Segments and Angles

Definition For three points and we say that is between and and we write if and only if and are distinct, collinear points, and

Theorem (3) If then and neither nor

Definition (Betweeness) If and are four distinct collinear points, then we write if and only if the composite function of all four betweeness relations and are true.

Theorem (4) If and hold, then is true.

Definition (Segment, Ray, Line,Angle) The following sets and notation are defined.

The segment of and denoted is defined as the set,

    

The ray of and denoted by is defined by the set,
    
    

The line of and denoted by is defined by the set,
    
    
    
The angle of the noncollinear points and denoted by   is defined by the set,

.
    
Further, points and are called the end points of segment and point is the endpoint (also, sometimes called the origin) of ray Point in is called the vertex of the angle, and rays and are called its sides.    

    The notation denotes the (geometric) point with its coordinate (real number) , as guaranteed by the Ruler Postulate.

Theorem (5) If and lie on line then if and only if or
Theorem (6) If lies on ray and then

Theorem (7) If there exists a unique point on ray such that and

Definition The midpoint of segment is the point on such that Also, any geometric object passing through is said to bisect the segment

Theorem (8) The midpoint of any segment exists, and is unique.

Theorem (9) If then

Theorem (10) If and are three distinct points, collinear points, then either or

Theorem (11) A segment cannot be ray.

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Cite this as:
Segments And Angles
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/segments-and-angles.html