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In parallelogram ABCD, two points P and...

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that `D P = B Q`. Show that:
(i) `DeltaA P D~= DeltaC Q B`
(ii) `A P = C Q`
(iii) `DeltaA B C`
(iv) `A Q = C P`
(v) APCQ is a parallelogram.

Text Solution

Verified by Experts

i) In ΔAPD and ΔCQB,
=>AD = CB (Opposite sides)
=>∠ADP = ∠CBQ (Alternate interior angles)
=> DP = BQ (Given)
So,ΔAPD ≅ ΔCQB (by SAS congruence rule)

(ii) Since ΔAPD ≅ ΔCQB,
∴ AP = CQ

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Knowledge Check

  • ABCD is a parallelogram, P and Q are the points on the diagonal AC=QC, the quadrilateral BPDQ is a :

    A
    trapezium
    B
    parallelogram
    C
    square
    D
    none of these

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