Question

What does denying Playfair's Postulate do to the notions of parallel lines, number of points of intersection of two lines, and the measurement of angles of intersection between lines? Pose your response relative to Euclidean geometry with Playfair's Postulate intact.

Answer 1

I believe that denying his postulate opens up a whole new branch of geometry called hyperbolic geometry. As our textbook states, denying it would mean saying that
"Given a line and a point not on the line, it is possible to construct more than one line through the given point parallel to the line." This postulate has been termed the hyperbolic postulate.

I believe denying it is exactly how Gauss and several other mathematicians developed non-Euclidean Geometry in the early 1800's. They were able to show that all of Euclid's postulates still could hold true without Euclid's fifth postulates which is just another way to state Playfair's postulate. When they tried to negate it, they found no way to contradict the first four postulates. In other words, they could not prove Euclid's 5th postulate (Playfair's Postulate). Several non-Euclidian geometries developed, including Riemann Geometry,

Also, even if Playfair's postulate is denied, a segment can still be constructed between two points, segments can still be extended indefinitely, circles can still be constructed, and all right angles are still congruent. In the non-Euclidian geometries, however, no longer are similar triangles possible as in Euclidian. They are either congruent or they are not congruent.