# Lesson 1

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```					Lesson 1.1 Sets, Statements, and Reasons
Statement: A statement is a set of words or symbols that make a claim that can be classified as
true or false.

Negation: The negation of a statement P makes a claim opposite of the original statement
(negation of the statement). In symbols this would be represented ~P.

Conjunction: P and Q (P is true and Q is true). A conjunction is true only when both statements
are true.

Disjunction: P or Q (P is true or Q is true). A disjunction is true when either of the statements is
true.

Conditional statement (implication): If P, then Q. P Q“P implies Q.” A conditional
statement consists of two clauses, one of which begins with the word “if” or “when” or some
equivalent word.

Hypothesis: In the conditional statement  P, then Q,” P is the hypothesis. The hypothesis is
“If
the “if” clause.

Conclusion: In the conditional statement “If P, then Q,” Q is the conclusion. The conclusion is
the “then” clause.

Inductive reasoning: Inductive reasoning uses experiments and observations to draw
conclusions. Experiments and observations form and confirm hypotheses. Inductive reasoning is
the method for drawing conclusions from a limited set of observations.

Deductive reasoning: Deductive reasoning uses formal argument to prove a tested theory.
Deductive reasoning uses logic to draw conclusions from statements accepted as true.

Intuition: Intuition involves statements made without applying a formal reasoning.

Deduction: Deduction is the type of reasoning in which the knowledge and acceptance of
selected assumptions guarantee the truth of a particular conclusion.

Premises: Premises are statements that are treated as facts in a valid argument (direct proof).

Conclusion: A conclusion is what must follow on the basis of the premises.
Law of detachment (chains of reason—syllogisms) On the basis of premises, a particular
conclusion must follow. A syllogism is an argument of the form,
ab
bc
therefore,
ac

If you accept the first two premises, you must accept the conclusion.

Counterexample: a counterexample disproves the validity of an argument

Example of a valid argument:
1) If P, then Q
2) P
conclusion) therefore () Q

Example of an invalid argument:
1) If P, then Q

2) Q
conclusion) therefore () P
This is not valid because the premises tell us nothing about what happens if Q is true.

Venn (Euler) diagrams: To make a Venn diagram for a conditional statement, draw two circles,
   one inside the other. The interior of the smaller circle represents the hypothesis; the interior of
the larger circle represents the conclusion.

Intersection: The intersection of two sets P and Q is exactly those elements common to both P
and Q. This is represented P Q.

Union: The union of two sets P and Q is the set of elements that are in either P or Q. This is
represented P Q.



1.2 Informal Geometry and Measurement
Point: A point has location, but no size. It is represented by a dot and labeled with upper case
letters.

Line: A line is an infinite collection of points that has length but no width. A line extends
infinitely in both directions and is labeled with a lower case letter (usually italicized) or with two
points on the line with arrows on either end.

Plane: A plane is two-dimensional. It has length and width but no thickness.

Collinear: If three points lie on the same line, they are said to be collinear. Points are collinear if
there is a line that contains all of them.

Noncollinear: Points are noncollinear if no single line contains them all.

Concurrent: Lines are concurrent if they contain the same point. Three or more lines that
intersect at one point are concurrent.

Betweeness of points: If one point lies between two others, then the points are collinear. This is
symbolized A-X-B, or B-X-A. This tells us that X is between A and B on AB.

Vertex: The point at which sides meet is called a vertex.

Line segment: A line segment consists of two points and every point between them. A line
segment is part of a line bounded by two endpoints.

Midpoint: If A-B-C and AB  BC , then B is the midpoint of AC. The midpoint of a line
segment divides a line segment into two equal segments.

Segment congruence: Segments can be congruent if the lengths of the segments (distances

between endpoints) are equal.

Parallel lines: Parallel lines lie in the same plane and never intersect.

Empty set: The empty set is the set of no elements.

Bisect: To bisect something means to separate it into two equal parts.

Straight angles: Straight angles are angles having a measure of 180

Right angles: Right angles are angles having a measure of 90 .

Perpendicular: Two lines are perpendicular if the intersect to form congruent adjacent angles.

using only a compass and straightedge.
Circle: A circle is the set of all points a given distance (radius) from a given point (center).

Arc: An arc is two points on a circle and all points on the circle between them. An arc is part of
a circle.

Radius: A radius is a line segment joining the center of a circle to a point on the circle.

Addition and subtraction property of equality: If a  b, then a  c  b  c. If
a  b, then a  c  b  c.

