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AABC algebra angle opening angle sum angles of parallelism assumption axiom CD meet concrete congruence consider cross-caps definition denoted dot products draw drawn E.T. Bell equation equivalent etry Euclid's fifth postulate Euclid's first four Euclid's postulates Euclidean geometry example factorable figure following theorem four postulates Geome try given graph Hence holes of type homeomorphic hypotenuse integers interior angles intersect Lafayette High School Legendre less line AC Lobatchewskian geometry logical manifolds mathematician midpoint monomial multiple non-Euc1idean geometry number of edges number theory odd nodes pair parallel lines pendicular perfect square perpendicular plane geometry Playfair polyhedra polyhedron Posidonius possible postulationa1 method problem Projective Plane proof reader regions result Riemannian geometry right angles Saccheri scalar set of postulates shown solved sphere straight line summit angles Suppose surface teacher theorem proven Theory of Numbers tion tulate undefined terms vectors vertex