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The measure of angle DF can be determined by subtracting the given angles from the total angle in a circle, which is 360 degrees.
Angle DF = 360 - 160 - 44
= 360 - 204
= 156 degrees
Therefore, the measure of angle DF is 156 degrees.
None of the provided answer choices match this calculation, so there may be a mistake in the question or the answer choices.