Transcript Conjecture - Angelfire

Conjecture: an educated guess
Inductive reasoning: Looking
at several specific situations
to arrive at a conjecture.
Juan, Pedro, Teres and Ines
are ISP students and they are
10, 17, 15 and 16 years old.
Conjecture: All ISP students
are 18 years old or less.
Counterexample
A conjecture may be true or false
It takes only one false
example to show a
conjecture is not true.
The false example is called
a counterexample.
Orlando is an ISP student, and
he is 19 years old. Then, not all
ISP students are 18 years old
or less.
Conditional Statement
Conditional statement (if-then statement) :
hypothesis + conclusion (p  q)
Example: If we do well on the test we are going
to improve our grade in the subject.
Converse
The converse of a
conditional is written
by interchanging the
hypothesis and the
conclusion. (q  p)
Example: If we improve our grade in this subject,
then we are going to do well on the test
Inverse
Negation: the denial of a
statement. (p)
Inverse can be formed by negating both, the
hypothesis and the conclusion. (p q)
Example: If we don’t do well on the test, then we are
not going to improve our grade in this subject.
Contrapositive
Contrapositive:
Negating the
hypothesis and the conclusion of the
converse of the given conditional.
(no q  no p)
Example: If we don’t improve our grade in this
subject, then we are not going to do well on the test.
Postulates
1. Through any two points there is exactly one
line
2. Through any three points not on the same line
there is exactly one plane
3. A line contains at least two points
4. A plane contains at least three points not on
the same line
5. If two points lie in a plane, then the entire line
containing those two points lies in that plane
6. If two planes intersect, then their intersection
is a line.
Deductive and
Inductive Reasoning
Inductive reasoning uses examples to
make a conjecture or rule. From
specific to general.
Deductive reasoning uses a rule to make
a conclusion. From general to specific.
All ISP students are 18 years
old or less. Juan is an ISP
student, so he is 18 years old
or less.
Law of detachment
Law of detachment: If p  q is a true
conditional and p is true, the q is true.
If two lines are parallel, they don´t
intersect each other.
(True statement)
l
m
If l and m are parallel, they don´t intersect
each other.
(Application of the law of detachment)
Law of Syllogism
Law of syllogism: If p  q and q  r are
true conditionals, the p  r is also true.
l
m
If two lines are perpendicular they form right
angles.
(True statement)
If two lines form right angles, they divide the
plane in four equal angles.
(True statement)
If two lines are perpendicular they divide the
plane in four right angles.
(Application of the law of Syllogism)
Some properties from algebra
applied to geometry
Property
Segments
Angles
Reflexive
PQ=QP
Symmetric
If AB= CD,
then CD = AB.
If m<A = m<B,
then m<B = m<A
Transitive
If GH = JK and
JK = LM, then
GH = LM
If m<1 = m<2 and
m<2 = m<3, then
m<1 = m<3
m<1 = m<1
Examples
Name the property of equality that justifies each statement.
Statement
Reasons
If AB + BC=DE + BC, then AB = DE
Subtraction property (=)
m<ABC= m<ABC
Reflexive property (=)
If XY = PQ and XY = RS,
then PQ = RS
If (1/3)x = 5, then x = 15
If 2x = 9, then x = 9/2
Substitution property (=)
Multiplication property (=)
Division property (=)
Example 2
Justify each step in solving 3x + 5 = 7
2
Statement
The previous
3x + 5 = 7
example is a
2
proof of the
2(3x + 5) = (7)2 conditional:
2
If 3x + 5 = 7,
2
3x + 5 = 14
then x=3
Reasons
Given
Multiplication property (=)
Distributive property (=)
3x = 9
Subtraction property (=)
x=3
Division property (=)
This type of proof is called a TWO-COLUMN PROOF
Verifying Segment Relationships
Five essential parts of a good proof:
•State the theorem to be proved.
•List the given information.
•If possible, draw a diagram to illustrate the given
information.
•State what is to be proved.
•Develop a system of deductive reasoning. (Use
definitions, properties, postulates, undefined terms, or
other theorems previously proved).
Theorem 2.1
Congruence of segments is reflexive,
symmetric, and transitive.
Reflexive property: AB  AB.
Symmetric property: If AB  CD, then CD  AB
Transitive property:
If AB  CD, and CD  EF, then AB  EF
Abbreviation: reflexive prop. of  segments
symmetric prop. of  segments
transitive prop. of  segments
Verifying Angle Relationships
Theorem 2-2 Supplement Theorem: If two angles form
a linear pair, then they are supplementary angles
Theorem 2-3: Congruence of angles is reflexive,
symmetric and transitive.
Abbreviation: reflexive prop. of  <s
symmetric prop. of  <s
transitive prop. of  <s
Theorem 2-4 and 2-5:
Theorem 2-4 Angles supplementary to the
same angle or to congruent angles are
congruent:
Abbreviation: <s supp. to same < or  <s are 
Theorem 2-5: Angles complementary to the
same angle or to congruent angles are
congruent.
Abbreviation: <s comp. to same < or  <s are 
Theorem 2-6, 2-7 and 2-8
Theorem 2-6 : All right angles are congruent
Abbreviation: All rt. <s are 
Theorem 2-7: Vertical angles are congruent.
Abbreviation: Vert. <s are 
Theorem 2-8: Perpendicular lines intersect to
form four right angles.
Abbreviation:  lines form 4 rt. <s