Question

In parallelogram ABCD, two points P and Q are taken on the diagonal BD such that DP=BQ. show that:
(1) ∆APD=~∆CQB
(2) AP=CQ
(3) ∆AQB=~∆CPD
(4) AQ=CP
(5) APCQ is a parallelogram
I will make you brainlist....​

Answer 1

Construction: Join AC to meet BD in O.

Therefore, OB=OD and OA=OC ...(1)

(Diagonals of a parallelogram bisect each other)

But BQ=DP ...given

∴OB–BQ=OD–DP

∴OQ=OP ....(2)

Now, in □APCQ,

OA=OC ....from (1)

OQ=OP ....from (2)

∴□APCQ is a parallelogram.

In △APD and △CQB,

AD=CB ....opposite sides of a parallelogram

AP=CQ ....opposite sides of a parallelogram

DP=BQ ...given

△APD≅△CQB ...By SSS test of congruence

∴AP=CQ ...c.s.c.t.

AQ=CP ...c.s.c.t. ...(3)

In △AQB and △CPD,

AB=CD ....opposite sides of a parallelogram

AQ=CP ...from (3)

BQ=DP ...given

∴△AQB≅△CPD ....By SSS test of congruence

Answer 2

Answer:

Your answer is given below,

Step-by-step explanation:

(i) In ΔAPD and ΔCQB,

DP = BQ (Given)

∠ADP = ∠CBQ (Alternate interior angles)

AD = BC (Opposite sides of a parallelogram)

Thus, ΔAPD ≅ ΔCQB [SAS congruency]

(ii) AP = CQ by CPCT as ΔAPD ≅ ΔCQB.

(iii) In ΔAQB and ΔCPD,

BQ = DP (Given)

∠ABQ = ∠CDP (Alternate interior angles)

AB = CD (Opposite sides of a parallelogram)

Thus, ΔAQB ≅ ΔCPD [SAS congruency]

(iv) As ΔAQB ≅ ΔCPD

AQ = CP [CPCT]

(v) From the questions (ii) and (iv), it is clear that APCQ has equal opposite sides and also has equal and opposite angles. , APCQ is a parallelogram.