The correct order is: C. IV, V, III, VI, VII
In the proof by contradiction for triangle XYZ, with XY = 4 and XZ = 8, an initial assumption is made that YZ is less than or equal to 4. The analysis then considers the possibilities: either YZ is less than 4, or YZ equals 4. Both cases contradict the Triangle Inequality Theorem, which states that XY + YZ must be greater than XZ.
Further examination demonstrates that if YZ is less than 4, then XY + YZ is less than XZ (since 4 + YZ < 8 when YZ < 4). Similarly, if YZ equals 4, then XY + YZ equals XZ (since 4 + 4 equals 8). This contradiction proves that the initial assumption of YZ being less than or equal to 4 is false, confirming the original statement that YZ must be greater than 4.