4.1: Euclidean Geometry

Overview

• Euclidean geometry is a branch of mathematics concerning the properties and relations of points, lines, surfaces, and solids in space.
• Also known as parabolic geometry, it follows Euclid's five postulates.
• It includes two types of geometry:
  • Plane geometry (2D)
  • Solid geometry (3D)

Basic Elements

Points

• Point: No dimension but has a location.
• Collinear points: Points on the same straight line.
• Line Segment: A portion of a line with two endpoints.
• Ray: A part of a line that starts at a point and extends infinitely.
• Intersection point: The point where two lines meet.
• Midpoint: The center point of a line segment.

Angles

• Angle measures are typically in degrees or radians.
• Right angle: 90 degrees.
• Obtuse angle: Greater than 90 degrees.
• Acute angle: Less than 90 degrees.
• Straight angle: 180 degrees.
• Reflex angle: Greater than 180 degrees.
• Adjacent angles: Share a vertex and a side.
• Complementary angles: Add up to 90 degrees.
• Supplementary angles: Add up to 180 degrees.
• Vertical angles: Equal angles formed by intersecting lines.
• Corresponding angles: Equivalent angles in corresponding positions.
• Alternate interior angles: Equivalent angles on opposite sides of a transversal intersecting parallel lines.

Lines

• Parallel lines: Do not intersect, extending infinitely.
• Perpendicular lines: Intersect at a 90-degree angle.
• Concurrent lines: Intersect at a single point.
• Skew lines: Non-intersecting, non-parallel lines in 3D.
• Perpendicular bisector: A line that bisects another at a 90-degree angle.

Planes

• Plane: A flat, two-dimensional surface extending infinitely.
• Cartesian plane: A coordinate plane for graphing functions.

Axioms

1. Unique line through two distinct points.
2. Plane through points lays entirely in the plane.
3. Plane contains three non-collinear points.
4. Intersection of planes is a line.

Euclid's Five Postulates

1. A line segment can be drawn joining any two points.
2. A line segment can extend indefinitely.
3. A circle can be drawn with any center and radius.
4. All right angles are congruent.
5. Only one parallel line can be drawn through a point not on a line.

This summary is based on the principles outlined in the Mathematics LibreTexts article on Euclidean geometry.