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प्रश्न
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that:
- ΔAPD ≅ ΔCQB
- AP = CQ
- ΔAQB ≅ ΔCPD
- AQ = CP
- APCQ is a parallelogram
उत्तर
(i) In ΔAPD and ΔCQB,
∠ADP = ∠CBQ ...(Alternate interior angles for BC || AD)
AD = CB ...(Opposite sides of parallelogram ABCD)
DP = BQ ...(Given)
∴ ΔAPD ≅ ΔCQB ...(Using SAS congruence rule)
(ii) As we had observed that ΔAPD ≅ ΔCQB,
∴ AP = CQ ...(CPCT)
(iii) In ΔAQB and ΔCPD,
∠ABQ = ∠CDP ...(Alternate interior angles for AB || CD)
AB = CD ...(Opposite sides of parallelogram ABCD)
BQ = DP ...(Given)
∴ ΔAQB ≅ ΔCPD ...(Using SAS congruence rule)
(iv) As we had observed that ΔAQB ≅ ΔCPD,
∴ AQ = CP ...(CPCT)
(v) From the results obtained in (ii) and (iv),
AQ = CP and
AP = CQ
Since opposite sides in quadrilateral APCQ are equal to each other, APCQ is a parallelogram.
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