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Postulates

Postulates are statements that are assumed true. 

Postulates of Euclidean Geometry

Point-Line-Plane Postulate:
a. unique line assumption: through any two points there is exactly one
line
b. dimension assumption: given a line in a plane, there exists a point 
in the plane not on the line
c. number line assumption: every line is a set of points that can be 
put in a one-to-one correspondence with real numbers, with any point
on it corresponding to zero and any other point corresponding to one
d. distance assumption: on a number line, there is a unique distance
between two points

Arithmatic and Algebra Postulates

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 Equality: if a = b, then ac = bc
Substitution Property of Equality: if a = b, then a may be substituted 
for b in any expression

Posulates 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 ac < bc
if a < b and c < 0, then ac > bc
Equation to Inequality Property: if a and b are positive and a + b = c, 
then c > a and c > b  
Transitive Property of Inequality: if a < b and b < c, then a < c

Postulates of Operations:
Communtative Property of Addition: a + b = b + a
Communtative Property of Multiplication: ab = ba
Distributive Property: a(b+c) = ab + ac

Triangle Inequality Postulate: the sum of the lengths of two sides of
any triangle is greater than the length of the 3rd side

Angle Measure Postulate:
a. unique measure assumption: every angle has a unique measure from 0
to 180 degrees
b. two sides of line assumption: given any ray VA and any number x 
between 0 and 180 there are unique rays VB and VC such that segment BC
intersects line VA and measure of angle BVA is equal to the measure of 
angle CVA that equals x
c. Zero Angle Assumption: if ray VA and ray VB are the same ray, then
the measure of angle AVB equals 0 
d. Straight Angle Assumption: if ray VA and ray VB are opposite rays,
then the measure of angle AVB equals 180 
e. Angle Addition Property: if ray VC (except for point V) is in the 
interior of angle AVB, then the measure of angle AVC + the measure of 
angle CVB = the measure of angle AVB

Corresponding Angles Postulate: if two coplaner lines are cut by a 
transversal so that two corresponding angles have the same measure, 
then the lines are parallel

Parallel Lines Postulate: if two lines are parallel and cut by a 
transversal corresponding angles have the same measure

Reflection Postulate: 
Under a Reflection:
a. There is a one-one correspondence between points and their images
b. If three points are collinear, then their images are collinear
c. If B is between A and C, then the image of B is between images of
A and C
d. The distance between two preimages equals the distance between 
their images
e. The image of an angle is an angle of the same measure
f. A polygon and its image, with vertices taken in corresponding 
order, have opposite orientations

Area Postulate:
a. Uniqueness Property: given a unit region, every polygonal region 
has a unique area
b. Rectangle Formula: the area of a rectangle with dimensions l and w 
lw
c. Congruence Property: congruent figures have the same area
d. Additive Property: the area of the union of two non-overlapping 
regions is the sum of the areas of the regions

Volume Postulate:
a. Uniqueness Property: given a unit cube, every polyhedral solid has 
unique volume
b. Box Volume Formula: the volume of a box with dimensions l,w, and h
is lwh
c. Congruence Property: congruent figures have the same volume
d. Additive Property: the volume of the union of two non-overlapping
solids is the sum of the volumes of the solids
e. Cavalieri's Principle: let I and II be solids included between 
parallel planes. If every place p parallel to the given planes 
intersects I and II in sections with the same area then volume (I) 
equals volume (II)

Postulates of Logic:
Law of Detachment: If you have a statement or given information p and 
the justification of the form if p, then q, you may conclude q
Law of Transtitivy: If if p, then q and if q, then r, then if p, 
then r
Law of the Contrapositive: A statement if p, then q and its 
contrapositive if not q, then not p are either both true or both false
Law of ruling out Possibilities: when p or q is true and q is not true
then p is true
Law of inderect reasoning: if reasoning from a statement p leads to 
a false conclusion, then p is false

Postulates of Euclid:
1. Two points determine a line segment
2. A line segment can be extended indefinitely along a line
3. A circle can be drawn with any center and any radius
4. All right angles are congruent
5. If two lines are cut by a transversal, then the interior angles 
on the same side of the transversl have a total measure less than 180, 
then the lines will intersect on that side of the transversal