1. Study Guide Timeline Euclid’s five axioms (300 BC) From Proclus (400AD) belief that the fifth axiom is derivable from the first four Saccheri (17 th century): Either HOA or HAA or HRA. Also the first four axioms imply either HRA or HAA. Discovery of non-Euclidean Geometry (1 st half of the 19 th century) by Gauß, Bolyai and Lobachevsky (1829) –The fifth axiom is independent and cannot be derived from the first four. –There is a dichotomy: either one or two Lobachevsky parallels Models for non-Euclidean geometry (2 nd half of the 19 th century) by Beltrami (1868), Klein (1871) and Poincaré.

2. Study Guide Triangles HOA Sum of interior angles of a triangle is > 180° HAA Sum of interior angles of a triangle is < 180° HRA Sum of interior angles of a triangle is = 180°

3. Study Guide Equivalent versions of the Parallel Postulate That, if a straight line falling on two straight lines make the interior angle on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. (Euclid ca. 300BC) For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l. (Playfair 1748-1819) The sum of the interior angles in a triangle is equal to two right angles. (Legendre 1752- 1833 )

4. Study Guide Angles and the Parallel Postulate Assuming the first four axioms there is a dichotomy: Euclidean or non-Euclidean 1.Euclidean: equivalently 1.The sum of the interior angles of a triangle is 180. 2.There is exactly one parallel to a given line through a point not on that line. 2.Non-Euclidean: equivalently 1.The sum of the interior angles of a triangle is strictly less than 180. 2.There is exactly two parallel lines, in the sense of Lobachevsky, to a given line through a point not on that line. 3.To a given line there are infinitely many lines through a through a point not on that line which do not intersect this line.

5. Study Guide Facts The dichotomy means that If one triangle has the sum of interior angles equal to 180° the so do all triangles And If one triangle has the sum of interior angles strictly less than 180° the so do all triangles Also If there is one line which through a given point has only one parallel than this statement is true for all lines and points. And If there is one line which through a given point has more than one parallel than this statement is true for all lines and points.

6. Sample Questions By whom and when was non-Euclidean geometry discovered? State Euclid’s version of the parallel postulate? Give other equivalent formulations! Which postulate equivalent to the parallel postulate did Legendre assume in his false deduction? State the first four axioms of Euclid! Show that the parallel postulate implies that the sum of the interior angels in a triangle is 180°!

7. Sample Questions Show the equivalence of Euclid’s parallel postulate and the Playfair’s version which states that there is a unique parallel. Draw a picture of a parallel at angle 45° in a model of non-Euclidean geometry! What values can the sum of interior angles of a triangle take in a non-Euclidean geometry? What is the contribution of Beltrami to non-Euclidean geometry? What is Gauss’s contribution? Is there a relationship between area and angles of a triangle in Euclidean geometry? How about in non- Euclidean geometry. And study the homework!