prove the postulate 5:
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there is exactly one parallel;there are no parallels;there are more than one parallel.
The three hypotheses are known as hypotheses of the right, obtuse, and, respectively, acute angles. The first one is Playfair's axiom and, thus, is equivalent to Euclid's fifth postulate. Assuming that Euclid's second postulate (A piece of straight line may be extended indefinitely.) requires straight lines to be infinitely long, he showed that (B) indeed leads to a contradiction. Based on (C), he proved several counterintuitive statements but couldn't formally obtain a logical contradiction. Probably to justify the title of the work he stated
The three hypotheses are known as hypotheses of the right, obtuse, and, respectively, acute angles. The first one is Playfair's axiom and, thus, is equivalent to Euclid's fifth postulate. Assuming that Euclid's second postulate (A piece of straight line may be extended indefinitely.) requires straight lines to be infinitely long, he showed that (B) indeed leads to a contradiction. Based on (C), he proved several counterintuitive statements but couldn't formally obtain a logical contradiction. Probably to justify the title of the work he stated
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