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Warm-up. Show that the following conjecture is not true by finding a counterexample. Conjecture : All odd numbers can be expressed as the sum of two primes. Directions: In a few sentences, describe as many of the words below. pointlineplane collinear coplanarsegment - PowerPoint PPT Presentation
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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.
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