Parallel Lines¶
Recall (doc/postdoc). Not a first course.
Job: Euclidean parallel classes = directions in affine plane; transversal angle dictionary.
Core¶
\(\ell\parallel m\): coplanar, non-intersecting (or identical, by convention).
Transversal \(t\) cuts \(\ell,m\):
| Pair | Euclidean \(\ell\parallel m\) |
|---|---|
| Corresponding | congruent |
| Alternate interior | congruent |
| Consecutive interior | supplementary |
Converse: equal corresponding (etc.) \(\Rightarrow\) parallel (Euclid).
Playfair: unique parallel to \(\ell\) through \(P\notin\ell\).
Coordinates: lines with equal slope (or both vertical) are parallel; direction vectors linearly dependent.
Used later for¶
- Coordinate geometry: slope classes
- Non-Euclidean: many/no parallels
- Affine geometry: parallel transport of free vectors
- Triangles: similar triangles via parallels
Check¶
In neutral geometry (absolute), which of corresponding angles equal \(\Rightarrow\) parallel still holds?
The implication “corresponding equal \(\Rightarrow\) parallel” is valid in absolute geometry; uniqueness of parallel is Euclidean-only.