Crossbar Theorem

(A-1) Each angle crossbar theorem _gr_1.gif] is associated with a unique real number between 0 and 180, called its measure and denoted crossbar theorem _gr_2.gif] No angle can have measure 0 nor 180.

Definition A point crossbar theorem _gr_3.gif] is an interior point of crossbar theorem _gr_4.gif] if an only if there exists a segment crossbar theorem _gr_5.gif] containing crossbar theorem _gr_6.gif] with crossbar theorem _gr_7.gif] and crossbar theorem _gr_8.gif] that extends from one side of the angle to the other ( crossbar theorem _gr_9.gif] and crossbar theorem _gr_10.gif] crossbar theorem _gr_11.gif] crossbar theorem _gr_12.gif]).

(A-2) If crossbar theorem _gr_13.gif] lies in the interior of crossbar theorem _gr_14.gif] then crossbar theorem _gr_15.gif] crossbar theorem _gr_16.gif] Conversely, if crossbar theorem _gr_17.gif] crossbar theorem _gr_18.gif] then crossbar theorem _gr_19.gif] is an interior point of crossbar theorem _gr_20.gif]

Definition For any three rays crossbar theorem _gr_21.gif] crossbar theorem _gr_22.gif] and crossbar theorem _gr_23.gif] (having the same endpoint) we say that crossbar theorem _gr_24.gif] lies between rays crossbar theorem _gr_25.gif] and crossbar theorem _gr_26.gif] and we write crossbar theorem _gr_27.gif] if and only if the rays are distinct and crossbar theorem _gr_28.gif]

(A-3) The set of rays crossbar theorem _gr_29.gif] lying on one side of a given line crossbar theorem _gr_30.gif] including ray crossbar theorem _gr_31.gif] may be assigned to the entire set of real numbers crossbar theorem _gr_32.gif] crossbar theorem _gr_33.gif] 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 crossbar theorem _gr_34.gif] is 0
    
     (iv) if rays crossbar theorem _gr_35.gif] and crossbar theorem _gr_36.gif] on crossbar theorem _gr_37.gif] have coordinates crossbar theorem _gr_38.gif] and crossbar theorem _gr_39.gif] then crossbar theorem _gr_40.gif]

Theorem (12) If the rays crossbar theorem _gr_41.gif] crossbar theorem _gr_42.gif] and crossbar theorem _gr_43.gif] have coordinates crossbar theorem _gr_44.gif] crossbar theorem _gr_45.gif] and crossbar theorem _gr_46.gif] relative to some half-plane, then crossbar theorem _gr_47.gif] if and only if either crossbar theorem _gr_48.gif] or crossbar theorem _gr_49.gif]

Definition We say ray crossbar theorem _gr_50.gif] is an angle bisector of angle crossbar theorem _gr_51.gif] when crossbar theorem _gr_52.gif] lies between crossbar theorem _gr_53.gif] and crossbar theorem _gr_54.gif] such that crossbar theorem _gr_55.gif]

Theorem (13) If crossbar theorem _gr_56.gif] there is a unique ray crossbar theorem _gr_57.gif] such that crossbar theorem _gr_58.gif] and crossbar theorem _gr_59.gif]

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

Definition Given crossbar theorem _gr_60.gif] then the two rays crossbar theorem _gr_61.gif] and crossbar theorem _gr_62.gif] 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.
    
crossbar theorem _gr_63.gif]

We will use a direct proof for the theorem with complementary angles.
    
crossbar theorem _gr_64.gif]
crossbar theorem _gr_65.gif]

(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 crossbar theorem _gr_66.gif] and crossbar theorem _gr_67.gif] 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 crossbar theorem _gr_68.gif] then crossbar theorem _gr_69.gif] and crossbar theorem _gr_70.gif] are perpendicular at crossbar theorem _gr_71.gif]

crossbar theorem _gr_72.gif]

Two lines crossbar theorem _gr_73.gif] and crossbar theorem _gr_74.gif] are perpendicular at crossbar theorem _gr_75.gif] then crossbar theorem _gr_76.gif]

crossbar theorem _gr_77.gif]
crossbar theorem _gr_78.gif]

Definition A set crossbar theorem _gr_79.gif] in crossbar theorem _gr_80.gif] is called convex provided it has the property that for all points crossbar theorem _gr_81.gif] and crossbar theorem _gr_82.gif] the segment joining crossbar theorem _gr_83.gif] and crossbar theorem _gr_84.gif] lies in crossbar theorem _gr_85.gif] that is, crossbar theorem _gr_86.gif]

(H-1) Let crossbar theorem _gr_87.gif] be any line lying in any plane crossbar theorem _gr_88.gif] The set of all points in crossbar theorem _gr_89.gif] not on crossbar theorem _gr_90.gif] consists of the union of two subsets crossbar theorem _gr_91.gif] and crossbar theorem _gr_92.gif] of crossbar theorem _gr_93.gif] such that

     (i) crossbar theorem _gr_94.gif] and crossbar theorem _gr_95.gif] are convex sets
     
     (ii) crossbar theorem _gr_96.gif] and crossbar theorem _gr_97.gif] have no points in common
     
     (iii) If crossbar theorem _gr_98.gif] lies in crossbar theorem _gr_99.gif] and crossbar theorem _gr_100.gif] lies in crossbar theorem _gr_101.gif] the line crossbar theorem _gr_102.gif] intersects the segment crossbar theorem _gr_103.gif]
     

Definition The two sets crossbar theorem _gr_104.gif] and crossbar theorem _gr_105.gif] in the axiom (H-1) are called the two sides of crossbar theorem _gr_106.gif] or also, half-planes determined by crossbar theorem _gr_107.gif]

Theorem (17) If crossbar theorem _gr_108.gif] then there exists a unique perpendicular to line crossbar theorem _gr_109.gif] at crossbar theorem _gr_110.gif]

    Proof. First we will prove the following statement using the direct method: if crossbar theorem _gr_111.gif] is any line then there is a perpendicular to line crossbar theorem _gr_112.gif] at crossbar theorem _gr_113.gif]

crossbar theorem _gr_114.gif]

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

crossbar theorem _gr_115.gif]

Thertefore, any perpendicular is unique. crossbar theorem _gr_116.gif]

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 crossbar theorem _gr_117.gif] and crossbar theorem _gr_118.gif] crossbar theorem _gr_119.gif]

crossbar theorem _gr_120.gif]
    
crossbar theorem _gr_121.gif]     

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

Theorem (20) If crossbar theorem _gr_122.gif] and crossbar theorem _gr_123.gif] are any three rays on one side of a line and having the same end point, then either crossbar theorem _gr_124.gif] crossbar theorem _gr_125.gif] or crossbar theorem _gr_126.gif]

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.

Cite this as:
Crossbar Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/crossbar-theorem.html
 
    
Library of Math
Library of Math
Online Math Organized by Subject Into Topics
math search
Library of Math AddThis Feed Button
Online Math
Math Books
Custom math gifts on sale now.
Share your enthusiasm for mathematics!
Library of Math
Math Gifts
The Library of Math - Online Math Organized by Subject Into Topics.
© 2005 - 2008 www.LibraryOfMath.com All rights reserved.
about us | feedback | privacy policy | terms of use | mision statement | help

Page copy protected against web site content infringement by Copyscape Valid CSS! Valid HTML 4.01 Transitional Subscribe to the Library of Math Feed