In parallelogram ABCD two-point P and Q are taken on diagonal BD such that DP = BQ. Then the relations are

(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. Then the relations are

(i) $△ APD ≅ △ CQB$

(ii) $AP = CQ$

(iii) $△ AQB ≅ △ CPD$

(iv) $AQ = CP$

(v) APCQ is a parallelogram

Accepted Answer

Given,ABCD = parallelogram in which,Diagonal = BD DP = BQHere,AD∥BC         (parallelogram opposite sides are parallel)BD = transversalSo,∠ADP=∠CBQ          (alternate angles) Now, in △APD and △CQB.AD = CB             (opposite sides of parallelogram are equal)∠ADP=∠CBQ          (Alternate angles) DP = BQ        (given)Thus, by side-angle-side criteria of congruency,△APD≅△CQB So, relation (i) is true.From corresponding parts of congruent triangles,AP = CQSo, relation (ii) is true. Now, we have,AB∥DC           (parallelogram opposite sides are parallel)BD  =transversalSo,In △AQB and △CPD.AB = CD           (opposite sides of parallelogram) ∠ABQ=∠CDP       (Alternate angles)BQ = DP      (given)Thus, by side-angle-side criteria of congruency,△AQB≅△CPDSo, relation (iii) is true.From corresponding parts of congruent triangles,AQ = CPSo, relation (iv) is true.Now we have,AP = CQ and AQ = CP.As, in APCQ both pairs of opposite sides are equal. Thus, APCQ = parallelogram.So, relation (v) is true.So, all the relations are true.Option 1 is correct.

In parallelogram ABCD two-point P and Q are taken on diagonal BD such that DP = BQ. Then the relations are

(i) $△ APD ≅ △ CQB$

(ii) $AP = CQ$

(iii) $△ AQB ≅ △ CPD$

(iv) $AQ = CP$

(v) APCQ is a parallelogram

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In parallelogram ABCD two-point P and Q are taken on diagonal BD such that DP = BQ. Then the relations are(i) △APD≅ △CQB(ii) AP=CQ(iii) △AQB≅ △CPD(iv) AQ=CP(v) APCQ is a parallelogram

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In parallelogram ABCD two-point P and Q are taken on diagonal BD such that DP = BQ. Then the relations are

(i) $△ APD ≅ △ CQB$

(ii) $AP = CQ$

(iii) $△ AQB ≅ △ CPD$

(iv) $AQ = CP$

(v) APCQ is a parallelogram