Transcript GeoNotes2-1,2-3,2-5

Reasoning, Conditionals, and
Postulates
Sections 2-1, 2-3, 2-5
Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
The next month is July.
Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
The next multiple is 35.
Reasoning
• Inductive Reasoning – To draw a
conclusion from a pattern.
• Conjecture – A statement you believe to
be true based on inductive reasoning.
• Counterexample – One example in which
the conjecture is not true; proves the
conjecture is false.
• Deductive Reasoning – To draw
conclusions from given facts, definitions,
and properties.
Example: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
n = –3 is a counterexample.
Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
0.0007
Determine if each conjecture is true. If false,
give a counterexample.
3. The quotient of two negative numbers is a positive
number. true
2
4. Every prime number is odd. false; false;
90° and
5. Two supplementary angles are not90°
congruent.
6. The square of an odd integer is odd. true
Conditional Statements
Example
Identify the hypothesis and conclusion of each
conditional.
A. If today is Thanksgiving Day, then today is
Thursday.
B. A number is a rational number if it is an
integer.
C. A number is divisible by 3 if it is divisible by 6.
Examples:
Determine if the conditional is true. If false,
give a counterexample.
1. If this month is August, then next month is
September.
2. If two angles are acute, then they are
congruent.
Related Conditionals:
• Conditional:
p → q (read as “if p then q”)
• Converse:
q → p (switch: “if q then p”)
• Inverse:
~p → ~q (“if not p then not q”)
• Contrapositive:
~q → ~p (“if not q then
not p”)
Example: Biology Application
Conditional: If an animal is an adult insect, then it
has six legs.
Converse: If an animal has six legs, then it is an adult
insect.
Inverse: If an animal is not an adult insect, then it does
not have six legs.
Contrapositive: If an animal does not have six legs,
then it is not an adult insect.
Lesson Quiz: Part I
Identify the hypothesis and conclusion of each
conditional.
1. A triangle with one right angle is a right triangle.
H: A triangle has one right angle.
C: The triangle is a right triangle.
2. All even numbers are divisible by 2.
H: A number is even.
C: The number is divisible by 2.
3. Determine if the statement “If n2 = 144, then
n = 12” is true. If false, give a counterexample.
False; n = –12.
Lesson Quiz: Part II
4. Write the converse, inverse, and contrapositive
of the conditional statement “If Maria’s birthday is
February 29, then she was born in a leap year.”
Converse: If Maria was born in a leap year, then
her birthday is February 29.
Inverse: If Maria’s birthday is not February 29,
then she was not born in a leap year.
Contrapositive: If Maria was not born in a leap
year, then her birthday is not February 29.
Postulate - A statement that
describes the relationship between
basic terms in Geometry. Postulates
are accepted as true without proof.
Examples of some Postulates:
• Through any 2 points there is exactly 1 line.
• Through any 3 noncollinear points there is
exactly 1 plane.
• A line contains at least 2 points.
• A plane contains at least 3 noncollinear
points.
Theorem
A conjecture or statement that can
be shown to be true. Used like a
definition or postulate.
Midpoint Theorem - If M is the
midpoint of AB, then AM  to MB.
A
M
B
Proof
A logical argument in which each
statement is supported by a
statement that is true (or accepted as
true).
Supporting evidence in a proof (the
reason you can make the statement)
are usually postulates, theorems,
properties, definitions or given
information.