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.
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
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:
- There is a unique line passing through two distinct points.
- If two points lie in a plane, then any plane containing those points lies in the plane.
- There is a unique plane containing three non-collinear points.
- 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
- It is possible to draw a straight line segment joining any two points.
- It is possible to indefinitely extend any straight line segment continuously in a straight line.
- Given any straight-line segment, it is possible to draw a circle with the segment as a radius and one endpoint as its center.
- All right angles are equal to each other or congruent
- Only one line can be drawn parallel to a given line through a given point, not on a given straight line.