Points, Lines, Planes and Angles

Objectives

• Students will be able to:• Define: Point, line, plane, collinear, coplanar,

line segment, ray, intersect, intersection• Name collinear and coplanar points• Draw lines, line segments, and rays• With proper labeling• Draw opposite rays• Sketch intersections of lines and planes and two

planes

Common Words

• What are “common words” we use in everyday conversation?

• Do you have to think about their meanings?

• What if you didn’t know these words?

• In today’s lesson you will learn about common words needed to speak the language of geometry.

Points, Lines, and Planes

A location in space, but has no size or shape

Called Point A

A

ZY

Extends without end in one dimension (two directions) and always straight

Called Or line l

l

A

B

C

MCalled plane ABC or plane M

Extends without end in two dimensions (all directions), always flat, and has no thickness

Space

The set of all points.

Collinear and CoplanarPoints on the same line

Points GFED

D, E, F, and G are collinear

AC

B

nl

A, B, and C are

coplanar points

Lines l and n are

coplanar lines“Co” means “together”

Naming Points

Name a point that is collinear with the given points

D and G:

C and H:

A and C:

G and B:

B and E:

H and E:

A

ED

CB

HI

G

FI

D

B

C

B

E

Example 1: Naming Points

H

P

F

D

G

E

Three points that are collinear:

D, E, F

Four points that are Coplanar:

D, E, F, G and D, E, F, H (plane not shown)

Three points that are not collinear:

H, E, G

Naming Points

H

E

G

DC

BA

F

S

R

Q

M, N, R:

M, N, O:

M, T, Q:

P, O, R:

T, Q, N:

T, S, R:

Q, S, O:

Name a point that is coplanar with the given points or not coplanar

P

T

O

P

Q

P

T

Line Segments and Rays

End points Y and Z and all points in between

Y A Z Line YZ

Y A Z Line segment

YZ YA AZ

AZYA,YZ,

Starting point and all points that extend from that point

A B

A B

Ray AB AB

Ray BA BA

Drawing Lines and Rays

Draw Points J, K, and L (non-Collinear)

J

L

K

Then, draw JK, KL, and LJ … try connecting them all…

IntersectionsTo cross at a common point.

(verb)

k

l

Lines k and l intersect at the point A, so both lines have this point in common.

A

(noun)

A set of points that are shared between two lines or planes

Postulate 1-3If two planes intersect, then they intersect

in exactly one line.

T

n

• Intersection: A set of points that are shared between two lines or planes

line n and plane P intersect at point T

P

• VISUALIZATION Name the geometric shape modeled by a colored dot on a map used to mark the location of a city.

A. point

B. line segment

C. plane

D. none of the above

VISUALIZATION Name the geometric shape modeled by the ceiling of your classroom. (In this particular case, it doesn’t extend forever)

A. point

B. line segment

C. plane

D. none of the above

Choose the best diagram for the given relationship. Plane D contains line a, line m, and line t, with all three lines intersecting

at point Z. Also point F is on plane D and is not collinear with any of the three given lines.

A. B.

C. D.

Interpret Drawings

How many planes appear in this figure?

Answer: There are two planes: plane S and plane ABC.

• Through any two points there is exactly one line.

Postulate 1-1

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

Postulate 1-2

A

Postulate 1-4

Through any three non-collinear points there is exactly one plane.

In the figure below, name three points

that are collinear and three points that

are not collinear.

Points Y, Z, and W lie on a line, so they are collinear.

Any other set of three points do not lie on a line, so no other set of three points is collinear.

For example, X, Y, and Z and X, W, and Z form triangles and are not collinear.

Name the plane shown in two different ways.

You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following:

plane RST

plane RSU

plane RTU

plane STU

plane RSTU

Shade the plane that

contains X, Y, and Z.

Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z.

Use the diagram below. What is the intersection

of plane HGC and plane AED?

As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED.

The back and left faces of the cube intersect at HD.

Planes HGC and AED intersect vertically at HD.

Closure

Name five of the nine common words we learned

How far does a line go in each direction? A plane?

How many directions does a ray have?

What is it called when points are on the same line? In the same plane?

Make a Conjecture

N 1 2 3 4 5

Q 5 7 9 11 13

a) Make a conjecture in terms of Q(n) for the following sequence, b) find the 100th term.

a) 2n + 3 b) 203