Which theorem or postulate is the construction of parallel lines based upon?

Question

I'm pretty sure it's one of these three:
Consecutive Interior Angles Converse Theorem, Corresponding Angles Converse Postulate, or the Alternate Interior Angles Converse Theorem.

Answer 1

Euclid's Fifth Postulate:

http://en.wikipedia.org/wiki/Fifth_postulate

Answer 2

It depends on the geometry, for example, in elliptic geometry you cant construct parallell lines at all!
But in good old euclidean (traditional) geometry, it is based on the alternte interior angles converse theorem, since it is the one that clearly states that two lines with the given condition (i.e having the same interior alternate angles) will be paralell.
A construction based on this theorem would go as follows:
Given a line l, draw any line m through point P on l not coinciding with l, next on a point Q on m trace a line such that the interior angles formed by m and n coincide with their corresponding alternate interior angles formed by l and m. By the converse of the Alternate interior angles theorem, l and n will be parallel.
Good Luck
Quick note, the construction of parallel does not depend on teh fifth postulate, it depends on the exterior angle theorem (euclids 20th proposition, book 1), you can see it from the proof of I.27 here http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI27.html

Answer 3

Well, Mike, there are several possible answers to this. The one that works is the one which involves allowable constructions. For example, you can construct a perpendicular to a line, and from any point on the perpendicular line, construct another perpendicular. That falls under alternate interior angles, I believe.

Answer 4

properly, does no longer this word to horizontal strains to boot? except you do no longer even count selection the horizontal strains for having the null set or regardless of...i myself do no longer understand the respond, yet perhaps you're able to desire to look that as much as help your seek. :) satisfied finding!