Yes, those inequalities would be corollaries of the Pythagoeran theorem, assuming two things: that c is the longest side and that angle x is the angle opposite the hypotenuse--the angle that would be the 90 degree angle in a 90 degree triangle. This is visually intuitive. If we start with a right triangle, and then "stretch" the hypotenuse longer, if the shape is to remain a triangle, the angle opposite the hypotenuse will get wider; likewise, if we shrink the hypotenuse shorter, the angle opposite the hypotenuse will have to close up.
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