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Foundations of Geometry

Preliminary study of basic concepts of Euclidean geometry

Marie Graham

on 24 August 2010

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Transcript of Foundations of Geometry

Double click anywhere & add an idea Euclid Greek mathematician most important work--13 volume book entitled Elements based his axiomatic system of Euclidean Geometry all on three undefined terms-the most basic parts of an axiomatic system point line plane Euclidean Geometry circa 300 BC no thickness
no width
contains an infinite number of points
infinite length
named by one lower case letter or
by two points that lie on the line with a
miniature line over the top infinitely small
represents a location in space or object
named using one capital letter
no thickness
extends infinitely in all directions
represents a flat surface
named using one upper case letter (not associated with a point) or by three points that lie in the plane From these three undefined terms, Euclid developed defined terms and postulates or axioms axioms/postulates: mathematical laws
that are presumed true and need no proof
Vocabulary space collinear non collinear coplanar noncoplanar parallel skew the set of all points,
boundless and three-
dimensional three or more points
that do not lie on the
same line three or more points
that do not lie on the
same plane set of points that
lie in the same
plane set of points that
lie on the same
line coplanar lines that
do not intersect noncoplanar lines
that do not intersect line segment ray "a piece of a line"
that consists of two
endpoints and all of
the points between
them looks like "half of a
line"; has one
endpoint and extends infinitely in one direction
***the endpoint is always named first*** opposite rays two rays that share an
endpoint and extend
away from each other
(looks like a line) Intersection the set of points created
where two objects meet perpendicular lines that intersect to form a right angle 1. Through any two points there exists exactly one line. II. A line contains at least two points. III. If two lines intersect, then their intersection is exactly one point. IV. Through any three noncollinear points, there exists exactly one plane. V. A plane contains at least three noncollinear points. VI. If two points lie in a plane, then the line containing them lies in the plane. VII. If two planes intersect, then
their intersection is a line. line perpendicular to a plane a line is perpendicular to
a plane if it is perpendicular
to every line in the plane
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