To prove the parallelogram is a rhombus you have to show that AB = BC = CD = DA.

To prove AB = DA and BC = CD, we need to prove triangles ABC and ADC are congruent. 
Because angle BAC = angle DAC, angle BCA = angle DCA, and AC = AC, so triangles ABC and ADC are congruent.
Hence AB = DA and BC = CD.

Next we need to show that AB = CD, this is obvious because they are the opposite sides of a parallelogram.

So AB = BC = CD = DA