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Study Notes 011:
By William E. Steinman:
Geometry 4 Postulates:
Let’s get to what we call Postulates.
A postulate is something assumed without proof as being self-evident or generally accepted.
So, a postulate is an assumption.
Another word, often used in logic and math, for these assumptions is axiom.
Again, an axiom is a statement accepted without proof as the basis for logically deducing other statements (theorems).
The difference between an axiom and a postulate is a bit subtle. An axiom is more general in nature while a postulate deals with specific subject matter, like, for example, geometry.
These little assumptions underlie all of mathematics. Without them, we could have no science of mathematics. So, for convenience we willingly accept these postulates even though we cannot prove them.
We have ten of these agreements, which are called Algebraic Postulates.
Postulate 1:
Things equal to the same or equal things are equal to each other. This is called the Transitive Postulate.
For example:
A quart is equal to two pints. A gallon is equal to 4 quarts.
Therefore 4 quarts is equal to 8 pints, 8 pints is equal to 1 gallon.
Self evident!
Postulate 2:
A quantity may be substituted for its equal in any expression or equation. This is called the Substitution Postulate.
Postulate 3:
The whole equals the sum of its parts. This is called the Partition Postulate.
Postulate 4:
Any quantity equals itself. This is called the Identity Postulate.
Postulate 5:
If equals are added to equals, the sums are equal. This is called the Addition Postulate.
For example:
X = Y
W = Z
X + W = Y + Z
Self evident!
Postulate 6:
If equals are subtracted from equals, the differences are equal. This is called the Subtraction Postulate.
Postulate 7:
If equals are multiplied by equals, the products are equal. Guess what this one is called.
Postulate 8:
If equals are divided by equals, the quotients are equal.
Postulate 9:
Like powers of equals are equal. The Powers Postulate.
If 3 x 5 = 15, then (3 x 5)n = 15n.
Postulate 10:
Like roots of equals are equal.
We have 9 additional postulates for dealing with geometry.
Postulate 11:
One and only one straight line can be draw through any two points.
Postulate 12:
Two lines can intersect at one and only one point.
Postulate 13:
The length of a segment is the shortest distance between two points.
I.E. The shortest distance between any two points is a straight-line segment.
Postulate 14:
One and only one circle can be draw with a given point as center and a given line segment as radius.
Postulate 15:
Any geometric figure can be moved without changing its size or shape.
Postulate 16:
A segment can have one and only one midpoint.
Postulate 17:
An angle has one and only one bisector.
Postulate 18:
Through any point on a line, one and only one perpendicular can be drawn to the line.
Postulate 19:
Through any point outside a line, one and only one perpendicular can be drawn to the line.
Now, how do we use these postulates? We use them to prove theorems. Huh? That’s right! We use postulates that we do not, indeed cannot, prove but accept as true to prove theorems.