lines, planes, and separation

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By Aldinette and By Aldinette and Jacinda Jacinda BSED Math students BSED Math students

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This is a slideshow of plane geometry and topic is about lines, planes and separation.

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Page 1: Lines, Planes, And Separation

By Aldinette and Jacinda By Aldinette and Jacinda BSED Math studentsBSED Math students

Page 2: Lines, Planes, And Separation

Postulate 4. The Line PostulatePostulate 4. The Line Postulate For every two points, there is exactly For every two points, there is exactly

one line that contains both points.one line that contains both points.

Postulate 5. Postulate 5. (a) (a) Every plane contains at least Every plane contains at least

three noncollinear points.three noncollinear points. (b) (b) Space contains at least four Space contains at least four

noncoplanar points.noncoplanar points.

Theorem 3-1Theorem 3-1 If two different lines intersect, their If two different lines intersect, their

intersection contains only one point.intersection contains only one point.

Page 3: Lines, Planes, And Separation

Flatness of PlanesFlatness of PlanesPostulate 6 Postulate 6 It two points of a line lie in a plane, It two points of a line lie in a plane,

then the line lies in the same plane.then the line lies in the same plane.

Theorem 3-2Theorem 3-2 If a line intersects a plane not If a line intersects a plane not

containing it, then the intersection containing it, then the intersection contains only one point.contains only one point.

Postulate 7. The Plane PostulatePostulate 7. The Plane Postulate Any three points lie in at least one Any three points lie in at least one

plane, and any three noncollinear plane, and any three noncollinear points lie in exactly one plane.points lie in exactly one plane.

Page 4: Lines, Planes, And Separation

Theorem 3-3Theorem 3-3 Given a line and a point not on the Given a line and a point not on the

line, there is exactly one plane line, there is exactly one plane containing both.containing both.

Theorem 3-4Theorem 3-4 Given two intersecting lines, there Given two intersecting lines, there

is exactly one plane containing is exactly one plane containing both.both.

Postulate 8Postulate 8 If two different planes intersect, If two different planes intersect,

then there intersection is a line.then there intersection is a line.

Page 5: Lines, Planes, And Separation

PointPoint

• 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

Page 6: Lines, Planes, And Separation

LineLine

• 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 letter.

A

B

m

Line AB or line

m

Page 7: Lines, Planes, And Separation

PlanePlane• A planeplane 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 non-collinear points.

A

BC

M

Plane ABC or plane M

Page 8: Lines, Planes, And Separation

SPACE

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

Page 9: Lines, Planes, And Separation

A few more basic concepts A few more basic concepts using these undefined terms using these undefined terms . . .. . . Collinear pointsCollinear points are points that are points that

lie on the same line.lie on the same line.

Coplanar pointsCoplanar points are points that are points that lie on the same plane.lie on the same plane.

Page 10: Lines, Planes, And Separation

Example 1:Example 1:

Name three points Name three points that are collinearthat are collinear

Solution: Solution:

D, E and F lie on D, E and F lie on the same line, so the same line, so they are they are collinear.collinear.

G

D E F

H

Page 11: Lines, Planes, And Separation

Example 2:Example 2:

Name four points Name four points that are that are coplanar.coplanar.

Solution: Solution:

D, E, F, and G D, E, F, and G lie on the same lie on the same plane, so they plane, so they are coplanar. are coplanar.

G

D E F

H

Page 12: Lines, Planes, And Separation

Example 3:Example 3:

Name three points Name three points that are not that are not collinear.collinear.

Solution: Solution:

points H, E, and G points H, E, and G do not lie on the do not lie on the same line.same line.

G

D E F

H

Page 13: Lines, Planes, And Separation

Intersections of Lines & Intersections of Lines & PlanesPlanes Two or more Two or more lines intersectlines intersect if they if they

have a have a common pointcommon point. . Two or more Two or more planes intersectplanes intersect if if

they have a they have a common linecommon line. .

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

Page 14: Lines, Planes, And Separation

Example 4:Example 4:

Sketching a line Sketching a line that intersects a that intersects a plane in one plane in one pointpoint

Page 15: Lines, Planes, And Separation

Example 5:Example 5:

Sketching two Sketching two planes that planes that intersect in a intersect in a lineline

Page 16: Lines, Planes, And Separation

U

X

V

W

Q

T

S

R

1.) Site 6 coplanar2.) Intersection of plane QRST and plane RSWV3.) Intersection of UV and plane QTXU4.) Lines that intersect at point S5.) Planes that intersect at XW6.) A point that is in the same plane as points U, S and R

Page 17: Lines, Planes, And Separation

Postulate 9.Postulate 9. ((Plane Separation Plane Separation PostulatePostulate)) Given a line and a plane Given a line and a plane containing it, the points of the plane that do containing it, the points of the plane that do not lie on the line form two sets such that: not lie on the line form two sets such that:

Each of the sets is convex; Each of the sets is convex;

If If PP is in one set and is in one set and QQ is in the other, then segment is in the other, then segment PQPQ intersects the line. intersects the line.

  

Postulate 10.Postulate 10. ((Space Separation Space Separation PostulatePostulate)) The points of space that do not The points of space that do not lie in a given plane form two sets such that: lie in a given plane form two sets such that:

Each of the sets is convex. Each of the sets is convex.

If If PP is in one set and is in one set and QQ is in the other, then segment is in the other, then segment PQPQ intersects the plane. intersects the plane.

Page 18: Lines, Planes, And Separation

CONVEXCONVEX In Euclidean space, an object is In Euclidean space, an object is

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

Page 19: Lines, Planes, And Separation
Page 20: Lines, Planes, And Separation
Page 21: Lines, Planes, And Separation

Half-planesHalf-planes

m

Q

P

Half-spacesHalf-spaces

P

Q

Page 22: Lines, Planes, And Separation