Lines, Planes, And Separation

By Aldinette and Jacinda BSED Math students

Postulate 4. The Line Postulate 

For every two points, there is exactly one line that contains both points.

Postulate 5. 



(a) Every plane contains at least three noncollinear points. (b) Space contains at least four noncoplanar points.

Theorem 3-1 

If two different lines intersect, their intersection contains only one point.

Flatness of Planes Postulate 6  It two 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 the intersection contains only one point. Postulate 7. The Plane Postulate  Any three points lie in at least one plane, and any three noncollinear 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  If two different planes intersect, then there intersection is a line.

Point • A point is simply a location. It has no dimension (shape or size), is usually represented by a small dot, and named by a capital letter.

A

Point A

Line • A line is a set of points and extends in one dimension. It has no thickness or width, is usually represented by a straight line with two arrowheads to indicate that it extends without end in both directions, and is named by two points on the line or a lowercase script

m

A

B

Line AB or line

m

Plane • A plane is a flat surface made up of points. It extends in two dimensions, is usually represented by a shape that looks like a tabletop or wall, and is named by a capital script letter or 3 noncollinear points.

A

M C

B

Plane

ABC

or plane

M

SPACE • Space is a boundless, three dimensional set of all points. It can contain points, lines, and planes.

A few more basic concepts using these undefined terms ...  Collinear

points are points that lie on the same line.

 Coplanar

points are points that lie on the same plane.

Example 1: Name three points that are collinear Solution: D, E and F lie on the same line, so they are collinear.

H G

D

E

F

Example 2: Name four points that are coplanar. Solution: D, E, F, and G lie on the same plane, so they are coplanar.

H G

D

E

F

Example 3: Name three points that are not collinear. Solution: points H, E, and G do not lie on the same line.

H G

D

E

F

Intersections of Lines & Planes  Two

or more lines intersect if they have a common point.  Two or more planes intersect if they have a common line.  Thus,

the intersection of any figures is the set of points the figures have in common.

Example 4:



Sketching a line that intersects a plane in one point

Example 5: 

Sketching two planes that intersect in a line

1.) Site 6 coplanar 2.) Intersection of plane

X W

U V T Q

S R

QRST and plane RSWV 3.) Intersection of UV and plane QTXU 4.) Lines that intersect at point S 5.) Planes that intersect at XW 6.) A point that is in the same plane as points U, S and R

Postulate 9. (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: Each of the sets is convex; If P is in one set and Q is in the other, then segment PQ intersects the line.

 

Postulate 10. (Space Separation Postulate) The points of space that do

not lie in a given plane form two sets such that: Each of the sets is convex. If P is in one set and Q is in the other, then segment PQ intersects the plane.

CONVEX  In

Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.

Half-planes

Half-spaces

m

P

P

Q

Q