Lesson (1.3) Points, Lines, and PlanesLesson (1.3) Points, Lines, and Planes

Students will…• understand basic terms. • understand basic postulates of geometry.

Evidence Outcome: Students will express properties with equations (coordinate geometry). (HS 4.3a)

Purpose: Photographers and surveyors use a tripod or a three-legged stand for their instrument.

Terms

A point is a location. It does not have a size.Space is the set of all points.

A line is a series of points going in two opposite directions. You can name a line by any two points on the line.Points that lie on the same line are collinear (opposite: noncollinear).

A

BName: AB or BA

A

B

C

Terms

A plane is a surface that has no thickness. It contains many lines going in all directions. A plane is named by one capital letter or at least three of its noncollinear points.

Points and lines in the same plane are coplanar.

Names:

P

Plane P Plane ABC or Plane BCA

cPostulates/Axioms

A postulate or axiom is an accepted statement of fact.

Postulate 1-1

Through any two points there is exactly one line.

Line t is the only line that passes through points A and B.

A B

Postulate 1-2

If two lines intersect, then they intersect in exactly one point.

AE and BD intersect at C. A C B

D E

cPostulates/Axioms

A postulate or axiom is an accepted statement of fact.

Postulate 1-3

If two planes intersect, then they intersect in exactly one line.

Postulate 1-4

Through any three noncollinear points, there is exactly one plane.

Plane RST and Plane STW intersect in ST. R

W

S

T

cTry these…

1. Identify collinear points in the room.

2. Identify a plane.

3. Find the intersection of two planes.

4. Draw a plane through three noncollinear points.

Lesson (1.4) Segments, Rays, Lesson (1.4) Segments, Rays, Parallel Lines and PlanesParallel Lines and Planes

Students will…• identify segments and rays.• recognize parallel lines.

Evidence Outcome: Students will express properties with equations (coordinate geometry). (HS 4.3a)

Purpose (Relevancy): On a compass, the directions north and south can be represented by opposite rays.

Terms

A segment is the part of a line consisting of two endpoints and the points between them.

A B

A ray is one endpoint and all of the points on one side of the endpoint.

Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line.

Naming

Name the segments and rays.

A

BB

C

Three segments:

Four rays:

Parallel… ??

•Visualize a plane (it’s flat).•Think of two lines that are both in the plane. •Now make sure that the two lines do not intersect. [PARALLEL]•Notice that the lines are coplanar (in the same plane).

•Now start over with a blank plane.•Think of one line in the plane.•Think of another line that can’t be included in the plane, even if you rotated the plane (remember: the plane has to be flat – it can’t turn a corner).•Now make sure that those two lines do not intersect.•This is different than parallel because the lines are noncoplanar (not in the same plane).

Terms

Parallel lines are coplanar lines that do not intersect.

Skew lines are noncoplanar; therefore, we can’t say they’re parallel, but we can say that they do not intersect.

Do skew lines “look” parallel?

Terms

Parallel planes are planes that do not intersect.

*Note: A line and a plane that do not intersect with each other are also parallel.

Example: 1. Identify two parallel planes. 2. Identify a plane and a line that are parallel.

A

B

C

D

J

G H

I