points, lines, and planes to understand basic terms of geometry. to understand basic postulates of...

Points, Lines, and Planes

To understand basic terms of geometry.

To understand basic postulates of geometry.

DefinitionsPoint –

Space –

Line –

Collinear points –

It is a location in space. A point is represented by a small dot and is named by a capital letter.

the set of all points.

A series of points that extends in two opposite directions.

Points that lie on the same line.

Segment – the part of a line consisting of two end points and all points between them.

Ray – the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.

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

Example 1

a. Are points E, F, and C collinear? If so, name the line of which they lie.

EF

C

PD

n

m

l

Answer

a. Points E, F, and C are collinear. They lie on line m.

EF

C

PD

n

m

l

Example 1

b. Are points E, F, and D collinear? If so, name the line of which they lie.

EF

C

PD

n

m

l

Answers

Points E, F, and D are not collinear.

EF

C

PD

n

m

l

Example 1: Naming Segments and Rays

Name the segments and rays in the figure at the right.

The three segments are LP, PQ, and LQ.

The four rays are LP or LQ, PQ, PL, and QP or QL.

Check Understandinga. Are points F, P,

and C collinear?

EF

C

PD

n

m

l

No, not on the same line

Check Understandingb. Name line m in 3 other ways.

EF

C

PD

n

m

l

, , or

Check UnderstandingC. Why do you think arrowheads are used when drawing a line or naming a line such as ?

EF

C

PD

n

m

l

Arrowheads are used to show that the line extends in opposite directions without end.

More DefinitionsPlane

– It is a flat surface that has no thickness. It contains many lines and extends without end in the direction of all its lines. You can name a plane by either a single capital letter or by at least 3 of its noncollinear points.

Coplanar - Points and lines in the same plane are coplanar.

Example 2

Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube.

The front plane can be named by any 3 of these four letters ABFE.

Check UnderstandingList three different names for the plane represented by the top of the ice cube.

HEF, HEFG, EFG, FGH, and GHE

to name a few.

Basic Postulates of GeometryA postulate or axiom is an accepted statement of fact.

You have used some of the following geometry postulates in algebra.

For example, you used “two points determine a line” when you graphed an equation such as y = -2x + 8.You plotted two points and then drew the line through those two points.

Through any two points there is exactly one line.

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

TWO POINTS DETERMINE A LINE.

TWO LINES INTERSECT IN A POINT.

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

AE and BD intersect at C.

TWO PLANES INTERSECT IN A LINE.

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

Plane RST and plane STW intersect in ST.

Example 3

What is the intersection of plane HGFE and plane BCGF?

They intersect in GF

Check understanding:

Name two planes that intersect in BF.

ABFE and BCGF

TRIPODA three-legged stand will always be stable. As long as the feet of the stand don’t lie in one line, the feet of the three legs will lie exactly in one plane

THREE POINTS DETERMINE A PLANE.

Through any three noncollinear points there is exactly one plane.

Example 4

Shade the plane that contains A, B, and C.

Example 4b. Shade the plane

that contains E, H, and C.

Check Understanding

a. Name another point that is in the same plane as points A, B, and C.

a. D

Check Understanding

b. Name another point that is coplanar with points E, H, and C.

b. B

Quiz

The following questions are a review of what we went over today in class. They are designed to help you assess for yourself whether or not you understood today’s lesson. If you do not do well and miss several of the questions it might be a good idea to come in and get some help!

1. Are O, N, and P collinear? If so, name the line on which they lie.

1. Yes, they lie on the line MP.2. Yes, they lie on the line NP.3. Yes, they lie on the line MO.4. No, the three points are not

collinear.

2. Name the line and plane shown in the diagram.

A. RS and plane RSUB. Line R and plane RSUC. RS and plane URD. SR and plane UT

3. What is the intersection of plane TUYX and plane VUYZ?

A. TXB. SWC. UYD. VZ

4. Name a fourth point in plane TUW. A. YB. ZC. WD. X

5. Plane ABC and plane BCE ____ be the same plane.

A. mustB. mayC. cannot