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Parallel lines
Parallel lines in absolute geometry
Parallel lines (definition)
Parallel straight lines are straight lines which, being in the same plane and being produced
indefinitely in both directions, do not meet one another in either direction. (Euclid)
Two distinct lines l and m are said to be parallel (and we write l||m) if and only if they lie
in the same plane and do not meet. (Kay)
Our current state is that parallel lines exist in absolute geometry, we have not yet proven any theorems
about parallel lines in absoute geometry (and there are some), and we lack a parallel postulate for
absolute geometry. The important point here is that a parallel postulate is one particular statement
about parallel lines (that we can’t make); there are still statements about them that we can make,
while remaining in absolute geometry.
Which statement is equivalent to Euclid’s parallel postulate?
S1: If two straight lines cut by a transversal are parallel, then measures of the alternate
interior angles must be equal.
S2:If two straight lines cut by a transversal have the measures of the alternate interior
angles equal, then the lines are parallel.
The distinction is the ordering of the if-then; the statements are converses of each other. Anything in
the form
If the lines are parallel, then you know various angle relationships hold.
is equivalent to Euclid’s Parallel Postulate. Anything in the form
If various angle relationships hold, then the lines are parallel.
is its converse.
The converse of the parallel postulate (in all its various forms) is a theorem of absolute geometry. It
can be proven from the axioms of absoute geometry, and does not need to be assumed as a postulate.
Theorem: Parallelism in absolute geomtry
If two lines in the same plane are cut by a transversal so that a pair of alternate interior
angles are congruent, then the lines are parallel.
Sketch it:
Prove it: