In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that(i) △APD≅△CQB(ii) AP=CQ(iii) △AQB≅△CPD(iv) AQ=CP(v) APCQ is a parallelogram
Question
In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that(i) △APD≅△CQB(ii) AP=CQ(iii) △AQB≅△CPD(iv) AQ=CP(v) APCQ is a parallelogram
Solution
Sure, here is the step by step solution:
(i) In triangle APD and triangle CQB, we have:
AD = BC (Opposite sides of a parallelogram are equal)
DP = BQ (Given)
∠ADP = ∠BCQ (Alternate angles as AB || DC)
So, by the ASA criterion, triangle APD is congruent to triangle CQB.
(ii) Since triangle APD is congruent to triangle CQB, their corresponding parts are equal. Therefore, AP = CQ.
(iii) In triangle AQB and triangle CPD, we have:
AQ = CP (From (ii))
BQ = DP (Given)
∠ABQ = ∠DCP (Alternate angles as AB || DC)
So, by the ASA criterion, triangle AQB is congruent to triangle CPD.
(iv) Since triangle AQB is congruent to triangle CPD, their corresponding parts are equal. Therefore, AQ = CP.
(v) In quadrilateral APCQ, we have:
AP = CQ (From (ii))
AQ = CP (From (iv))
So, opposite sides of quadrilateral APCQ are equal, which means APCQ is a parallelogram.
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