Question

☺☺☺☺ABCD is a parallelogram. A straight line is drawn parallel to the diagonal BD , cutting BC and CD in P and Q. Prove that ∆ABP and ∆ADQ are equal in areas☺☺☺☺

Answer 1

Parallelogram :

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

.A
quadrilateral is a parallelogram if

i)Its
opposite sides are equal

ii)
its opposite angles are equal

iii)
diagonals bisect each other

iv)
a pair of opposite sides is equal and parallel.

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

DP=BQ

 

To show:

 (i) ΔAPD ≅ ΔCQB 

(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram
 

Proof:

(i) In ΔAPD and ΔCQB,


DP = BQ (Given)


∠ADP = ∠CBQ (Alternate interior angles)



AD = BC (Opposite sides of a ||gm



Thus, ΔAPD ≅ ΔCQB                                (by SAS congruence rule)

 

(ii)  since, ΔAPD ≅ ΔCQB.

AP = CQ                                            ( by CPCT )





(iii) In ΔAQB and ΔCPD,



BQ = DP (Given)


∠ABQ = ∠CDP (Alternate
interior angles)


AB = CD (Opposite sides of a ||gm)


Thus, ΔAQB ≅ ΔCPD                       (by SAS congruence rule)



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



(v)  From (ii)  and (iv),

AP=CQ & AQ=CP

 it is clear that APCQ has equal
opposite sides also it has equal opposite angles. 

Hence,APCQ is a ||gm.



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$hope \: it \: helps \: u$

Answer 2

Answer:

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Step-by-step explanation:

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