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Answers
Given :
- ABCD is a parallelogram and P and Q are points on diagonal BD such that DP = BQ.
To Prove :
- (i) ∆APD ≅ ∆CQB
- (ii) AP = CQ
- (iii) ∆AQB ≅ ∆CPD
- (iv) AQ = CP
- (v) APCQ is a parallelogram
Proof :
(i) In ∆APD and ∆CQB, we have
AD = BC [Opposite sides of a ||gm]
DP = BQ [Given]
∠ADP = ∠CBQ [Alternate angles]
∴ ∆APD ≅ ∆CQB [SAS congruence]
(ii) ∴ AP = CQ [CPCT]
(iii) In ∆AQB and ∆CPD, we have
AB = CD [Opposite sides of a ||gm]
DP = BQ [Given]
∠ABQ = ∠CDP [Alternate angles]
∴ ∆AQB ≅ ∆CPD [SAS congruence]
(iv) ∴ AQ = CP [CPCT]
(v) Since in APCQ, opposite sides are equal, therefore it is a parallelogram.
#Proved.
Answer:
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Given :
ABCD is a parallelogram and P and Q are points on diagonal BD such that DP = BQ.
To Prove :
(i) ∆APD ≅ ∆CQB
(ii) AP = CQ
(iii) ∆AQB ≅ ∆CPD
(iv) AQ = CP
(v) APCQ is a parallelogram
Proof :
(i) In ∆APD and ∆CQB, we have
AD = BC [Opposite sides of a ||gm]
DP = BQ [Given]
∠ADP = ∠CBQ [Alternate angles]
∴ ∆APD ≅ ∆CQB [SAS congruence]
(ii) ∴ AP = CQ [CPCT]
(iii) In ∆AQB and ∆CPD, we have
AB = CD [Opposite sides of a ||gm]
DP = BQ [Given]
∠ABQ = ∠CDP [Alternate angles]
∴ ∆AQB ≅ ∆CPD [SAS congruence]
(iv) ∴ AQ = CP [CPCT]
(v) Since in APCQ, opposite sides are equal, therefore it is a parallelogram.
#Proved.
Step-by-step explanation:
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