BASIC GEOMETRY
The Basic Postulate
The Basic Postulate
Postulate #1 Midpoint Postulate
"every segment has exactly one midpoint"
Postulate #2 Points Postulate
A line contains infinite points
Postulate #3 Lines Postulate
For every two points there is exactly one line that contains both points
Postulate #4 Plane Postulate
Any 3 points lie in at least one plane
Any 3 non collinear points lie in exactly one plane
Postulate #5 Space Postulate
A space contains at least 4 non collinear points
Postulate #6 Flat Plane Postulate
If two points of a line lie in a plane, then the line lies in the same plane
Postulate #7 Plane Intersection Postulate
If two different planes intersect then their intersection is a line
Postulate #8 Distance Postulate
To every pair of different points there corresponds a unique positive real number
Definition:
The distance between two points is the number given by the Distance Postulate. If the points are P and Q, then the distance is demoted as PQ or QP
Postulate #9 The Ruler Postulate
The points of a line can be placed in correspondence with the real numbers such that to every...
> point of the line there corresponds exactly one real number
> number there corresponds exactly one point of the line; and the distance between any two points is the absolute value of the difference of the corresponding real number.
-> Any real number corresponding to a given point is called the coordinate of a point. The one to one correspondence between the point of a line and the set or real numbers is called a coordinate system
Postulate #10 The Ruler Placement Postulate
Given two points P and Q of a line, the coordinate system can be chosen such that the coordinate of P is zero and the coordinate of Q is positive
Postulate #11 Segment Construction Postulate
Let a RD be a ray, and let x be a positive number then there's exactly one point P of RP such that RP=x.
"every segment has exactly one midpoint"
Postulate #2 Points Postulate
A line contains infinite points
Postulate #3 Lines Postulate
For every two points there is exactly one line that contains both points
Postulate #4 Plane Postulate
Any 3 points lie in at least one plane
Any 3 non collinear points lie in exactly one plane
Postulate #5 Space Postulate
A space contains at least 4 non collinear points
Postulate #6 Flat Plane Postulate
If two points of a line lie in a plane, then the line lies in the same plane
Postulate #7 Plane Intersection Postulate
If two different planes intersect then their intersection is a line
Postulate #8 Distance Postulate
To every pair of different points there corresponds a unique positive real number
Definition:
The distance between two points is the number given by the Distance Postulate. If the points are P and Q, then the distance is demoted as PQ or QP
Postulate #9 The Ruler Postulate
The points of a line can be placed in correspondence with the real numbers such that to every...
> point of the line there corresponds exactly one real number
> number there corresponds exactly one point of the line; and the distance between any two points is the absolute value of the difference of the corresponding real number.
-> Any real number corresponding to a given point is called the coordinate of a point. The one to one correspondence between the point of a line and the set or real numbers is called a coordinate system
Postulate #10 The Ruler Placement Postulate
Given two points P and Q of a line, the coordinate system can be chosen such that the coordinate of P is zero and the coordinate of Q is positive
Postulate #11 Segment Construction Postulate
Let a RD be a ray, and let x be a positive number then there's exactly one point P of RP such that RP=x.
THEOREMS
2.1 Theorem Line Intersection Theorem
If two different lines intersect, their intersection contains only one point
2.2 Line-Plane Intersection Theorem
If a line intersects a plane not containing it, then the intersection contains only one point.
2.3 Point-Line Plane Theorem
Given a line and a point not on the line there is exactly one plane containing both
2.4 Intersecting Lines Planes Theorem
Given two intersecting lines, there is exactly one plane containing both
2.1 Theorem Line Intersection Theorem
If two different lines intersect, their intersection contains only one point
2.2 Line-Plane Intersection Theorem
If a line intersects a plane not containing it, then the intersection contains only one point.
2.3 Point-Line Plane Theorem
Given a line and a point not on the line there is exactly one plane containing both
2.4 Intersecting Lines Planes Theorem
Given two intersecting lines, there is exactly one plane containing both
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