Construction skills:

   Copying a segment, Perpendicular bisector construction (midpoint)
1.3 Early definitions and Postulates
Mathematical system: A mathematical system is a system of undefined terms (starting points),
definitions, assumptions, and principles that follow from logic.

Axioms/postulates (assumptions): Properties of a system that are apparent—they do not need to
be proved. A postulate is a statement that is assumed to be true without proof.

Theorems: Theorems follow by logic from the definitions and postulates. Theorems must be
proved. A theorem is a statement that is proved by reasoning deductively from already accepted
statements.

Definition: A good definition 1) names the term being defined, 2) places the term into a set or
category, 3) distinguishes it from other terms, 4) it is reversible. Both a b, and b  a . This
can be written as a  b which means “a, if and only if b.” When we define a word in geometry,
the word and its definition are understood to have exactly the same meaning.

Definition of an isosceles triangle: An isosceles triangle is a
                                                     triangle that has two congruent
sides.

Definition of a line segment: A line segment is part of a line segment that consists of two points
known as endpoints and all the points between them. A line segment is part of a line bounded
by two endpoints.

Postulate 1) (two points determine a line): Through two distinct points there is exactly one line.

Postulate 2) (ruler postulate): the measure of any line segment is a unique positive number.

Distance between two points: The distance between two points A and B is the length of the line
segment AB that joins the two points.

AB represents line AB

AB represents line segment AB
   AB represents the length of segment AB (this distance between A and B).

Postulate 3) (segment addition postulate): If X is a point of AB and A-X-B, then AX  XB  AB

Congruent segments: Congruent (  ) segments are two segments that have the same length.
                     
Midpoint: The midpoint of a line segment is the point that separates the line segment into two
congruent parts. The midpoint is equidistant from the endpoints of a segment.

Bisector: A bisector is any line, ray, or segment that separates a segment into two congruent
parts. A bisector passes through the midpoint of a segment.

Ray: Ray AB, symbolized by AB, is the union of AB and all points X on AB such that B is
between A and X (A-B-X). A ray is part of a line that extends endlessly in one direction.

Opposite rays: Opposite rays are two rays that share a common endpoint and form a line.
                                        
Intersection: The intersection of two figures is the set of all points that the two figures have in
common.

Postulate 4) (two points determine a line): If two lines intersect, they intersect at one point.

Parallel lines: Parallel lines are lines that lie in the same plane and never intersect.

Coplanar: Points are coplanar if there is a plane that contains all of them. Coplanar points are
points that lie in the same plane.

Postulate 5) (three points determine a plane): Through three noncollinear points there is exactly
one plane.

Space: Space is the set of all possible points.

Postulate 6) (the intersection of two planes is a line): If two distinct planes intersect, then their
intersection is a line.

Postulate 7): Given two distinct points in a plane, the line containing them also lies in the plane.

Therorem 1.3.1: The midpoint of a line segment is unique. (This will be proved later).
1.4 Angles and their relationships
Angle: an angle is the union of two rays that share a common endpoint. The rays are called the
sides of the angle and their common endpoint is called the vertex of the angle.

Postulate 8) (protractor postulate): The measure of an angle is a unique positive number.

Acute angle: An acute angle is an angle whose measure is less than 90 .

Right angle: A right angle is an angle whose measure is exactly 90 .

Obtuse angle: An obtuse angle is an angle whose measure is greater than 90 and less than 180 .

Straight angle: A straight angle is an angle whose sides for opposite rays (a straight line). A
straight angle has a measure of exactly 180 .
                 
Reflex angle: A reflex angle is an angle whose measure is greater than 180 and less than 360 .

Postulate 9) (angle addition postulate): If a point D lies in the interior of ABC , then
mABD  mDBC  mABC .
                
Congruent angles: Congruent angles (  ' s ) are two angles with the same measure. Congruent

angles are angles that coincide exactly when superimposed.

Bisector of an angle: The bisector of an angle is the ray that separates the given angle into two

congruent angles.

For example, if NP is the angle bisector of MNQ, then mMNP  mPNQ. It also
1
follows that mMNP  mMNQ.
2
Complementary angles: Two angles are complementary if the sum of their measures is 90 .
                                       
Each angle in the pair is said to be the complement of the other angle.

Supplementary angles: Two angles are supplementary if the sum of their measures is 180 .
Each angle in the pair is said to be the supplement of the other angle.   

Vertical angles: When two lines intersect, the pairs of nonadjacent angles formed are known as
vertical angles.                                                              

Theorem 1.4.1 (the angle bisector of an angle is unique): There is one, and only one, angle
bisector for a given angle. (This theorem will be proved later).

Construction Skills:

Copying an angle, Construction of an angle bisector

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