Question

ABCD-is-a-parallelogram-P-and-Q-are-two-points-on-the-diagonal-BD-such-that-DP-QB-Prove-that-APCQ-is-a-parallelogram?

Answer 1

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

Step-by-step explanation:

A quadrilateral is said to be  parallelogram if

• If its opposite sides are equal
• If the  opposite angles are equal
• If the diagonals bisect each other
• If  a pair of opposite sides is equal and parallel.

From the question given above

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

DP=QB

In ΔAPD and ΔCQB,--------------------------(i)

DP = QB (Given)

∠ADP = ∠CBQ (Alternate interior angles)

AD = BC

(Hence it is proved that -Opposite sides of a parallelogram are equal)

so, ΔAPD ≅ ΔCQB    (As per  SAS congruence rule)

 If , ΔAPD ≅ ΔCQB.-------------------------(ii)

AP = CQ               ( by CPCT )

 In ΔAQB and ΔCPD,-----------------------(iii)

BQ = DP (Given)

∠ABQ = ∠CDP (Alternate interior angles)

AB = CD  (Opposite sides of a parallelogram)

so, ΔAQB ≅ ΔCPD                       (As per  SAS congruence rule)

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

 From (ii)  and (iv) equation ,we can say that

AP=CQ ,

AQ=CP

 It is proved that APCQ has equal opposite sides also it has equal opposite angles. Hence,APCQ is a Parallelogram