Question

9. In parallelogram ABCD, two points P and Q are
taken on diagonal BD such that DP = BQ
(see Fig. 7.20). Show that:
(i) A APD=ACQB
(ii) AP=CQ
(iii) A AQBEACPD
(iv) AQ=CP
(v) APCQ is a parallelogram
Fig. 7.20
10 ABCD is a parallelogram and AP and co​

Answer 1

Figure:-

Given:-

• In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ.

To find:-

• <APD = <CQB
• AP=CQ
• <AQB and <CPD
• AQ=CP
• APCQ is a parallelogram

Solutions:-

(i) In ∆APD and ∆CQB,

<ADP = <CBQ (Alternate internal angle for BC//AD

AD = CB (opposite sides of parallelogram ABCD)

DP = BQ (Given)

.:. ∆APD ~ ∆CQB (using SAS Congruence rule)

(ii). As observed that ∆APD ~ ∆CQB,

.:. AP = CQ (CPCT)

(iii). In ∆AQB and ∆CPD,

<ABQ = <CDP (alternate interior angle for AB//CD)

AB = CD (opposite sides of parallelogram ABCD)

BQ = DP (Given)

.:. ∆AQB ~ ∆CPD (using SAS Congruence)

(iv). As observed that ∆AQB ~ ∆CPD,

.:. AQ = CP (CPCT)

(v) from the results obtained in (ii). and (iv).

AQ = CP and AP = CQ

Hence, Opposite sides in quadrilateral APCQ are equal to each other, APCQ is a parallelogram.