-P-and-Q-are-two-points-on-the-diagonal-BD-such-that-DP-QB-Prove-that-APCQ-is-a-parallelogram?

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Jade 4 days 2021-11-25T17:01:33+00:00 1 Answer 0 views 0

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    2021-11-25T17:03:19+00:00

    A quadrilateral in which both pairs of opposite sides are parallel is called a Parallelogram

    Step-by-step explanation:

    A quadrilateral is said to be  parallelogram if

    • If its opposite sides are equal
    • If the  opposite angles are equal
    • If the diagonals bisect each other
    • If  a pair of opposite sides is equal and parallel.

    From the question given above

    ABCD is a parallelogram and P and Q are points on BD such that

    DP=QB

    In ΔAPD and ΔCQB,————————–(i)

    DP = QB (Given)

    ∠ADP = ∠CBQ (Alternate interior angles)

    AD = BC

    (Hence it is proved that -Opposite sides of a parallelogram are equal)

    so, ΔAPD ≅ ΔCQB    (As per  SAS congruence rule)

     If , ΔAPD ≅ ΔCQB.————————-(ii)

    AP = CQ               ( by CPCT )

     In ΔAQB and ΔCPD,———————–(iii)

    BQ = DP (Given)

    ∠ABQ = ∠CDP (Alternate interior angles)

    AB = CD  (Opposite sides of a parallelogram)

    so, ΔAQB ≅ ΔCPD                       (As per  SAS congruence rule)

    (iv) AQ = CP              (According to  CPCT as ΔAQB ≅ ΔCPD.)

     From (ii)  and (iv) equation ,we can say that

    AP=CQ ,

    AQ=CP

     It is proved that APCQ has equal opposite sides also it has equal opposite angles. Hence,APCQ is a Parallelogram

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