Download Inductive Reasoning Reasoning to a conclusion based upon the

Reasoning to a conclusion based upon the
perceived pattern. Conclusions are NOT
certain using Inductive Reasoning, as more
data may change the pattern. Examples:
Inductive Reasoning
Reasoning logically to a conclusion based
upon FACTs in evidence. Done correctly,
conclusions are CERTAIN using Deductive
Reasoning. Examples:
Deductive Reasoning
Undefined Terms
To begin our vocabulary in Geometry, we
must "accept" the meanings of certain terms
as building blocks, upon which we can build
the rest of our vocabulary: POINT, LINE and
PLANE.
Points are a location in space. They have no
size, color or weight. They are infinitely
small.
They
are named using single, capital, print letters.
Point (features, naming)
Line (features, naming)
Plane (features. Naming)
Ray (definition, naming)
Segment (definition, naming)
A line is straight. It is infinitely long (no end
points). It is infinitely thin. It has no color,
weight, etc. It is made up of infinite points.
It is named one of two ways: a) by using
TWO points on the line, with a two arrowsided line above the letters or b) by using a
single, lower case, cursive letter. (l)
A plane is flat. It is infinitely long and wide
(it has no end points). It is infinitely thin, it
has no color, weight, etc. It is made up of
infinite points, or infintely many lines
stacked side by side. It is named using
either three points that are non-collinear
(plane ABC) or a single CAPITAL cursive
letter (R).
A ray is a part of a line. It has one end
point, and the other end extends infinitely.
It is named, like a line, using two points on
the Ray, however, be sure to list the
Endpoint FIRST, followed by any other point
on the line. Above the letters, draw an
arrow with an endpoint on the left, and an
arrow on the right.
A segment is a part of a line.It has two
endpoints. It is named, like a line or ray,
using the two end points. On top is a line
segment (with end points, not arrows on the
ends). Because a segment has a finite
length, it also has a way of "naming" its
length...either mAB with segment over top,
or AB (Shorthand for distance between A
and B).
0 Dimensional means there is no room for
movement. The only object demonstrating
zero dimesions is a point.
0 dimensional
1 Dimensional means there is room for
movement in only one direction (forward and
back). A line represents a 1 dimensional
space.
1 dimensional
2 Dimensional means there is room for
movement in two directions (forward and
back as well as left and right). A plane
represents a 2 dimensional space.
2 dimensional
3 dimensional
3 Dimensional means there is room for
movement in three directions (forward and
back as well as left and right as well as up
and down). Space represents a 3
dimensional space.
Collinear
Coplanar
Two or more points are collinear if a
(straight) line CAN be drawn between them
to connect them. ANY two points are
ALWAYS collinear. Three or more points are
SOMETIMES collinear.
Two or more points are coplanar if a (flat)
plan CAN be drawn through them. ANY two
points are ALWAYS coplanar. ANY three
points are ALWAYS coplanar. Four or more
points are SOMETIMES collinear. A line and
point not on the line are ALWAYS coplanar.
Two intersecting lines are ALWAYS coplanar.
Two non-intersecting lines are SOMETIMES
COPLANAR
Lines that are parallel are: Coplanar AND
never intersect
Parallel
Lines that are perpendicular are: Coplanar
and their intersection is a right angle.
Perpendicular
Lines are skew if they are: NON-coplanar
(and thus never intersect).
Skew
Two objects intersect if they share one or
more point.
Intersecting
Counter Example
A counter example is an example which
disproves a statement. If the statement is in
an IF hypotheis THEN conclusion, format,
the counter example can be stated by
saying: Counterexample is hypothesis but
NOT conclusion. Example: If teenager, then
15 years old. Counterexample: 14 years
old is a teenager, but is not 15 years old.
I know
an Example
Always, Sometimes, Never
Always
I know
a CounterExample
Sometimes
Never
Pencil, straight edge (no ruler markings),
compass
Construction Tools
Bisector
BFF51: Copy segment
BFF52: Segment Sum/Difference
Show Drawing. Include bisector and
perpendicular markings.
