$\begin{align} &\text {Parallelograms are b,d,f,j,k,m,n,r}\\[5pt] \end{align}$
$\begin{align}(i)&BCP \text { }\triangle \text { }\text{and}\text { } DPE \text { }\triangle \\[5pt] &PC=DP\text{(Midpoint)}\\[5pt] &P\hat BC=D\hat EP\text{(Alternate angles)}\\[5pt] &B\hat PC=D\hat PE\text{(Vertically opposite angles)}\\[5pt] &BCP \text { }\triangle \text { }\equiv DPE \text { }\triangle\text{(AAS)}\\[5pt] (ii)&DE=BC\text{(Corresponding elements are equal)}\\[5pt] &DE\parallel BC\\[5pt] &\text{BCED is a parallelogram}\\[5pt] \end{align}$
$\begin{align}&OD=OC\text{(Radius of circle)}\\[5pt] &AO=OB\text{(Radius of circle)}\\[5pt] &\text{ACBD is a parallelogram (Diagonals are bisected each other)}\\[5pt] \end{align}$
$\begin{align}(i)&AXD\text { }\triangle \text { }\text{and}\text { } BYC \text { }\triangle \\[5pt] &D\hat XA=C\hat YB= 90^0\text{(Given)}\\[5pt] &AD\parallel BC\\[5pt] &D\hat AX=B\hat CY\text{(Alternate angles)}\\[5pt] &AD=BC\text{(Opposite sides of ABCD parallelogram)}\\[5pt] &AXD \text { }\triangle \text { }\equiv BYC \text { }\triangle\text{(AAS)}\\[5pt] (ii)&DX=BY\text{(Corresponding elements are equal)}\\[5pt] (iii)&D\hat XY =90^0=B\hat YX\text{(Alternate angles are equal)}\\[5pt] &DE\parallel BY\\[5pt] &\text{BYDX is a parallelogram (A pair of side parallel and equal)}\\[5pt] \end{align}$
$\begin{align}&\text {QPRS quadrilateral}\\[5pt] &AB=PQ\text{(ABPQ Parallelogram -Opposite sides are equal)}\\[5pt] &AB=SR\text{(ABRS Parallelogram -Opposite sides are equal)}\\[5pt] &PQ=SR\\[5pt] &PQ\parallel SR\\[5pt] &\text{QPRS is a parallelogram (A pair of side parallel and equal)}\\[5pt] \end{align}$
$\begin{align}&\text {ABCD parallelogram}\\[5pt] &AO=OC -->(1)\text{(Diagonals bisect)}\\[5pt] &AE=FC -->(2)\text{(Given)}\\[5pt] &(1)-(2)\\[5pt] &AO-AE=OC-FC\\[5pt] &EO=OF\\[5pt] &OD=OB\text{(Diagonal of ABCD parallelogram)}\\[5pt] &\text {DEBF quadrilateral}\\[5pt] &EO=OF\text{(Diagonal bisect each other)}\\[5pt] &OD=OB\text{(Diagonal bisect each other)}\\[5pt] &\text{EBFD is a parallelogram}\\[5pt] \end{align}$
$\begin{align}(i)&\text {ABCD parallelogram}\\[5pt] &AB=DC \text{(Opposite sides are equal)}\\[5pt] &DA=BC \text{(Opposite sides are equal)}\\[5pt] &DA=AX\\[5pt] &BC=AX\\[5pt] &\text {AXBC quadrilateral}\\[5pt] &BC=AX\text{(Opposite sides)}\\[5pt] &BC\parallel AX\\[5pt] &\text{AXBC is parallelogram (A pair of sides equal and parallel)}\\[5pt] (ii)&\text {ABYC quadrilateral}\\[5pt] &AB\parallel DY\text{(Given)}\\[5pt] &AC\parallel XY\text{(AXBC parallelogram)}\\[5pt] &\text{ABYC is a parallelogram (Opposite sides are parallel)}\\[5pt] (iii)&\text {ABCD parallelogram}\\[5pt] &CD=AB \\[5pt] &\text {ABYC parallelogram}\\[5pt] &CY=AB\\[5pt] &CY=CD\\[5pt] \end{align}$
$\begin{align}(i)&\text {PQRS parallelogram}\\[5pt] &OP=OR \text{(Diagonals bisect)}-->(1)\\[5pt] &PM=TR \text{(Given)}-->(2)\\[5pt] &(1)-(2)\\[5pt] &OP-PM=OR-TR\\[5pt] &OM=OT\\[5pt] (ii)&\text {PQRS quadrilateral}\\[5pt] &OS=OQ\text{(Diagonals bisect each other)} -->(1)\\[5pt] &SN=LQ \text{(Given)}-->(2)\\[5pt] &(1)-(2)\\[5pt] &OS-SN=OQ-LQ\\[5pt] &ON=OL\\[5pt] &\text {LMNT quadrilateral}\\[5pt] &OM=OT\\[5pt] &ON=OL\\[5pt] &\text{LMNT is a parallelogram (Diagonals bisect each other)}\\[5pt] (iii)&\text{MSTQ quadrilateral}\\[5pt] &OM=OT\\[5pt] &OS=OQ\text{(PQRS parallelogram)}\\[5pt] &\text{MSTQ is a parallelogram (Diagonals bisect each other)}\\[5pt] \end{align}$

By studying this lesson you will be able to

Identify the conditions that need to be satisfied for a quadrilateral to be a parallelogram.