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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
• 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”
•
•
•
•
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”
•
•
•
•
•
•
•
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. There is exactly one line that contains
any two distinct points
2. If two points lie in a plane, then the line
containing the points lies in the plane.
3. If two distinct planes intersect, then their
intersection is a line.
4. There is exactly one plane that contains
any three distinct noncollinear points.
Properties (cont)
5. A line and a point not on the line
determine a plane.
6. Two parallel lines determine a plane
7. 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
F
I
J
<I
<GFE
M
Ex.
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)°
A
D
x = 75
C
m<ABC = 82°
m<ABD = (x – 13)°
m<DBC = (x + 7)°
x = 44
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
•
•
•
•
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
• 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.