Playfair's parallel postulate, which is equivalent to Euclid's fifth postulate, states: If & is a line...

 

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Playfair's parallel postulate, which is equivalent to Euclid's fifth postulate, states: If & is a line and P is a point not on l, then there exists exactly one line through P that is parallel to l. In hyperbolic geometry, Playfair's postulate is replaced by the following statement: If is a line and P is a point not on l, then there exist at least two lines through P that are parallel to l. (i) This statement contradicts the Euclidean parallel postulate. Does this mean that none of the theorems from Euclidean geometry are valid in hyperbolic geometry? (ii) The hyperbolic version of the postulate states that there are at least two parallel lines. Is it possible that there are exactly two lines? Playfair's parallel postulate, which is equivalent to Euclid's fifth postulate, states: If & is a line and P is a point not on l, then there exists exactly one line through P that is parallel to l. In hyperbolic geometry, Playfair's postulate is replaced by the following statement: If is a line and P is a point not on l, then there exist at least two lines through P that are parallel to l. (i) This statement contradicts the Euclidean parallel postulate. Does this mean that none of the theorems from Euclidean geometry are valid in hyperbolic geometry? (ii) The hyperbolic version of the postulate states that there are at least two parallel lines. Is it possible that there are exactly two lines?