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4.1: Euclidean geometry

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Euclidean geometry, sometimes called parabolic geometry, is a geometry that follows a set of propositions that are based on Euclid's five postulates.

There are two types of Euclidean geometry: plane geometry, which is two-dimensional Euclidean geometry, and solid geometry, which is three-dimensional Euclidean geometry.

The most basic terms of geometry are a point, a line, and a plane. A point has no dimension (length or width) but has a location. A line is straight and extends infinitely in opposite directions. A plane is a flat surface that extends indefinitely.

Points

Definitions:

Please refer to the image below for examples.

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Collinear points: points that lie on the same straight line or line segment. Points A, B, and C are collinear.

Line Segment: a straight line with two endpoints. Lines AC, EF, and GH are line segments.

Ray: a part of a straight line that contains a specific point. Any of the below line segments could be considered a ray.

Intersection point: the point where two straight lines intersect or cross. Point I is the intersection point for lines EF and GH.

Midpoint: a point in the exact middle of a given straight-line segment. Point B is the midpoint of line AC.

Think Out Loud

How many different lines can you draw through a fixed point? How many different lines can you draw through two fixed points? How many different lines can you draw through three fixed points?

Angles

Definitions:

Angle: ACB. Normally, the angle is measured in degrees (0) or in radians rad).

Right angle: Angles which measure 90° - ABC

Obtuse angle: Angles which measure > 90° - CDE

Acute angle: Angles which measure < 90° - FDE

Straight angle: Angles which measure 180° CDF

Reflex angle: A reflex angle is an angle that is measured > 180°, which adds to an angle to make 360° - CDE's reflex angle is CDF+FDE

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Adjacent angles: Have the same vertex and share a side. HRL,HRO are adjacent.

Complementary angles: add up to 90°. PRQ,QRI are complementary angles.

Supplementary angles: add up to 180°. JSN,NSK are supplementary angles.

Vertical angles (X property): Angles which share line segments and vertexes are equivalent. JSR,OST are vertical angles. They share the same degree value.

Corresponding angles (F property): Angles that share a line segment that intersects with parallel lines and is in the same relative position on each parallel line are equivalent. IRQ,KUQ are corresponding angles. They share the same degree value.

Alternate interior angles (Z property): Angles that share a line segment that intersects with parallel lines and is in opposite relative positions on each parallel line are equivalent. HRS,RST are alternate interior angles. They share the same degree value.

Bisecting an Angle: To bisect an angle, draw a concurrent line through the angle's vertex, which splits the angle exactly in half. This is possible using a compass and an unmarked straightedge.

Trisecting an Angle: To trisect an angle, use the same procedure as bisecting an angle, but use two lines and split the angle exactly in thirds. This is an ancient impossibility - it is impossible to accomplish using a compass and an unmarked straightedge.

Lines

Definitions:

Parallel lines: Lines drawn on a 2-dimensional plane may extend forever in either direction without ever intersecting. Lines HI and JK are parallel.

Perpendicular lines: Lines which intersect at exactly a 90° angle. Lines HI and MP are perpendicular.

Concurrent lines: Lines that all intersect at the same point. Lines HI,LQ,MP,NO are concurrent.

Skew lines: Lines drawn in a 3-dimensional space are neither parallel nor perpendicular and do not intersect.

Perpendicular Bisector: A line that is perpendicular to a given line and bisects it is called a perpendicular bisector.

Example 4.1.1:

Draw a cube and connect all the edges. Can you find two skew lines?

Planes

Definition:

A plane is a two-dimensional space that extends infinitely in all directions. For example, the graph of functions takes place on a Cartesian plane or a plane with coordinates. The plane continues in both the x and y directions.

Axioms

The points, lines, and planes are objects with the relations given by the following axioms:

  1. There is a unique line passing through two distinct points.
  2. If two points lie in a plane, then any plane containing those points lies in the plane.
  3. There is a unique plane containing three non-collinear points.
  4. If two planes meet, then their intersection is a line.

Euclid's Five Postulates

Euclid's Five Postulates 

Euclid's five postulates can be stated as follows

  1. It is possible to draw a straight line segment joining any two points.
  2. It is possible to indefinitely extend any straight line segment continuously in a straight line.
  3. Given any straight-line segment, it is possible to draw a circle with the segment as a radius and one endpoint as its center.
  4. All right angles are equal to each other or congruent
  5. Only one line can be drawn parallel to a given line through a given point, not on a given straight line.

 


This page titled 4.1: Euclidean geometry is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.

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