Triangle property

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.

In obtuse triangle, one of the angles will be more than 90 degrees. Hence x could be > 90. This answer 1st question.

In acute triangle, none of the angles will be more than 90 degrees. Hence all angles will be < 90. So, x is < 90. This answer 2nd question