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Math 3379 – Chapter 3, Venema
Neutral Geometry in the Plane
Undefined terms:
point, line, distance, halfplane, angle measure, area
Axioms:
Axiom 1
The Existence Postulate
The collection of all
points
forms a nonempty set.
There is more than
one point in the set.
p. 36
Axiom 2
The Incidence Postulate
Every
line
is a set of
points
. For every pair of distinct
points
A
and
B
there is exactly one
line
l
such that
A
∈
l
and
B
∈
l.
p. 36
Axiom 3
The Ruler Postulate
For every pair of
points
P
and
Q
there exists a real number PQ, called the
distance from
P
to
Q
.
For each
line
l
there is a onetoone correspondence
from
l
to
R
such that if
P
and
Q
are
points
on the line that correspond to
the real numbers
x
and
y
, respectively, then PQ =
x
y

.
p. 37
Axiom 4
The Plane Separation Postulate
For every
line
l
, the
points
that do not lie on
l
form two disjoint,
nonempty sets
1
H
and
2
H
, called
halfplanes
bounded by
l
, such that the
following conditions are satisfied.
1.
Each of
1
H
and
2
H
is convex.
2.
If P
∈
1
H
and Q
∈
2
H
, then
PQ
intersects
l
.
p. 46
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The Protractor Postulate
For every angle
∠
BAC
there is a real number
μ( ∠
BAC
), called the
measure
of
∠
BAC
, such that the following conditions are satisfied.
1.
0º
≤ ∠
BAC
< 180 º for every angle
BAC.
2.
μ( ∠
BAC
) = 0º if and only if
AB
AC
=
uuur
uuur
.
3.
(Angle Construction Postulate) For each real number
r
,
0 <
r
< 180, and for each halfplane
H
bounded by
AB
uur
s
there exists
a unique ray
AE
uuur
such that
E
is in
H
and
μ∠
(
BAE
) =
r
º.
4.
(Angle Addition Postulate) If the ray
AD
uuur
is between rays
AB
uuur
and
AC
uuur
, then
μ( ∠
BAD
) +
μ( ∠
DAC
) =
μ( ∠
BAC
).
p. 51
Axiom 6
The SideAngleSide Postulate (SAS)
If
∆
ABC
and
∆
DEF
are two triangles such that
AB
DE
2245
,
ABC
DEF
∠
2245 ∠
, and
BC
EF
2245
, then
∆
ABC
2245
∆
DEF.
p. 64
Axiom 7
The Parallel Postulate
p. 66
(Chapter 5 and beyond)
Euclidean Parallel Postulate
:
For every line
l
and for every point
P
that does not lie on
l
, there is exactly one line
m
such that
P
lies on
m
and
m is parallel to
l
.
Elliptic Parallel Postulate
:
For every line
l
and for every point
P
that does not lie on
l
, there is no line
m
such that
P
lies on
m
and
m
is
parallel to
l
.
Hyperbolic Parallel Postulate
:
For every line
l
and for every
point
P
that does not lie on
l
, there are at least two lines
m
and
n
such that
P
lies on both
m
and
n
and both
m
and
n
are
parallel to
l
.
For Chapters 3 and 4 we will work only with Axioms 1 – 6.
Starting with Chapter 5 we
will make decisions about which parallel postulate we are going to explore.
In Chapter 7
we will explore the undefined term
area
and add axioms about area to our list.
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 Spring '08
 Staff
 Geometry

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