Short Answer 
The analysis of parallelogram ABCD reveals that its diagonals bisect each other, leading to the conclusion that triangles ABC and BCD are isosceles with equal sides. Thus, all adjacent sides are equal, confirming that ABCD is a rhombus.
Step 1: Understand the Parallelogram Properties
Begin by recognizing the characteristics of the parallelogram ABCD in the diagram. A key property is that the diagonals of a parallelogram bisect each other, which means that segments AM and MC are equal, as are segments BM and MD. This fact sets the stage for further analysis of the triangles formed by the diagonals.
Step 2: Analyze the Triangles
Focus on triangle ABC, where BM acts as both an altitude (perpendicular to AC) and a median (since AM=MC). This dual role indicates that triangle ABC is isosceles, leading to the conclusion that sides AB and BC are equal. A similar analysis applies to triangle BCD, confirming that sides BC and CD are also equal.
Step 3: Conclude the Shape of Quadrilateral ABCD
Now, use the properties of parallelograms to establish the lengths of the sides. Note that opposite sides are equal:
- AB = DC
- BC = DA