Yes it is possible to prove this figure is a parallelogram
Refer to the attached image
Step 1) mark the corner points as A, B, C, D
Step 2) Draw a diagonal from A to C (shown in red)
Step 3) Through the SSS (side side side) property, we can prove the two triangles ABC and ADB are congruent
Step 4) By CPCTC (corresponding parts of congruent triangles are congruent), we can say the angles of the triangles are congruent as well. So angle ACB is congruent to angle CAD (as shown in green)
Step 5) Through the converse of the alternate interior angle theorem, this means that the BC is parallel to AD
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If you repeat those steps, but instead of drawing diagonal AC you draw diagonal BD, then you can prove that AB || CD
Once you know that AB || CD and BC || AD, then you have proven the figure is a parallelogram. By definition, a parallelogram is a four-sided figure with opposite sides that are parallel.