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# Foundations of Geometry

Preliminary study of basic concepts of Euclidean geometry

by

Tweet## Marie Graham

on 24 August 2010#### 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

Full transcriptno 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