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Neutral geometry: All axioms except the fifth axiom (about the number of parallel lines). Could be proven(neutral geometry): 1. SSS. 2. ASA. 3. SAS. 4. Pons Asinorum (about angles of isoceles triangle). 5. The sum of two angles of a triangle is < 180° 6. The sum of angles in a triangle is ≤ 180° 7. The sum of angles in a quadrilateral is ≤ 360° 8. We can always draw the perpendicular line to the given line through the given point Hide Caption Hide Captions Hide Caption Hide Caption Eucledian geometry= neutral+ we can draw only one line parallel to the given line though the given point 6. The sum of angles in a triangle is =180° 7. The sum of angles in a quadrilateral is = 360° Hyprbolic geometry= neutral+ we can draw more than one line parallel to the given line though the given point 6. The sum of angles in a triangle is <180°. (For any number ε>0 there exists a triangle with the sum of angles <ε.) 7. The sum of angles in a quadrilateral is < 360° Other corollaries: No rectangles. No similar triangles. There exist three non collinear points such that it is impossible to draw a circle through them. Locus of points equidistant from a line is not a line.