Math 7 geometry 02 postulates and theorems on points, lines, and planes

The session shall begin
shortly…

Basics of Geometry 2
Postulates and Theorems Related
to Points, Lines and Planes

Foundations of Geometry
Definitions
Undefined Terms
Postulates and Theorems on Points, Lines, and Planes
Postulates
Theorems
Euclid
Father of Geometry

Definitions
DEFINITIONS are words that can be
defined by category and characteristics
that are clear, concise, and reversible.
Example: Definition of Line Segment
B
A LINE SEGMENT (or segment) is a set of
points consisting of two points on a line, and all
the points on the line between the two points
C

Postulates
POSTULATES statements accepted
as true without proof.
They are accepted on faith alone.
They are considered self-evident
statements.
They are also called AXIOMS.

Ruler Postulate
PART 1: There is a one-to-one
correspondence between the points of a
line and the set of real numbers.
This means that every point on the number
line corresponds to a UNIQUE real number

Ruler Postulate
PART 2: the distance between any two
points equals the absolute value of the
difference of their coordinates.
a b
a bDistance =

Segment Addition Postulate
If B is a point between A
and C, then
AB + BC = AC
A B C
Note that B must be on AC.

Definition of “Betweenness”
If A, B, and C are points
such that AB + BC = AC,
then B is between A and
C.
A B C

Examples
1. has length 10 cm and has
length 8 cm. If A is between P and
K, find the length of
PA AK
PK

Examples
2. Is on a number line and O is
between B and X. If the
coordinates of B and O are 3 and
8, respectively, and BX = 12, what
is the coordinate of X?
BX

The Midpoint of a Line Segment
The MIDPOINT of a line segment
is a point that divides the segment
into two equal segments.
A M B
M is the midpoint ofAB
 
1
2
AM MB AB

Examples
3. has length 10 cm. If J is the
midpoint of , what are the
lengths of the following?
KL
KL
a. KJ b. JL

Examples
4. Find the coordinate of the
midpoint of on the number line
if the coordinates of L and N are –3
and 7, respectively.
LN

Line Postulate
Through any two points
there is exactly one line.
Restated: 2 points determine a unique line.

Plane Postulate
Part 1: Through any three
points there is at least one
plane.
Part 2: Through any three non-
collinear points there is exactly
one plane.

Three collinear points
can lie on multiple
planes.
M
While three non-
collinear points can lie
on exactly one plane.
(Three noncollinear
points determine a
unique plane)
Plane Postulate

With 3 non-collinear points, there is only one
plane – the plane of the triangle.
B
A C
Plane Postulate

Flat Plane Postulate
If two points of a line are in a
plane, then the line
containing those points in
that plane. M
A
B

Intersection of Planes Postulate
If two planes intersect, then their
intersection is a line.
Remember, intersection means points in common or in both sets.

H
G
F
E
D
CB
A

Final Thoughts on Postulates
 Postulates are accepted as true on
faith alone. They are not proved.
 Postulates need not be memorized.
 Those obvious simple self-evident
statements are postulates.
 It is only important to recognize
postulates and apply them
occasionally.

Theorems
Theorems are important
statements that are proved
true.
These are statements that
needs to be proven using
logical valid steps.
The principles and ideas used in proving theorems
will be discussed in Grade 8 

Intersection of Lines Theorem
If two lines intersect, then they
intersect in exactly one point.
This is very obvious.
To be more than one the line
would have to curve.
But in geometry,
all lines are straight.

Theorem
Through a line and a point not on the line
there is exactly one plane that contains
them.
A
Restatement: A line and a point not on the line
determine a unique plane.

Theorem
Through a line and a point not on the line
there is exactly one plane that contains
them. WHY?
A
B C
If you take any two points
on the line plus the point off
the line, then…
The 3 non-collinear points
mean there exists a exactly
plane that contain them.
If two points of a line are in the plane, then line is in
the plane as well.

If two lines intersect, there is exactly one
plane that contains them.
Theorem
Restatement: Two intersecting lines determine a
unique plane.

If you add an
additional point from
each line, the 3
points are
noncollinear.
Through any three noncollinear points there is
exactly one plane that contains them.
If two lines intersect, there is exactly one
plane that contains them. WHY?
Theorem

Foundations of Geometry:
1 Undefined terms: Point, Line & Plane
2 Definitions
3 Postulates
4 Theorems
Statements accepted without proof.
Statements that can be proven true.
Primitive terms that defy definition due to circular definitions.
Words that can be defined by category and characteristics
that are clear, concise, and reversible.
Summing it up!

Math 7 geometry 02 postulates and theorems on points, lines, and planes

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