The Pythagorean Theorem depends on the parallel postulate. So, it does not hold for the non-Euclidean geometries. In relation to the above proofs There are no similar triangles in the non-Euclidean geometries. We cannot prove that the above triangles have half the area of the square or rectangle. But, mainly we have a problem because there are no squares in the non-Euclidean geometries.
Congruence applies to geometric figures, including line segments and angles.
Equality applies to numbers, such as areas, lengths of line segments, and the measures of angles. A square and a rectangle are not equal, their areas are equal. See Rik Littlefield's proof , which is interesting. The Pythagorean Theorem is also mentioned in Isosceles Triangles . Given the right triangle at the bottom of this diagram, duplicate it as shown, and complete the trapezoid as shown.