§2.1 Introductory Material
The student will learn about:
definitions of basic terms,
and the beginning
postulates to be used in
this course.
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Assumptions
We are assuming knowledge of all of the
arithmetic and algebra properties that are
known.
Basic terms such as point, line and
plane will not be defined.
The postulate system we will be
using is attributed to David Hilbert.
1862 –1943
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The Distance Postulate
Postulate 1. The Distance Postulate.
To every pair of different points there
corresponds a unique positive number.
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The Ruler Postulate
Postulate 2. The Ruler Postulate.
The points of a line can be placed in
correspondence with the real numbers in such a
way that
1) To every point of the line there corresponds
exactly one real number;
2) To every real number there corresponds exactly
one point of the line; and
3) The distance between any two points is the
absolute value of the difference of the
corresponding numbers.
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The Ruler Placement Postulate
Postulate 3. The Ruler Placement Postulate.
Given two points P and Q of a line, the
coordinate system can be chosen in such a
way that the coordinate of P is zero and the
coordinate of Q is positive.
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Betweenness
Definition
If A, B and C are points on the same line,
then B is between A and C iff
AB + BC = AC
And we write A – B – C or C – B – A.
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Theorem 2 - 1
If A, B, and C are three different points of the
same line, then exactly one of them is between
the other two.
Proof assigned for homework.
Hint: show that exactly one of the following must
hold:
AB + BC = AC or AC + CB = AB or
BA + AC = BC
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The Line Postulate
Postulate 4. The Line Postulate.
For every two points there is exactly one
line that contains both points.
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Some Definitions
Definition.
For any two points A and B, the segment
AB is the set whose points are A and B,
together with all the points between A and B.
the points A and B are called the endpoints of
AB
Definition.
The number AB is called the length of
segment AB .
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Some Definitions
Definition.
Let A and B be points on line L. The ray AB is
the set which is the union of the segment AB
and the set of all points C for which it is true
that A – B – C. the point A is called the end
point of AB .
Definition.
If A is between B and C , then AB and AC
are opposite rays.
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Segments, Rays, Lines & Angles.
Segment AB: AB   X : A  X  B, X  A, or X  B
Ray AB: AB   X : A  X  B, A  B  X, X  A, or X  B
Line AB: AB  X : X  A  B, A  X  B, A  B  X, X  A, or X  B
Angle ABC: ABC  BA  BC
(A, B and C noncollinear)
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Some Definitions
Definition.
A point B is called a midpoint of a segment AC if
A – B – C and AB = BC. The midpoint is said
to bisect the segment.
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Theorem
Every segment has exactly one midpoint.
Given: B is the midpoint of segment AC. Prove B is
unique.
What is our strategy?
A
BD
C
Assume that D is also a midpoint of segment AC.
Arrive at a contradiction.
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Theorem
Every segment has exactly one midpoint.
Given: B is the midpoint of segment AC. Prove B is
unique. Assume that D is also a midpoint of
segment AC. Arrive at a contradiction. A B D
Statement
1. AB + BC = AC and AB = BC
Reason
Given, B is midpoint.
2. AB = ½ AC
3. AD + DC = AC and AD = DC
4. AD = ½ AC
Arithmetic
Assumed, D is midpoint
Arithmetic
C
→← of the ruler postulate. Hence there is only one midpoint.
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Some Definitions
Definition.
The set of all points is called space.
Definition.
A set of points is collinear if there is a line which
contains all the points of the set.
Definition.
A set of points is coplanar if there is a plane which
contains all the points of the set.
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Postulate
Postulate 5.
(a) Every plane contains at least three
noncollinear points.
(b) Space contains at least four nonplaner
points.
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Theorem
If two distinct lines l and m meet,
their intersection is a single point.
l
Given: Distinct lines l and m with l ∩ m = A
Prove: A is unique.
m
A
Proof for homework.
Strategy?
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Postulate
Postulate 6.
If two points of a line lie in a plane, then
the line lies in the same plane.
Theorem
If a line intersects a plane not containing it,
then the intersection contains only one
point.
This theorem is a direct result of the postulate.
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Postulate
Postulate 7. The plane postulate.
any three points lie in at least one plane,
and any three noncollinear points lie in
exactly one plane.
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Theorems
Theorem.
given a line and a point not on the line,
there is exactly one plane containing both.
This theorem is a direct result of the postulate.
Theorem
Given two intersecting lines, there is
exactly one plane containing both.
This theorem is a direct result of the postulate.
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Some Definitions
Definition.
The interior of BAC is the set of all points P in
the plane of BAC such that P and B are on
the same side of AC and P and C are on the
same side of AB . The exterior of BAC is
the set of all points of the plane of BAC
that lie neither on the angle nor in its interior.
C
P
A
B
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Postulate
Postulate 8. The Angle Measurement Postulate.
To every angle BAC there corresponds a
real number between 0 and 180.
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Postulate
Postulate 9. The Angle Construction Postulate.
For any two angles ABC and DEF such
that m ABC < m DEF, there is a unique
ray EG such that m ABC = m GEF and
D
A
C
G
B
E
F
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Postulate 10
Postulate 10. The Angle Addition Postulate.
If D is in the interior of BAC, then
m BAC = m BAD + m DAC
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Some Definitions
Definition.
If AB and AD are opposite rays, and AC is any
other ray, then BAC and CAD form a
linear pair.
Definition.
If the sum of the measures of two angles is 180,
then the angles are called supplementary, and
each is called a supplement of the other.
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Postulate 11
Postulate 11. The Supplement Postulate
If two angles form a linear pair, then they are
supplementary.
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Some Definitions
Definition.
If two angles in a linear pair have the same
measure, then each of them is a right angle.
Definition.
If AB and AC form a right angle, then they are
called perpendicular rays, and we write
AB  AC
Definition.
If AB  AC , then the lines AB and AC are
called perpendicular.
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Some Definitions
Definition.
If the sum of the measures of two angles is 90,
then they are called complementary, and each
of them is called a complement of the other.
An angle with measure less than 90 is called
acute. And angle with measure greater than
90 is called obtuse.
Definition.
Two angles with the same measure are called
congruent.
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Some Definitions
Definition.
Two angles are vertical angles if their sides form
two pairs of opposite rays.
There are many theorems involved with
supplementary, complementary, vertical
angles and perpendicularity. Some will be
assigned for homework.
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Assignment: §2.1