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## Euclid’s Postulates

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**Euclid’s Postulates**• Two points determine one and only one straight line • A straight line extends indefinitely far in either direction 3. A circle may be drawn with any given center and any given radius 4. All right angles are equal 5. Given a line k and a point P not on the line, there exists one and only one line m through P that is parallel to k**Euclid’s Fifth Postulate (parallel postulate)**• If two lines are such that a third line intersects them so that the sum of the two interior angles is less than two right angles, then the two lines will eventually intersect**Saccheri’s Quadrilateral**He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles.**Non-Euclidean Geometry**• The first four postulates are much simpler than the fifth, and for many years it was thought that the fifth could be derived from the first four • It was finally proven that the fifth postulate is an axiom and is consistent with the first four, but NOT necessary (took more than 2000 years!) • Saccheri (1667-1733) made the most dedicated attempt with his quadrilateral • Any geometry in which the fifth postulate is changed is a non-Euclidean geometry**Lobachevskian (Hyperbolic) Geometry**• 5th: Through a point P off the line k, at least two different lines can be drawn parallel to k • Lines have infinite length • Angles in Saccheri’s quadrilateral are acute**Riemannian (Spherical) Geometry**• 5th: Through a point P off a line k, no line can be drawn that is parallel to k. • Lines have finite length. • Angles in Saccheri’s quadrilateral are obtuse.