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
