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CHAPTER TWO
 Conditional Statement a logical statement that is dependent upon
some other factor.
 Conditional Statements are written in an “If – Then” format.
 First part of a Conditional Statement is called the hypothesis. This is
the assumption. The hypothesis always follows directly after the “If”
in the “If-Then” format. “If” is not part of the hypothesis. The
hypothesis is symbolized by the italicized lower case p.
 Second part of a Conditional Statement is called the conclusion. This
is what happens if the assumption is true. The conclusion always
follows directly after the “Then” in the “If-Then” format. “Then” is
not part of the conclusion. The conclusion is symbolized by the
italicized lower case q.
 Conditional Statements can be either true or false. This is called its
Truth Value.
 A true hypothesis does not dictate a true Conditional Statement and
vice versa.
 To show that the Conditional Statement is false, you have to show
only one counterexample.
 Statements that are based on a given conditional statement are called
Related Conditionals and they have special names (Converse, Inverse,
Contrapositive).
 “Then” is abbreviated with a small arrow to the right . If is not
included in the symbolic representation of the Conditional or its
Related Conditionals.
 Converse of a conditional statement is formed by switching the
hypothesis and conclusion.
 The Converse of a conditional statement may not always be true even
if the original Conditional Statement is true.
 Negation is the negative of a statement. It is formed by adding the
word “not”. It is symbolized by a swoop in front of the p, ~ p.
 If a statement is true than its negation is false and vice versa. The
truth values can be organized in a truth table.
p ~p
T F
F T
 Inverse of a Conditional Statement is formed by negating both the
hypothesis and conclusion of a conditional statement.
 Contrapositive of a Conditional Statement is formed by negating the
Converse of the Conditional Statement.
 Law of Contrapositive is the relationship of the truth values of a
conditional and its contrapositive.
 Equivalent Statements or Logically Equivalent Statements are when
the two statements are both true or both false.
Conditional Statement = Contrapositive
Inverse = Converse
IF p, THEN q
Conditional Statement
IF q, THEN p
Converse
IF not ~p, THEN not ~q
Inverse
IF not ~q, THEN not ~p
Contrapositive
 Perpendicular Lines are two lines who intersect to form four right
angles.
 If two lines intersect to form four right angles then the lines are
perpendicular
 Line Perpendicular to a Plane is a line that intersect the plane in a
point and is perpendicular to every line in the plane that intersects it.
  is the symbol for “perpendicular”
 If a Conditional Statement is true and its Converse is true then you
can combine the two into a Biconditional Statement.
 A Biconditional Statement is a statement that contains the phrase “if
and only if”.
 Biconditional Statements can be true or false.
 All definitions can be written as true biconditional statements.
 Unlike definitions, not all postulates can be written as true
biconditional statements.
 “if he committed a crime, then he was at the scene of the crime”
 “If and Only If” can be abbreviated “iff”
 In a Biconditional Statement, the “if and only if” can be symbolized
as a two way arrow .
pq
Conditional Statement
qp
Converse
~p  ~q
Inverse
~q  ~p
Contrapositive
pq
Biconditional
 The negation of a negative statement is a positive statement.
 Deductive Reasoning uses facts, definitions and accepted properties in
a logical order to write a logical argument.
 Deductive Reasoning is using definite information to form a definitive
conclusion while Inductive Reasoning uses definite information or
circumstantial information to form a educated guess as to the
conclusion.
 One method of solving problems using Deductive Reasoning is called
matrix logic where a grid or table is made to solve the problem.
 A Conjunction is a compound statement formed by joining two
statements with the word “and”. A Conjunction is symbolized by  .
 A conjunction is true only when both the statements are true.
 A Disjunction is a compound statement formed by joining two
statements with the word “or”. A Disjunction is symbolized by  .
 A disjunction is false only when both statements are false.
 The Law of Detachment states that if the conditional statement is true
then the hypothesis is true and the conclusion is true.
 When you apply the Law of Detachment, make sure that the
conditional is true before you test the validity of the conclusion.
