Library of Math
Online Math Organized by Subject Into Topics
Subscribe to the Library of Math Feed
PRINT LINKED RELATED

Interior of Angles

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

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

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

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

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

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

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

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

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

Definition Given interior of angles _gr_60.gif] then the two rays interior of angles _gr_61.gif] and interior of angles _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.
    
interior of angles _gr_63.gif]

We will use a direct proof for the theorem with complementary angles.
    
interior of angles _gr_64.gif]
interior of angles _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 interior of angles _gr_66.gif] and interior of angles _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 interior of angles _gr_68.gif] then interior of angles _gr_69.gif] and interior of angles _gr_70.gif] are perpendicular at interior of angles _gr_71.gif]

interior of angles _gr_72.gif]

Two lines interior of angles _gr_73.gif] and interior of angles _gr_74.gif] are perpendicular at interior of angles _gr_75.gif] then interior of angles _gr_76.gif]

interior of angles _gr_77.gif]
interior of angles _gr_78.gif]

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

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

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

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

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

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

interior of angles _gr_114.gif]

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

interior of angles _gr_115.gif]

Thertefore, any perpendicular is unique. interior of angles _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 interior of angles _gr_117.gif] and interior of angles _gr_118.gif] interior of angles _gr_119.gif]

interior of angles _gr_120.gif]
    
interior of angles _gr_121.gif]     

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

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

Geometry Books

Calculus (With Analytic Geometry)(8th edition) Product Image
List Price: $209.95
Buy Used: $71.79
You Save: $138.16 (66%)
New (56) Used (173) from $71.79Designed for the three-semester calculus course for math and science majors, Calculus continues to offer instructors and students new and innovative teaching and learning resources. This was the first (more)
The Misbehavior of Markets: A Fractal View of Financial Turbulence Product Image
List Price: $17.95
Buy New: $16.99
You Save: $0.96 (5%)
New (5) Used (4) from $7.48From the inventor/founder of fractal geometry, the award-winning book that turns modern financial theory on its head Mathematical superstar and inventor of fractal geometry, Benoit Mandelbrot, has spent (more)
The Fractal Geometry of Nature Product Image
Buy New: $34.99
New (6) Used (15) from $27.90Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary (more)
The Golden Ratio: The Story of PHI, the World's Most Astonishing Number Product Image
List Price: $14.95
Buy Used: $4.81
You Save: $10.14 (68%)
New (40) Used (51) Collectible (1) from $4.81Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale (more)
Trigonometry (9th Edition) Product Image
List Price: $138.67
Buy New: $85.32
You Save: $53.35 (38%)
New (33) Used (29) from $85.32Over the years, the text has been shaped and adapted to meet the changing needs of both students and educators. As always, special care was taken to respond to the specific suggestions of users and reviewers (more)
Geometry Product Image
List Price: $94.88
Buy Used: $6.95
You Save: $87.93 (93%)
New (29) Used (117) from $6.95Geometry (more)
Geometry For Dummies (For Dummies (Math & Science)) Product Image
List Price: $19.99
Buy New: $8.52
You Save: $11.47 (57%)
New (53) Used (10) from $8.52Learning geometry doesn?t have to hurt. With a little bit of friendly guidance, it can even be fun! Geometry For Dummies, 2nd Edition, helps you make friends with lines, angles, theorems and postulates. (more)
College Algebra with Modeling and Visualization (3rd Edition) (Rockswold Series) Product Image
List Price: $138.67
Buy Used: $20.00
You Save: $118.67 (86%)
New (25) Used (127) from $20.00Gary Rockswold focuses on teaching algebra in context, answering the question, "Why am I learning this?" and ultimately motivating the students to succeed in this class. In addition, the author's understanding (more)
Euclid's Elements Product Image
List Price: $29.95
Buy New: $25.99
You Save: $3.96 (13%)
New (2) from $25.99Green Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's "Elements" In keeping with Green Lion's design commitment, diagrams have been placed (more)
The Symmetries of Things Product Image
List Price: $75.00
Buy New: $60.00
You Save: $15.00 (20%)
New (7) Used (4) from $59.99Start with a single shape. Repeat it in some way translation, reflection over a line, rotation around a point and you have created symmetry. Symmetry is a fundamental phenomenon in art, science, and nature (more)

Cite this as:
Interior Of Angles
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/interior-of-angles.html
about us contact us privacy policy terms of use mision statement lom help
The Library of Math - Online Math Organized by Subject Into Topics. © 2005 - 2009 www.LibraryOfMath.com All rights reserved. math rss