Question

ABCD is a quadrilateral in AB = CD and AD=BC .Show that it is a parallelogram.

Answer 1

see the attached figure ,
given , 
AB = CD 
AD = BC
to prove that : ABCD is a IIgram
proof :- AB = CD (from the given )
              BC = AD (from the given )
              BD = BD (common side )
∴ Δ ADB ≡ Δ DBC by SSS congruence rule . 
therefore from CPCT we can write that ∠ADB = ∠ DBC (alternate interior angle )
since the alternate interior angles and opposite sides are equal in a quadrilateral therefore it is a IIgram . 
  i hope it helps ...........................^_^

Answer 2

Step-by-step explanation:

REF.image.

Given:In □ABCD

AB=DCandAD=BC

To prove: ABCD is a parallelogram

Construction: Draw diagonal AC and DB.

Proof: in △ABC and △ADC

AD=BC [Given]

AB=DC [Given]

AC=AC [Common side]

∴ By SSS property

△ADC≅△ACB

∴∠DAC=∠DCA

∴AB∣∣DC [By theorem]

In△ABDand△DCB

DB=DB [Common side]

AD=BC [Given]

AB=DC [Given]

∴△ABD≅△DCB

∴AD∣∣BC

Since AB∣∣DC and AD∣∣BC.

△ABCD is parallelogram