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Read 2.7, 3.1, 3.2 Section 2.7, Non Euclidean Geometries Hyperbolic Parallel Postulate: There exists a line and a point P such that at least two distinct lines pass through P that are parallel to . Poincare Half Plane Model: 1 Consequences: Through a given point not on there are infinitely many parallels There are no triangles with angle sum 180 There exist no triangles that are similar but not congruent There exist triangles that cannot be circumscribed The sum of the interior angles of a triangle varies and is always less than 180 No rectangles exist 2 There is an upper limit to the area of a triangle The larger a triangles area, the smaller its angle sum The distance between certain pairs of parallel lines approaches 0 in one direction and in the other If two parallel lines are crossed by a transversal, alternate interior angles may or may not be congruent Elliptic Parallel Postulate: Given any line and point P not on there is no line through P parallel to 3 3.2 Neutral Geometry Notation: AB, AB or AB , ABC, mABC, P S Q, AB Def. Two segments are congruent iff their measures are equal, two angles are congruent iff their measures are equal. Def. Two polygons are congruent iff there exists a 1-1 correspondence between vertices such that all corresponding sides and angles are congruent. 4 Thm: The relations of line segment congruence, angle congruence, and polygon congruence are equivalence relations. Thm: Pasch’s Axiom If a line intersects PQR at a point S such that P S Q Proof: Thm: Every line segment has exactly one midpoint. Every angle has exactly one bisector. The Crossbar Thm: If X is a point in the interior of UVW, then UX intersects WV at a point Y such that W Y V. Thm: Supplements and complement of the same or congruent angles are congruent. Isoceles Triangle Thm: If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Thm: Vertical angles are . 5 6 Thm: A point is on the perpendicular bisector of a line segment iff it is equidistant from the endpoints of that segment. Def. An angle that is both supplementary and adjacent to an angle of a triangle is called an exterior angle. Exterior Angle Theorem: An exterior angle of a triangle is greater in measure than either of the nonadjacent interior angles. 7