Denote the vertices of our quadrilateral as ABCD. Draw a diagonal BD. Consider triangles BDA and DBC (the order of points is important). They have all three sides equal -- AB = BC, AD = DC by the condition of the problem, BD is a common side. Thus, these triangles are equal an therefore corresponding angles are equal.
Angles BDA and DBC are equal, then lines AD and BC are parallel. Similarly, angles ABD and CDB are equal, then lines AB and DC are parallel. And this is a definition of parallelogram.
Actually, this is a rhombus.
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