# Geometry/Angles

An angle is the union of two rays with a common endpoint, called the vertex. The angles formed by vertical and horizontal lines are called right angles; lines, segments, or rays that intersect in right angles are said to be perpendicular.
Angles, for our purposes, can be measured in either degrees (from 0 to 360) or radians (from 0 to $2\pi$). Angles length can be determined by measuring along the arc they map out on a circle. In radians we consider the length of the arc of the circle mapped out by the angle. Since the circumference of a circle is $2\pi$, a right angle is ${\frac {\pi }{2}}$ radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.

## Naming Conventions

Angles are named in several ways.
* By naming the vertex of the angle (only if there is only one angle formed at that vertex; the name must be non-ambiguous) $\angle B$
* By naming a point on each side of the angle with the vertex in between. $\angle ABC$
* By placing a small number on the interior of the angle near the vertex.

## Classification of Angles by Degree Measure

Acute Angle
* an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.
Right Angle
* an angle is said to be right if it measures 90 degrees.
* notice the small box placed in the corner of a right angle, unless the box is present it is not assumed the angle is 90 degrees.
* all right angles are congruent
Obtuse Angle
* an angle is said to be obtuse if it measures between 90 and 180 degrees, exclusive.

## Special Pairs of Angles

* adjacent angles
** adjacent angles are angles with a common vertex and a common side.
** adjacent angles have no interior points in common.
* complementary angles
** complementary angles are two angles whose sum is 90 degrees.
** complementary angles may or may not be adjacent.
** if two complementary angles are adjacent, then their exterior sides are perpendicular.
* supplementary angles
** two angles are said to be supplementary if their sum is 180 degrees.
** supplementary angles need not be adjacent.
** if supplementary angles are adjacent, then the sides they do not share form a line.
* linear pair
** if a pair of angles is both adjacent and supplementary, they are said to form a linear pair.
* vertical angles
** angles with a common vertex whose sides form opposite rays are called vertical angles.
** vertical angles are congruent.

### Navigation

* Geometry
* Part I—Euclidean Geometry:
** Chapter 1. Geometry/Points, Lines, Line Segments and Rays
** Chapter 2. Geometry/Angles
** Chapter 3. Geometry/Properties
** Chapter 4. Geometry/Inductive and Deductive Reasoning
** Chapter 5. Geometry/Proof
** Chapter 6. Geometry/Five Postulates of Euclidean Geometry
** Chapter 7. Geometry/Vertical Angles
** Chapter 8. Geometry/Parallel and Perpendicular Lines and Planes
** Chapter 9. Geometry/Congruency and Similarity
** Chapter 10. Geometry/Congruent Triangles
** Chapter 11. Geometry/Similar Triangles
** Chapter 12. Geometry/Quadrilaterals
** Chapter 13. Geometry/Parallelograms
** Chapter 14. Geometry/Trapezoids
** Chapter 15. Geometry/Circles/Radii, Chords and Diameters
** Chapter 16. Geometry/Circles/Arcs
** Chapter 17. Geometry/Circles/Tangents and Secants
** Chapter 18. Geometry/Circles/Sectors
** Appendix A. Geometry/Postulates & Definitions
** Appendix B. Geometry/The SMSG Postulates for Euclidean Geometry
* Part II—Coordinate Geometry:
** Geometry/Synthetic versus analytic geometry
* Two and Three-Dimensional Geometry and Other Geometric Figures
** Geometry/Perimeter and Arclength
** Geometry/Area
** Geometry/Volume
** Geometry/Polygons
** Geometry/Triangles
** Geometry/Right Triangles and Pythagorean Theorem
** Geometry/Polyominoes
** Geometry/Ellipses
** Geometry/2-Dimensional Functions
** Geometry/3-Dimensional Functions
** Geometry/Area Shapes Extended into 3rd Dimension
** Geometry/Area Shapes Extended into 3rd Dimension Linearly to a Line or Point
** Geometry/Polyhedras
** Geometry/Ellipsoids and Spheres
** Geometry/Coordinate Systems (currently incorrectly linked to Astronomy)
* Traditional Geometry:
** Geometry/Topology
** Geometry/Hyperbolic and Elliptic Geometry
** Geometry/Projective Geometry
** Geometry/Neutral Geometry
** Geometry/Inversive Geometry
* Modern geometry
** Geometry/Unified Angles
** Geometry/Groups
** Geometry/Algebraic Geometry
** Geometry/Differential Geometry
** Geometry/Algebraic Topology
** Geometry/Noncommutative Geometry
* Geometry/An Alternative Way and Alternative Geometric Means of Calculating the Area of a Circle