Question

"Question 9 In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

Class 9 - Math - Quadrilaterals Page 147"

Answer 1

·Parallelogram :

A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram

.A quadrilateral is a parallelogram if

i)Its opposite sides are equal

ii) its opposite angles are equal

iii) diagonals bisect each other

iv) a pair of opposite sides is equal and parallel.

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Given: ABCD is a parallelogram and P and Q are points on BD such that

DP=BQ

 

To show:

 (i) ΔAPD ≅ ΔCQB 

(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram

 

Proof:

(i) In ΔAPD and ΔCQB,

DP = BQ (Given)

∠ADP = ∠CBQ (Alternate interior angles)

AD = BC (Opposite sides of a ||gm

Thus, ΔAPD ≅ ΔCQB                                (by SAS congruence rule)

 

(ii)  since, ΔAPD ≅ ΔCQB.

AP = CQ                                            ( by CPCT )

(iii) In ΔAQB and ΔCPD,

BQ = DP (Given)

∠ABQ = ∠CDP (Alternate interior angles)

AB = CD (Opposite sides of a ||gm)

Thus, ΔAQB ≅ ΔCPD                       (by SAS congruence rule)

(iv) AQ = CP                                      (by CPCT as ΔAQB ≅ ΔCPD.)

(v)  From (ii)  and (iv),

AP=CQ & AQ=CP

 it is clear that APCQ has equal opposite sides also it has equal opposite angles. 

Hence,APCQ is a ||gm.

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Hope this will help you...

Answer 2

Answer:

(iv)

Step-by-step explanation:

Here, ABCD is a parallelogram.

AB∥DC and BD is a transversal.

∴ ∠ABQ=∠CDP [ Alternate angles ] ---- ( 1 )

In △AQB and △CPD,

⇒ AB=CD [ Opposite sides of parallelogram are equal ]

⇒ ∠ABQ=∠CDP [ From ( 1 ) ]

⇒ BQ=DP [ Given ]

∴ △AQB≅△CPD [ By SAS congruence ]

⇒ AQ=CP [ CPCT ]