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. We also have the problem that 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.