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Crossbar Theorem

(A-1) Each angle is associated with a unique real number between 0 and 180, called its measure and denoted No angle can have measure 0 nor 180.

Definition A point is an interior point of if an only if there exists a segment containing with and that extends from one side of the angle to the other ( and ).

(A-2) If lies in the interior of then Conversely, if then is an interior point of

Definition For any three rays and (having the same endpoint) we say that lies between rays and and we write if and only if the rays are distinct and

(A-3) The set of rays lying on one side of a given line including ray may be assigned to the entire set of real numbers called coordinates, in such a manner that

    (i) each ray is assigned to a unique coordinate
    
    (ii) no two rays are assigned to the same coordinate
    
    (iii) the coordinate of is 0
    
     (iv) if rays and on have coordinates and then

Theorem (12) If the rays and have coordinates and relative to some half-plane, then if and only if either or

Definition We say ray is an angle bisector of angle when lies between and such that

Theorem (13) If there is a unique ray such that and

Theorem (14) The bisector of any angle exists and is unique.

Definition Given then the two rays and are called opposing rays.  

Definition Two angles are said to form a linear pair if and only if they have one side in common and the other two sides are opposite rays.

Definition Any two angles whose angle measure sum to 180 is called a supplementary pair and any two angles whose angle measures sum to 90 is called a complementary pair.

Theorem (15) Angles supplementary (or complementary) to the same angles have the same measure.

    Proof. We will use a direct proof for the theorem with supplementary angles.
    


We will use a direct proof for the theorem with complementary angles.
    

(A-4) A linear pair of angles is supplementary pair.

Definition A right angle is any angle having measure 90. An acute angle is any angle whose measure is less than 90 and an obtuse angle is any angle who measure is greater than 90.

Definition Two distinct lines and are called perpendicular lines if and only if they contain the sides of a right angle.

    For convenience, segments are perpendicular if and only if they lie, respectively, on perpendicular lines. Similar terminology applies to segment and ray, two rays, and so.

Theorem (16) If then and are perpendicular at



Two lines and are perpendicular at then

Definition A set in is called convex provided it has the property that for all points and the segment joining and lies in that is,

(H-1) Let be any line lying in any plane The set of all points in not on consists of the union of two subsets and of such that

     (i) and are convex sets
     
     (ii) and have no points in common
     
     (iii) If lies in and lies in the line intersects the segment
     

Definition The two sets and in the axiom (H-1) are called the two sides of or also, half-planes determined by

Theorem (17) If then there exists a unique perpendicular to line at

    Proof. First we will prove the following statement using the direct method: if is any line then there is a perpendicular to line at



Next we will show, using an indirect method, that the perpendicular is unqiue.



Thertefore, any perpendicular is unique.

Definition Two angles having the sides of one opposite the sides of the other are called vertical angles.

Theorem (18) Vertical angles have equal measures.

    Proof. We will prove the statement: for any vertical angles and


    
     

Theorem (19) Bisectors of a linear pair of angles are perpendicular.

Theorem (20) If and are any three rays on one side of a line and having the same end point, then either or

Theorem (21) If two angles have a side in common that passes through an interior point of the angle formed by the other two sides, then the other two sides are perpendicular if and only if the given angles are complementary.

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