In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:
(i) Δ APD ≅ Δ CQB
(ii) AP = CQ
(iii) Δ AQB ≅ Δ CPD
(iv) AQ = CP
(v) APCQ is a parallelogram

Solution:

Given: ABCD is a parallelogram and DP = BQ

(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) Since ΔAPD ≅ ΔCQB,

∴ AP = CQ (By CPCT)

(iii) In ΔAQB and ΔCPD,

AB = CD (Opposite sides of parallelogram ABCD)

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

BQ = DP (Given)

∴ ΔAQB ≅ ΔCPD (Using SAS congruence rule)

(iv) Since ΔAQB ≅ ΔCPD,

∴ AQ = CP (CPCT)

(v) From the result obtained in (ii) and (iv), AQ = CP and AP = CQ

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

☛ Check: NCERT Solutions Class 9 Maths Chapter 8

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Video Solution:

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 9

Summary:

If in parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ, then ΔAPD ≅ ΔCQB by SAS congruence, AP = CQ, ΔAQB ≅ ΔCPD by SAS congruence, AQ = CP, and APCQ is a parallelogram.

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