Geometry July 7, 2008 1st Qtr

Other Theorems & Postulates

Distance Postulate
> to every pair of different points there corresponds a unique positive real number.

definition of Distance Postulate:
the distance between two points is the number given by the Distance Postulate. If the points are P & Q, then the distance is denoted as PQ/QP.

The Ruler Postulate
The point of a line can be placed in correspondence with the real numbers such that:

1) to every point of the line there corresponds exactly one real number.
2) to every real number there corresponds exactly one point of the line, and
3) there distance between any two points is the absolute value of the difference of the corresponding real number

definition of Ruler Postulate:
any real number corresponding to a given point is called the coordinate of a point. The one-to-one correspondence between the points of a line and the set/real numbers is called a coordinate system.

Ruler Placement Postulate
given two points P & Q of a line, the coordinate system can be chosen such that the coordinate of P is zero & the coordinate of Q is positive. (assigning points)

Segment Construction Postulate
Let RD be a ray, & let X be a positive number. Then there is exactly one point P of RD such that RP=X
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CONVEX SETS

> a set of convex if for any two points in A & B in it, the whole segment AB is also entirely in it. (should always be shaded)

Line Separation Postulate
> a point separates a line into two half lines, each of which is a convex set

Plane Separation Postulate
> a line separates a plane into two half-planes each of which is a convex set

Space Separation Postulate
> a plane separates space into two half-spaces each of which is a convex set

Angles & Angles Measurements

Angle
> union of two non collinear rays with a common endpoint

Angle Measurement Postulate
> to every angle there corresponds a unique positive real number between 0-180. This number is the degree measure of the angle

Angle Construction Postulate
> let H be half plane edge, AB. There is exactly one ray AC with B in H such that angle CAB has a given measure between 0-180.

DEFINITIONS:
Congruent Angles
> measures are equal

Angle Bisector
> ray bisects an angle if it divides the angle into two congruent parts

Angle Bisector Postulate
> every angle has exactly one bisector

Angle Addition Postulate
> if P is the interior of angle MAN, then m angle MAN=m angle PAM+m angle PAN

DEFINITIONS:
> two lines are perpendicular lines if they intersect to form right angles
> a line, ray, segment, or plane is the perpendicular bisector of a segment if it is perpendicular to the segment at its midpoint