SOLUTION: The question is: Can you think of any way to prove from Euclid's postulates that for every line l (a) There exists a point lying on l? (b) There exists a point not lying on l?

Algebra ->  Geometry-proofs -> SOLUTION: The question is: Can you think of any way to prove from Euclid's postulates that for every line l (a) There exists a point lying on l? (b) There exists a point not lying on l?       Log On


   

Question 642658: The question is:
Can you think of any way to prove from Euclid's postulates that for every line l
(a) There exists a point lying on l?
(b) There exists a point not lying on l?
I'm thinking it has something to do with Euclid's parallel postulate, but I'm not sure. I have never written proofs before, so I am just having a difficult time trying to formulate this properly. Any help would be appreciated. Thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)

Any line is uniquely determined by 2 points. So if you have a line, then you know it has at least two points on it.

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b)

Parallel lines will NEVER intersect in euclidean geometry (in the plane). This means that parallel lines will not have any points in common. So the existence of parallel lines shows us that if you have a line, then there are points that do not lie on this given line (since these points will lie on the parallel lines).