Download Ch 1 ASN

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NAME ______________________________
Teacher: ____________________________
Class and Period:______________________
Date: _______________________________
Introduction to Geometry Axiomatic System Notebook (Ch 1ASN)
Ch. Sec.
1.1
Term, Example, Proof
A _____________ is a place in space. It has no length, width, or height. We use capital
letters to name points.
Two points determine a _______________. All __________ are straight. It has infinite
length (but no width of height). It has ____ dimension. We name a line by a lower case
script letter, or by any two points on the line.
A ________________________________ is a part of a line. It has finite length. It is named
by the endpoints.
A ___ has one endpoint, and infinite length (in the direction opposite the endpoint). The
endpoint is always noted first in the name.
An ___________ is formed by two rays that share the endpoint. The shared endpoint is
called the _____________. The rays are called the ______. Names of angles should
include 3 points with the vertex noted as the MIDDLE letter. Sometimes angles are
numbered.
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A _________________ has three segments as its sides. It also has three angles. It is
named by the three points that are the vertices.
A _________________ has length and width (that is it is 2 dimensional), and is defined by
3 points.
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A triangle is the _____ () of three segments. Union implies () OR, which is always
INCLUSIVE in mathematics.
Another example: Meet me on 35th OR Wisconsin: (35th St.)  (Wisconsin Av.)
An ____________________ () of two sides of a triangle is the vertex.
Intersection  AND, which is exclusive. It’s the overlap.
Another example: Meet me on 35th AND Wisconsin: (35th St.)  (Wisconsin Av.)
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1.2
Angles are measured in degrees (), minutes (‘), and seconds (“).
There are ____ minutes in a degree, and _____ seconds in a minute.
An _____________ angle is an angle whose measure is greater than 0 and less than 90
degrees.
A _____________ angle is an angle that measures 90 degrees.
An __________ angle is an angle whose measure is greater than 90 and less than 180
degrees.
A __________straight angle is an angle whose measure is 180 degrees. A straight angle is
also a line.
Def
____________________________ are angles that have the same measure.
Def
_____________________________ are segments that have the same length.
___________________________ are used to indicate congruent angles or segments.
1.3
Def
Points that lie on the same line are __________________________.
Points that do not lie on the same line are _______________________.
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Betweenness of points: In order for us to say that a point is ____________ two other
points, all three points must be _________________.
Triangle Inequality:
The sum of two sides is always (greater than/equal to/ less than) the third side.
Interpreting a Diagram:
You should Assume
Straight lines and angles
Collinearity of points
Betweenness of points
Relative position of points
1.7
You should NOT assume
Right angles
Congruent segments
Congruent angles
Relative sizes of segments and angles
A ___________________________ is an axiomatic system that contains 4 elements:
A ________________ is an unproven assumption. A postulate is NOT always reversible.
A _________________ states the meaning of a term or idea. A definition is always
reversible.
Points lie on the same line  collinear
Collinear  points lie on the same line
A _______________ is a mathematical statement that can (and should) be proven.
Theorems are NOT always reversible.
Poodle  dog
Dog  poodle
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1.8
If you are not willing to learn, no one can help you.
If you are determined to learn, no one can stop you.
Conditional Statement
Negation
Converse
Inverse
Contrapositive
Truth tables get a little more complicated when
conjunctions and disjunctions of statements
are included.
^ is INTERSECTION, v is UNION
^ is AND, v is OR
This is a brief introduction to Boolean Algebra,
and is the logic used in search engines.
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The truth table for an implication, or
conditional statement looks like this:
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1.4
Purposes of proof
Mnemonic: Very Easy Class Don’t Cheat
VERIFY the truth of a mathematical statement.
EXPLAIN why the statement is true.
COMMUNICATE mathematical knowledge.
DISCOVER new math (that is axioms)
CREATE an axiomatic system (or a system of truths)
A proof is the logical process, not the format of the argument.
A _________________________________________ is one format of a proof.
Thm
Thm
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1.5
Def
A point (or segment, ray, or line) that divides a segment into two congruent segments
___________________ the segment. The bisection point is called the ________________
of the segment.
Def
Two points (or segments, rays, or lines) that divide a segment into three congruent
segments ___________________ the segmetn. The two points at thich the segment is
divided are called the ______________________ of the segment.
Def
A ray that divides an angle into two congruent angles ______________ the angle. The
dividing ray is called the ________________________.
Def
Two rays that divide an angle into three congruent angles __________________ the
angle. The dividing rays are called _____________________________.
1.9
Two Basic Steps for Probability Problems
1. Determine all possiblities in a logical manner. Count them.
2. Detemine the number of these possibiliites that are “favorable.” We shall call
these winners.
number of winners
Probability = number of possibilities
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