The Basic Postulates & Theorems of Geometry
These are the basics when it comes to postulates and theorems in Geometry. These
are the ones that you
have
to know
.
Postulates
Postulates are statements that are assumed to be true without proof. Postulates
serve two purposes

to explain undefined terms, and to serve as a starting point for
proving other statements.
Point

Line

Plane Postulate
A) Unique Line Assumpt
ion: Through any two points, there is exactly one line.
Note:
This doesn't apply to
nodes
or
dots
.
B) Dimension Assumption: Given a line in a plane, there exists a point in the plane
not on that line. Given a plane in space, there exists a line or a point in space not on
that plane.
C) Number Line Assumption: Every line is a set of points that can be
put into a one

to

one correspondence with real numbers, with any point on it corresponding to
zero and any other point corresponding to one.
Note:
This doesn't apply to
nodes
o
r
dots
. This was once called the Ruler Postulate.
D) Distance Assumption: On a number line, there is a unique distance between two
points.
E) If two points lie on a plane, th
e line containing them also lies on the plane.
F) Through three noncolinear points, there is exactly one plane.
G) If two different planes have a point in common, then their intersection is a line.
Euclid's Postulates
A) Two points determine a line se
gment.
B) A line segment can be extended indefinitely along a line.
C) A circle can be drawn with a center and any radius.
D) All right angles are congruent.
Note:
This part has been proven as a theorem. See
below
,
proof
.
E) If two lines are cut by a transversal, and the interior angles on the same side of
the transversal have a total measure of less
than 180 degrees, then the lines will
intersect on that side of the transversal.
Polygon Inequality Postulates
Triangle Inequality Postulate: The sum of the lengths of two sides of any triangle is
greater than the length of the third side.
Quadrila
teral Inequality Postulate: The sum of the lengths of 3 sides of any
quadrilateral is greater than the length of the fourth side.
Theorems
Theorums are statements that can be deduced and proved from definitions,
postulates, and previously proved theorum
s.
Euclid's First Theorem: The triangle in the picture is an equilateral triangle.
Note:
D. Joyce's Elements page (the link above) is where you'll find anything else
you need to know about Euclid's ideas, postulates, and theorems.
Line Intersection Th
eorem: Two different lines intersect in at most one point. For
proof see
Unique Line Assumption
Betweenness Theorem: If C is between A and B and on
, then AC + CB = AB.
Related Theorems:
Theorem: If A, B, and C are distinct points and
AC + CB = AB, then C lies on
Theorem: For any points A, B, and
C, AC + CB
.
Pythagorean Theorem: a
2
+ b
2
= c
2
, if c is the hypotenu
se.
Right Angle Congruence Theorem: All right angles are congruent. See
proof
.
Note: While you can usually get away with not knowing the names of theorems,
your Geometry teacher will g
enerally require you to know them.
Algebra Postulates
Here are the basic postulates of equality, inequality, and operations. Dave didn't ge
t a
chance to write them, and I needed them for my section on the
basic postulates of
Geometry
(review is always good). Have a blast!
.
Postulates of Equality
Reflexive Property of Equality:
a = a
Symmetric Property of Equality:
if a = b, then b = a
Transitive Property of Equality:
if a = b and b = c, then a = c
.
Postulates of Equality and Operations
Addition Property of Equality:
if a = b, then a + c = b + c
Multiplication Property of Equa
lity:
if a = b, then a * c = b * c
Substitution Property of Equality:
if a = b, then a can be substituted for b in any
equation or inequality
Subtraction Property of Equality: if a = b, then a

c = b

c
.
Postulates of Inequality and Operations
Addition
Property of Inequality: if a < > b, then a + c < > b + c
Multiplication Property of Inequality: if a < b and c > 0, then a * c < b * c
if a < b and c < 0, then a * c > b * c
Equation to Inequality Property: if a and b are positive, and a + b = c, then c >
a
and c > b
if a and b are negative, and a + b = c, then c < a and c < b
Subtraction Property of Inequality: if a < > b, then a

c < > b

c
Transitive Property of Inequality: if a < b and b < c, then a < c
.
Postulates of Operation
Commutative Property
of Addition: a + b = b + a
Commutative Property of Multiplication: a * b = b * a
Distributive Property: a * (b + c) = a * b + a * c and vice versa
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