Question

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that:

1. ΔAPD ≅ ΔCQB
2. AP = CQ
3. ΔAQB ≅ ΔCPD
4. AQ = CP
5. APCQ is a parallelogram

Answer 1

(i) In ΔAPD and ΔCQB,

∠ADP = ∠CBQ     ...(Alternate interior angles for BC || AD)

AD = CB              ...(Opposite sides of parallelogram ABCD)

DP = BQ          ...(Given)

∴ ΔAPD ≅ ΔCQB        ...(Using SAS congruence rule)

(ii) As we had observed that ΔAPD ≅ ΔCQB,

∴ AP = CQ       ...(CPCT)

(iii) In ΔAQB and ΔCPD,

∠ABQ = ∠CDP       ...(Alternate interior angles for AB || CD)

AB = CD                ...(Opposite sides of parallelogram ABCD)

BQ = DP      ...(Given)

∴ ΔAQB ≅ ΔCPD        ...(Using SAS congruence rule)

(iv) As we had observed that ΔAQB ≅ ΔCPD,

∴ AQ = CP           ...(CPCT)

(v) From the results obtained in (ii) and (iv),

AQ = CP and

AP = CQ

Since opposite sides in quadrilateral APCQ are equal to each other, APCQ is a parallelogram.