SOLUTION: Hello, I have submitted the following (please open web link for my work)twice for grading, and my grader keeps telling me this: "Through a point outside a line, there is exa

Question 173835: Hello,
I have submitted the following (please open web link for my work)twice for grading, and my grader keeps telling me this:
"Through a point outside a line, there is exactly one line parallel to the given line." but this is Euclid's postulate and as such does not require proof. Please provide the theorem that can be proved in Euclidean geometry but not in non-Euclidean geometry."
http://www.taskstream.com/ts/brechtel/MGA5Task2.html-Web link for my work
Thanks for your help!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Euclid's fifth postulate is "If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough." This postulate is equivalent to what is known as the parallel postulate, which is the statement you call a theorem. In fact, no one has ever been able to prove the fifth or the parallel postulates using the first four postulates. So your choice of a thing to show as a theorem that cannot be proven in Non-Euclidian Geometry was inappropriate. What you need to do is find a Theorem whose proof is based upon the fifth or parallel postulate, and show that the Theorem cannot be proven in an elliptic or hyperbolic geometry. I suggest you start looking in Euclid's Elements at Proposition 29.

Good luck.
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