In writing the Elements Euclid collected and extended many of the ideas of other Greek mathematicians before him. The Elements is basically a chain of 465 propositions encompassing most of the geometry, number theory, and geometric algebra of the Greeks up to that time. The definitions of Book I include those of points, lines, planes, angles, circles, triangles, quadrilaterals, and parallel lines. A line segment extended infinitely in both directions produces a straight line. A circle is determined by a center and distance.
All right angles are equal to one another.
If a straight line falling an two straight lines forms interior angles on the same side less than 180 degrees, the two straight lines, if produced indefinitely, will meet on that side. Hyperbolic geometry was invented by the Russian mathematician Nicolai Ivanovitch Lobachevsky. Lobachevsky's geometry grew out of his unsuccessful attempts to prove Euclid's parallel postulate. He claimed that one must postulate that two distinct straight lines cannot have a segment in common. I are fallacious.