 Law of Detachment symbolically is as follows:
[(p  q)  p]  q
 The Law of Syllogism states That if a hypothesis leads to a conclusion
and that conclusion in turn forms a hypothesis that leads to another
conclusion then the original hypothesis can be said to lead to the
second conclusion.
If p  q and q  r are true conditional statements, then p  r is
true.
 Law of Syllogism symbolically is as follows:
[(p  q)  (q  r)]  (p  r)
 The original Conditional Statement must be true to use the Law of
Syllogism.
 Addition Property
If a =b, then a + c = b + c
 Subtraction Property
If a =b, then a - c = b – c
 Multiplication Property If a = b, then ac = bc
 Division Property
If a = b and c 0, then a  c = b  c
 Reflexive Property
a=a
 Symmetric Property
If a = b, then b = a
 Transitive Property
If a = b and b = c, then a = c
 Substitution Property
If a = b, a + 5 = 7 can be written b + 5 = 7
 Distributive Property
a(b + c) = ab + ac
 Commutative Property of Addition doesn’t matter what order you
add the numbers the result will be the same
 Commutative Property of Multiplication
doesn’t matter what
order you multiply the numbers the result will be the same
 Associative Property of Addition
doesn’t matter how the
numbers are grouped before you add the numbers the result will be the
same
 Associative Property of Multiplicationdoesn’t matter how the
numbers are grouped before you multiply the numbers the result will
be the same
 Deductive Argument is a group of algebraic steps used to solve
problems
 Theorem is a true statement that follows as a result of other true
statements.
 Two-Column Proof is a two-column table that has numbered
statements and reasons that show the logical order of an argument.
 Each step of a two-column proof is called a statement.
 In a two-column proof, the first column on the left is always the
statements and labeled as such.
 In a two-column proof, the second column on the right is always the
reasons that justify the statements and labeled as such.
 In a two-column proof, you cannot write a statement unless you can
give a reason for it.
 Paragraph Proof , also called an informal proof, is a statements and
reasons with a logical order of argument written as a paragraph.
 Before writing a proof, you should have a plan. One strategy is to
work backward. Start with what you want to prove and work
backward step by step until you reach the given information.
 5 essential parts of a good proof:
1.
State the theorem or conjecture to be proven.
2.
List the given information.
3.
If possible, draw a diagram to illustrate the given
information.
4.
State what is to be proved.
5.
Develop a system of deductive reasoning.
 Ruler Postulate states the points on any line or line segment can be
paired with real numbers like on a ruler so that given any two points A
and B on a line, A corresponds to zero, and B corresponds to a positive
real number.
 Remember AB with no line over it is the symbol for the measure of a
line segment and not the name of the line segment itself.
 Reflexive Property
a  a
 Symmetric Property
If a  b, then b  a
 Transitive Property
If a  b and b  c, then a  c
 Right Angle Congruence Theorem states that all right angles are
congruent.
 Congruent Supplements Theorem states that if two angles are
supplementary to the same angle (or to congruent angles) then they
are congruent.
If m1 + m2 = 180 and m2 + m3 = 180, then 1  3.
 Notice then when talking about measure you use the equal sign and
when you are talking about the angles themselves you use the
congruence sign.
 Complement Theorem states that if the non-common sides of two
adjacent angles form a right angle, then the angles are complementary
angles.
 Congruent Complements Theorem states that if two angles are
complementary to the same angle (or to congruent angles) then they
are congruent.
 Linear Pair Postulate states that if two angles form a linear pair, then
they are supplementary.
 Vertical Angles Theorem state that all vertical angles are congruent.
 Protractor Postulate Given ray AB and a number r between 0 and
180, there is exactly one ray with endpoint A, extending on either side
of ray AB, such that the measure of the angle formed is r.
 Angle Addition Postulate states that two adjacent angles can add
together to make one larger angle so that the sum of the measures of
each of the smaller two adjacent angles equals the angle measure of
the larger angle.

P
Q
R
S
If mPQR + mRQS = mPQS, then the ray QR is in the interior of
mPQS