PRINT LINKED RELATED

ASA Congruence Criterion

(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.

Geometry Books

Calculus (With Analytic Geometry)(8th edition) List Price: $209.95 Buy Used: $69.00 You Save: $140.95 (67%) New (57) Used (180) from $69.00Designed 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 List Price: $17.95 Buy New: $16.15 You Save: $1.80 (10%) New (4) Used (5) from $7.47From 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 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 List Price: $14.95 Buy Used: $4.83 You Save: $10.12 (68%) New (40) Used (50) Collectible (1) from $4.83Throughout 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)

Geometry For Dummies (For Dummies (Math & Science)) List Price: $19.99 Buy New: $8.62 You Save: $11.37 (57%) New (51) Used (10) from $8.62Learning 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)

Trigonometry (9th Edition) List Price: $138.67 Buy New: $85.49 You Save: $53.18 (38%) New (33) Used (33) from $85.49Over 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)

Baby Einstein: See and Spy Shapes List Price: $5.99 Buy Used: $0.01 You Save: $5.98 (100%) New (25) Used (97) Collectible (1) from $0.01Shapes invites babies and young children to identify different shapes in bold, graphic illustrations featuring the Baby Einstein characters. Playful poems will inspire children to seek out shapes in the (more)

Geometry List Price: $94.88 Buy Used: $6.95 You Save: $87.93 (93%) New (29) Used (122) from $6.95Geometry (more)

College Algebra with Modeling and Visualization (3rd Edition) (Rockswold Series) List Price: $138.67 Buy New: $20.00 You Save: $118.67 (86%) New (26) Used (129) from $19.99Gary 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 List Price: $29.95 Buy New: $24.45 You Save: $5.50 (18%) New (5) from $24.45Green 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)