Angles¶
Recall (doc/postdoc). Not a first course.
Job: measure of pairs of rays; full structure is \(S^1=\mathbb{R}/2\pi\mathbb{Z}\) (oriented) or \([0,\pi]\) (unordered).
Core¶
| Type | Definition |
|---|---|
| Adjacent | share ray, interiors disjoint |
| Vertical | opposite equal (two lines) |
| Complementary / supplementary | sum \(\pi/2\) / \(\pi\) |
| Linear pair | adjacent on a line \(\Rightarrow\) supplementary |
Measure: degrees or radians; \(2\pi\) rad = full turn. Right / straight: \(\pi/2\), \(\pi\).
Oriented angle: element of \(\mathbb{R}/2\pi\mathbb{Z}\); undirected interior angle in polygons \(\in(0,\pi)\) for convex.
Bisector: ray equidistant (angle metric) from sides; locus equidistant from two rays.
Used later for¶
- Trigonometry: angle as arc parameter
- Parallel lines: corresponding / alternate equal
- Differential geometry: angle between curves via metrics
- Complex analysis: \(\arg z\)
Check¶
Is angle measure additive under concatenation for oriented angles on \(S^1\)?
Yes — addition in \(\mathbb{R}/2\pi\mathbb{Z}\). Undirected interior angles of polygons are not a group law.