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 ). 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 , a right angle is radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.
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)
- By naming a point on each side of the angle with the vertex in between.
- By placing a small number on the interior of the angle near the vertex.
Classification of Angles by Degree Measure
- an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.
- 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
- 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.
- 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:
- Two and Three-Dimensional Geometry and Other Geometric Figures
- Geometry/Perimeter and Arclength
- Geometry/Right Triangles and Pythagorean Theorem
- 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/Ellipsoids and Spheres
- Geometry/Coordinate Systems (currently incorrectly linked to Astronomy)
- Traditional Geometry:
- Modern geometry