Consider that the diagonal of a parallelogram divides the parallelogram into two congruent triangles. The Triangle Inequality, which states that the sum of any two side lengths of a triangle must be greater than the length of the third side, can therefore be applied. We know two lengths, 3 and 4, of the congruent triangles into which the parallelogram can be divided. We must therefore establish bounds on the third side, the length of which we will denote x. Using this inequality, we find that 1<x<7. Since the length of the other side of the triangle is 3, the range for the perimeter is 8<x<20.
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