BFF 53: Perpendicular bisector
Postulate
Theorem
Congruence / Congruent
Equality / Equal
Overlap
Ruler Postulate
Angle Addition Postulate
Angle Overlap Theorem
Midpoint Formula
Midpoint
Angle
Naming Angles
Interior
Exterior
Vertex
Classify angles
Zero Angle
Acute
Right
Obtuse
Straight Angle
Reflex
Measure an angle with protractor
Draw an angle with protractor
ATM 50: Construct congruent angle
ATM 51: Construct Angle Sum/Diff.
ATM 52: Angle Bisector
Conditional Statement
A logical statement where a situation's
existence requires the existance of another
situation.
They
are phrased in format: IF…hypothesis,
THEN…conlcusion.
Identify Hypothesis, Conclusion
In a conditional statement, the hypothesis
immediately follows the indicator IF. The
conlcusion immediately follows the indicator
THEN. If written in symbolic notation
(p=>q), Hypothesis (p) precedes the
conditional arrow, while Conclusion (q)
follows it.
Create Logic Chain
If GLORP, then FRIZL
If
SATCZ then GLORP
If
FRIZL then WZIK
Can be rearranged to:
If
SATCZ then GLORP
If
GLORP, then FRIZL
If
FRIZL then WZIK
and
combined to make the summative
statement: if SATCZ then WZIK
Converse
Given a conditional statement "p=>q", the
Converse is of the form:
q=>p
Note: Converse is only sometimes true.
Inverse
Given a conditional statement "p=>q", the
Inverse is of the form:
~p=>~q
Note: Inverse is only sometimes true.
Contrapositive
Given a conditional statement "p=>q", the
Contrapositive is of the form:
~q=>~p
Note: If Conditional is true, Contrapositive is ALSO true.
Validity of argument
If an argument is phrased using IF…THEN
statements, and conditionals are valid, then
contrapositive statements are also valid, but
converse and inverse are NOT necessarily
valid.
Biconditional
A biconditional is written:
Hypothesis "if and only if" Conclusion
or p iff q or p<=>q
A
biconditional is VALID/TRUE only if both the
conditional and converse are true.
Is it a valid/good definition?
Opposite / Logical Negation
A definition is good ONLY if the biconditional
is valid/true. So write it is a conditional, see
if converse is true also.
All cases NOT covered by the other. If X=5,
the negation of that is X≠5. If A=>B, the
negation is A=>~B
Conclusion
Euler Diagram
Hypothesis
Contradiction
If I am working to validate A=>~B, a
contradiction is to find that ~A is true. Both
A and ~A cannot be true. They MUST be
mutually exclusive events.
∠A ≅ ∠A or
Reflexive Property of Congruence
AB ≅ AB
if ∠A ≅ ∠B, then ∠B ≅ ∠A or
if AB ≅ BC then BC ≅ AB
Symmetric Property of Congruence
if ∠A ≅ ∠B, and
∠B ≅ ∠C then
Transitive Property of Congruence
∠A ≅ ∠C ,
If a = b and c=d then a+c = b+d
Addition Property of Equality
or
If a
= b and c=c, then a+c = b+c
If a = b and c=d then a - c = b - d
Subtraction Property of Equality
or
If a = b and c=c, then a - c = b - c
If a = b and c=d then (a)(c) = (b)(d)
(…where neither c nor d = 0)
Mutliplication Property of Equality
or
If a= b and c=c, then (a)(c) = (b)(c )
(…where c≠0)
If a = b and c=d then (a)/(c) = (b)/(d)
(…where neither c nor d = 0)
Division Property of Equality
Substitution Property of Equality
or
If a = b and c=c, then (a)/(c) = (b)/(c )
(…where c≠0)
If a = b, and a = c ,then b = c
If a = b and b = c, then a = c
(unlike Transitive, order isn't required to work)
or
m∠A = m∠A or
Reflexive Property of Equality
AB ≅ AB
if m∠A ≅ m∠B, then m∠B ≅ m∠A or
Symmetric Property of Equality
if AB ≅ BC then BC ≅ AB
Distributive Property
If a(b + c), then ab + ac
A process of validating something using
deductive reasoning. Done correctly, the result is
irrefutable.
Given: A
Two Column Proof
Prove:
Statement
Reason
A
Given
B
B