Download 9-1

Math 310
Section 9.1
Geometry Introduction
Axiomatic System
A logical system which possesses an explicitly
stated set of axioms from which theorems can
be derived.
(mathworld.wolfram.com)
www.xkcd.com
Undefined Terms
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Point
Line
Plane
“One must be able to say at all
times—instead of points, straight
lines, and planes—tables, chairs,
and beer mugs.”
- David Hilbert
Linear “Notions”
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Collinear points
Is between
Line segment (segment)
Ray
Collinear points (& non collinear)
D
B
E
C
Is Between
B
E
C
Line Segment
F
G
Ray
I
H
Planar “Notions”
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Coplanar points
Noncoplanar points
Coplanar lines
Skew lines
Intersecting lines
Concurrent lines
Parallel lines
Coplanar & Noncoplanar Points
D
B
A
C
Coplanar lines
I
E
D
F
G
H
Skew Lines
I
E F
H
Intersecting Lines
J
L
K
Concurrent Lines
J
M
K
L
Parallel Lines
O
N
P
Q
Properties of “tables, chairs and beer
mugs”
1.
2.
3.
4.
There is exactly one line that contains any two
distinct points
If two points lie in a plane, then the line
containing the points lies in the plane.
If two distinct planes intersect, then their
intersection is a line.
There is exactly one plane that contains any
three distinct noncollinear points.
Properties (cont)
5.
6.
7.
A line and a point not on the line determine a
plane.
Two parallel lines determine a plane
Two intersecting lines determine a plane.
Property 1
B
A
A
B
Property 2
Property 3
B
A
Property 4
Property 5
Property 6
Property 7
Intersecting Planes


Parallel
Along a line
Parallel Planes
Planes Intersecting along a line
Angle, Vertex, Side
Def
When two rays share an endpoint, an angle is formed.
The common initial point of the rays is the vertex of
the angle. Each ray is called a side of the angle.
Ex.
A
B
Angle: <ABC
Side
Vertex
C
Ex.
E
H
E
G
F
<EBF
B
I
F
J
<I
<GFE
Ex.
M
J
K
N
<MJK
Name all six
angles.
<NKL
<OLJ
L
O
<KJL
<LKJ
<JLK
Angle Measure
To measure an angle we use the unit degree. It
measures the “opening” of the angle. The
largest angle measure is 360° and the smallest is
0°. A complete rotation about a point is 360°.
For more accuracy, angles can be further
measure in minutes, and seconds. Each degree
is divided into 60 minutes, and each minute is
divided into 60 seconds.
Ex.
Add: 45°23’47” and 62°36’51”
45°23’47” + 62°36’51” = 108°0’38” or 108°38”
Add: 145°17’4” and 220°31’32”
145°17’4” + 220°31’32” = 365°48’36” = 5°48’36”
Ex.
Solve for x.
m<ABC = 80°
m<ABD = 30°
m<DBC = (x – 25)°
x = 75
A
C
m<ABC = 82°
m<ABD = (x – 13)°
m<DBC = (x + 7)°
x = 44
D
B
Protractor
A protractor is a standard tool for measuring
angles. To use, line the vertex up with the
center of the base of the protractor and line one
side of the angle up with the 0° mark. Now
measure from 0°, increasing, until you see the
other side of the angle and read the mark.
Types of Angles
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Obtuse
Acute
Right
Straight
E
Ex.
H
B
F
acute
I
H
G
C
B
J
obtuse
F
straight
A
right
Perpendicular Lines
Def.
Two lines are perpendicular if they intersect and
form right angles.
perpendicular
not
perpendicular
Line Perpendicular to a Plane
A line is perpendicular to a plane if it is
perpendicular to every line, contained in the
plane, passing through the point of intersection.
Ex.
Ex.
Questions
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Is it possible for a line intersecting a plane to be
perpendicular to exactly one line in the plane
through its intersection with the plane?
Can a line intersecting a plane be perpendicular
to exactly two distinct lines in the plane going
through the point of intersection?
Yes
No
Line Perpendicular to a Plane
Thrm
A line perpendicular to two distinct lines in the
plane through its intersection with the plane is
perpendicular to the plane.