Geometry Chapter 3 Flashcards
Space
Space is the set of all points
Collinear
A set of points is collinear if there is line which contains all the points of the set
Coplanar
A set of points is coplanar if there is a plane which contains all of a set
Postulate 4
The Line Postulate
State the Line Postulate
For every two different points there is exactly one line that contains both points
Theorem 3-1
If two different lines intersect, their intersection contains only one point
Exactly one? Only One?
- 1 or 0
Postulate 5
The Plane Space Postulate
State the Plane Space Postulate
A) Every plane contains at least 3 different non-collinear points
B) Space contains at least 4 different non-coplanar points
Postulate 6
The Flat Plane Postulate
State the Flat Plane Postulate
If 2 points of a line lie in a plane, then the line lies in the same plane
Theorem 3-2
If a line intersects a plane not containing it, then their intersection contains only one point
Postulate 7
The Plane Postulate
State the Plane Postulate
Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane
Theorem 3-3
Given a line and a point not on the line, there is exactly one plane containing both
Theorem 3-4
Given two intersecting lines, there is exactly one plane containing both
Postulate 8
Intersection of Planes Postulate
State the Intersection of Planes Postulate
If two different planes intersect, then their intersection is a line
Convex
A set M is called convex if for every two points P and Q of the set, the entire segment (line on top PQ) lies in M
Postulate 9
The Plane Separation Postulate
State the Plane Separation Postulate
Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that
1) each of the sets is convex, and
2) if P is in one of the sets and Q is in the other, then the segment (line PQ) intersects the line
State the Plane Separation Postulate
Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that
1) each of the sets is convex, and
2) if P is in one of the sets and Q is in the other, then the segment (line PQ) intersects the line
Half planes pt 1
Given a line L and a plane E containing it, the two sets described in the Plane Separation Postulate are called half planes or sides of L, and L is called the edge of each of them.
Half planes pt2
If P lies in one of the half planes and Q lies in the other, then we say that P and Q lie on opposite sides of L
Half planes pt 1 and 2
Given a line L and a plane E containing it, the two sets described in the Plane Separation Postulate are called half planes or sides of L, and L is called the edge of each of them. If P lies in one of the half planes and Q lies in the other, then we say that P and Q lie on opposite lines of L
Postulate 10
The Space Separation Postulate
State the Space Separation Postulate
The points of space that do not lie in a given plane for two sets such that
1) each of the sets is convex, and
2) If P is in one of the sets and Q is in the other, then the segment (line on top PQ) intersects the plane
Half spaces
If two sets described in the Space Separation Postulate are called half spaces, and the given plane is called the fact of each of them
