Download Read 1.4, 2.6 Incidence Axiom 1. For each two distinct points there

 http://osu.orst.edu/instruct/mth338/garity
 Read 1.4, 2.6
Incidence Axiom 1. For each two
distinct points there exists a unique
line on both of them.
Incidence Axiom 2. For every line
there exist at least two distinct
points on it.
Incidence Axiom 3. There exist at
least three distinct points.
Incidence Axiom 4. Not all points
lie on the same line.
Any geometry that satisfies all four
incidence axioms will be called an
incidence geometry.
1
Incidence Theorem 1. If two
distinct lines intersect, then the
intersection is exactly one point.
Proof.
Incidence Theorem 2. For each
point there exist at least two lines
containing it.
Proof.
Incidence Theorem 3. There exist
three lines that do not share a
common point.
Proof
2
If we consider any line  and any
point P, where P is not on , then
three possibilities exist for a
parallel axiom.
1. There exist no lines on P that
are parallel to ,
2. There exists exactly one line on
P that is parallel to , or
3. There exists more than one line
on P parallel to .
3
THE SMSG POSTULATES FOR
EUCLIDEAN GEOMETRY
Undefined Terms
1. Point
2. Line
3. Plane
Postulate 1. Given any two distinct
points, there is exactly one line that
contains them.
2. The Distance Postulate. To
every pair of distinct points, there
corresponds a unique positive
number. This number is called the
distance between the two points.
4
3. The Ruler Postulate. The
points of a line can be placed in a
correspondence with the real
numbers such that
 To every point of the line there
corresponds exactly one real
number,
 To every real number there
corresponds exactly one point of
the line, and
 The distance between two distinct
points is the absolute value of the
difference of the corresponding real
numbers.
5
4. Ruler Placement . Given 2
points P and Q of a line, the
coordinate system can be chosen
in such a way that the coordinates
of P and Q are 0 and  0.
5. (a) Every plane contains at least
3 noncollinear points. (b) Space
contains at least 4 noncoplanar
points.
6. If 2 points lie in a plane, then the
line containing these points lies in
the same plane.
7. Any 3 points lie in at least one
plane, and any 3 non- colinear
points lie in exactly one plane.
6
8. If two planes intersect, then that
intersection is a line.
9. Plane Separation. Given a line
and a plane containing it, the
points of the plane that do not lie
on the line form two sets such that
 each of the sets is convex and
 if P is in one set and Q is in the
other, then segment P Q intersects
the line.
10. Space Separation.
11. Angle Measurement. To every
angle there corresponds a real
number between 0 and 180.
7
l2.Angle Construction. Let AB be
a ray on the edge of the half-plane
H. For every r between O and 180,
there is exactly one ray AP with P
in H such that mPAB   r.
13. Angle Addition. If D is a point
in the interior of BAC, then
mBAC   mBAD  
mDAC .
l4.Supplements. If two angles
form a linear pair,then they are
supplementary.
8
15. SAS. Given a 1-1
correspondence between two
triangles (or between a triangle
and itself): If two sides and the
included angle of the first triangle
are congruent to the corresponding
parts of the second triangle, then
the correspondence is a
congruence.
16. The Parallel Postulate.
Through a given external point
there is at most one line parallel to
a given line.
17-22. Area and Volume
Postulates
9