Sum of angles of a triangle The relation between angular defect and the triangle's area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small (but positive). For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal
Hypotenuse is the square root of 25, that is, 5. The word "hypotenuse" is derived from Greek (sc. or ), meaning "[side] subtending the right angle" (Apollodorus), The Greek term was loaned into Late Latin, as "hypotēnūsa". Adoption as a learned Latinism used in modern languages dates to the 16th century. The spelling in "-e", as "hypotenuse", is French in origin (Estienne de La Roche 1520). The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to