Squares as areas
With legs a and b and hypotenuse c, modern notation is a²+b²=c². The two smaller side-squares together have the area of the hypotenuse-square. Adding the leg lengths themselves would express a different, incorrect relationship.
I.47 builds squares and uses auxiliary lines, congruent triangles and equal-height area relationships to establish the equality. Its argument applies to an arbitrary right triangle meeting the hypotheses rather than to measurements of one drawing.
References: [1]
The converse as a criterion
The converse starts from the squared-side relationship and concludes that the angle opposite the longest side is right. I.48 constructs a right triangle and uses equal corresponding sides to identify the original angle. The two propositions begin with different hypotheses.
Sides three, four and five satisfy nine plus sixteen equals twenty-five. Five is the hypotenuse. When numbers come from physical measurements, approximate equality supports an error-sensitive assessment rather than the exact mathematical conclusion.
References: [2]
Conditions and generalization
A right angle and the Euclidean setting are conditions of the theorem. Merely assigning side letters to an arbitrary triangle does not establish them.
VI.31 replaces the squares with similarly constructed rectilinear figures on the three sides. The hypotenuse-figure again has the sum of the other areas. Similarity and correspondence remain essential; the figures cannot be chosen independently.