Understanding Transitional Lines to Parallel and Non-Parallel Lines

[lightbender]

Hi,
I need a little help understanding transitional lines on parallel and non parallel
lines. Can you please help me?
I am trying to refresh my memory!
Thank you!

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[HallsofIvy]

I think you mean what I would call a "transecting" line- a line crossing each of two other given lines.
The most famous reference to such is "Euclid's parallel postulate" which said "If a line crossing to given lines makes angles on one side whose measures sum to less than 180 degrees, then the lines meet on that side". Since it the angles on one side sum to more than 180 degrees, the angles on the other side must sum to less than 180 degrees, an immediate result is that "if a line crosses two parallel lines, then the angles it make on one side must sum to exactly 180 degrees" which further implies that it is impossible for more than one line, through a given point to both be parallel to a third line.