Final answer:
In triangle DEF, if DG is the median, the Triangle Inequality Theorem allows us to conclude that DE + EF + FD is greater than 2DG.
Step-by-step explanation:
In a triangle, the sum of the lengths of any two sides is greater than the length of the third side. This is known as the Triangle Inequality Theorem.
In triangle DEF, if DG is the median, that means it connects the midpoint of one side (EF) to the opposite vertex (D). Therefore, the lengths DE + EF + FD would represent the perimeter of the triangle.
Since DG is less than or equal to DE (because it's a segment of DE) and also less than or equal to FD, we can conclude that 2DG is less than or equal to DE + FD. Also, because EF is a side of the triangle itself, it is also greater than DG. Hence, when we add EF to both sides of the inequality DE + FD > 2DG, we get DE + EF + FD > 2DG + EF.
From the triangle inequality, we know that 2DG + EF is less than DE + EF + FD. Therefore, the original inequality holds: DE + EF + FD > 2DG.
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