Given: In quadrilateral ABCD :
It is given that, AB=DC and AD=BC.
To Prove: ABCD is a parallelogram.
Construction: Join AC.
Proof: In ΔABC and ΔCDA,
AB=CD (Given)
BC=DA (Given)
AC=CA (Common)
therefore, ΔABC ≅ ΔADC (by SSS)
So, ∠BAC = ∠DCA (CPCT) … (i)
∠BCA = ∠DAC (CPCT) … (ii)
from equations (i) and (ii), we get
AB || DC and AD || BC
Thus, In ABCD
Both pairs of opposite sides are parallel
Hence, ABCD is a parallelogram.