Warm-upShow that the following conjecture is not true

by finding a counterexample.

Conjecture: All odd numbers can be expressed as the sum of two

primes.

point line plane collinear coplanar segment

endpoint initial point ray

opposite ray

Directions: In a few sentences, describe as many of the words below.

Chapter 1.2: Points, Lines, and Planes

• Students will identify and apply basic definitions of geometry.

A point is an exact location in space.

You are here.

A true point has no length, no width, and no height.

In fact, you cannot see a true point.

A point is named by a letter.

P

Point P

Lines are 1-dimensional objects that have only length. Lines

continue forever in both directions.

Because a line has no width or height, you cannot see a true line.

A line is defined by 2 points.

A

Bline AB or line BA

Collinear Points

Collinear points are points that lie on the same line. (The line does not have to be visible.)

A B C

CollinearA

B

C

Non collinear

A ray has an initial point but no ending point.

initial point

A ray of light has an initial point (like the sun) and continues

forever in the same direction.

Of course, if the ray hits an object, the light could be absorbed or reflected.

MIRROR

A ray is also defined by 2 points.

C

DRay CD

If C is between A and D, then CD and CA are opposite rays

C

Dray CD and ray CA

A

A line segment has a starting point and an ending point. Line

segments can be measured.

A line segment also defined by 2 points.

E

F Line Segment EF or FE

Lines can do various things.

Lines can intersect at a point to form angles.

X

Y

Z

Angles are defined by 3 points.

X

Z

Y

XYZ

These angles can be right angles.

90

These angles can be right angles.

c

R

S TU

V

m RST = 90⁰m USR = 90⁰

m USV = 90⁰ m TSV = 90⁰

When lines intersect to form right angles, they are said to be perpendicular.

V

R

TU

cS

UT RV

Lines can intersect to form acute and obtuse angles.

135

obtuse

45 acute

Lines can also run into each other to form straight angles.

180 is a straight angle

Lines do not always intersect.

Lines can be parallel.

Parallel lines have the same slope or steepness.

Is it possible for lines not to intersect and not be parallel

either?

Believe it or not, this is possible.Let’s consider a 3-dimensional

rectangular prism.

These 2 lines are not parallel, but they are not intersecting either.

These lines are called skew lines.

You might have heard the word “skewer” the last time you had a

barbecue.

                     

   

S

K

E

W

E

R

Lines can be…

intersecting

perpendicular

90º

parallelskew

A plane

is a flat

surface that has

length & width

but no height.

You can see a plane only if you view it at a certain angle.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

Different planes in a figure:A B

CD

EF

GH

Plane ABCD

Plane EFGH

Plane BCGF

Plane ADHE

Plane ABFE

Plane CDHG

Etc.

Other planes in the same figure:

Any three non collinear points determine a plane!

H

E

G

DC

BA

F

Plane AFGD

Plane ACGE

Plane ACH

Plane AGF

Plane BDG

Etc.

Coplanar Objects

Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

H

E

G

DC

BA

F

Are the following points coplanar?

A, B, C ?A, B, C, F ?H, G, F, E ?E, H, C, B ?A, G, F ?C, B, F, H ?

YesNo

YesYesYesNo

Planes can intersect.

Intersection of Two Planes is a Line.

P

R

A

B

Plane P and Plane R intersect at the line AB

An X-Wing fighter from Star Wars has wings that intersect.

Planes can be perpendicular.

Planes can be parallel.

A Tie Fighter from Star Wars has wings that are parallel.

Planes can be…intersecting

perpendicular

parallel

The end.