<Li> Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent). </Li> <P> Spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce (extend) a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three: contrary to the first postulate, there is not a unique shortest route between any two points (antipodal points such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line . </P> <P> A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180 ° . Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere . The sum of the angles of a triangle on a sphere is 180 ° (1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle . For any positive value of f, this exceeds 180 ° . </P>

Sum of angles of a triangle on a sphere