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

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

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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