Incidence Geometry: confirm that model satisfies Playfair's

KLS2111

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Mar 8, 2007
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Hello,
I am confirming that a model satisfies Playfair's postulate by checking all possible cases "For any line l and any point P not lying on l, there exists a unique line m through P parallel to l." This is using four points. I have to tell how many instances or cases there are. I guess I am confused as to how I determine how many cases there are and what exactly I do to check them.

Here is what I did:
POints A B C D
Lines ab, bc, bd, ac, ad, cd

Since there can't be any points in common i determined ab||cd bc||ad bd||ac Is that the check? How many cases were there to check 15? ab to bc ab to bd ab to ac etc? I thought someone said there was a formula to determine the number of cases. Thank you for any guidance you can give me
 
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