Joseph A. Montagna
Postulates
are statements which are accepted without proof. They are especially useful in doing geometry proofs. Students must learn these. It is important that they be included in their geometry folders and studied often. Each postulate is simply listed with an example. Obviously, the teacher will have to take the time to discuss each in class.
Postulates of Equality
Addition postulate: If a=b and c=d, then a+c=b+d
Subtraction postulate: If a=b and c=d, then a-c=b-d
Multiplication postulate: If a=b, then ac=bc
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Division postulate: If a=b and y = 0, then a
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b
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— = —
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y
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y
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Substitution postulate: If a=b, then a can be substituted for b in any
equation
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Postulates of Geometry
Before moving on to these postulates we should introduce one more term to those that students already have,
plane
. A plane is suggested by a flat surface. The abstract idea in the definition of a plane is that they have no edge, they continue infinitely. When we draw a plane it appears to have corners, represented by points. Drawings are, therefore, misleading. A term used in discussing planes is
coplanar
. Points that lie in the same plane are coplanar.
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Postulate 1: Through any two points there is only one line.
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Postulate 2: Through any three
non-collinear
points there is only one plane.
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Postulate 3: If two points lie in a plane, then the line connecting them lies in that plane.
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Postulate 4: If two planes intersect, then their intersection is a line.
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Postulate 5: This postulate is also called the ruler postulate. Each point on a line can be paired with only one real number called its
coordinate
. The distance between these two points is the positive difference between their coordinates.
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Postulate 6: This postulate is also called the protractor postulate. If point C is on line ST, then all the rays that have C as an endpoint and lie on the same side of ST can be paired with exactly one real number between 0 and 180.
