Transcript Chapter 2 - Catawba County Schools

Reasoning and Proof
Chapter 2
2.1 – Conditional Statements
• Conditional statements –
If, then form
• If – hypothesis
• Then – conclusion
• Negation of a statementthe opposite of a
statement
• Biconditional statement –
A statement in the form of
if and only if (iff)
The Forms of Conditional
Statements
• Conditional – (original) If, then.
If p, then q.
• Converse – Switch the
hypothesis and conclusion.
If q, then p.
• Inverse – The negation of the
conditional statement.
If  p, then  q.
• Contrapositive – The negation of
the converse.
If  q, then  p.
Examples
• Con – If two angles are vertical,
then they are congruent.
• True or False?
• Converse
• Inverse
• Contrapositive
Example
• Write the statement in If, then
form: All monkeys have tails.
• Converse
• Inverse
• Contrapositive
Example
• Conditional – If it is cloudy,
then it is raining.
• Converse
• Inverse
• Contrapositive
Example
• Conditional – If two angles
are congruent, then they
have equal measure.
• Converse
• Inverse
• Contrapositive
Logically Equivalent
• A pair of statements with the
same truth value.
• CONDTIONAL  CONTRAPOSITIVE
– Both have the same truth value.
• CONVERSE  INVERSE – Both
have the same truth value.
Point, Line, and Plane Postulates
• Through any two points there
exists exactly one line.
• A line contains at least two
points.
• If two lines intersect, then their
intersection is exactly one point.
• Through any three noncollinear
points there is exactly one
plane.
More Postulates
• A plane contains at least three
noncollinear points.
• If two points lie in a plane,
then the line containing them
lies in the plane.
• If two planes intersect, then
their intersection is a line.
• If there is a line and a point not
on the line, then exactly one
plane contains them.
2.2 Definitions and Biconditional
Statements
• Definition of Perpendicular
Lines – Two lines are
perpendicular  iff they
intersect to form a right angle.
• Definition of a line  to a
plane – If a line is  to a plane,
then it is  to every line in that
plane that intersects it.
Biconditional statements
• All biconditional statements
are written using if and only if
(iff).
• Writing a biconditional
statement is equivalent to
writing a conditional
statement and its converse.
• A biconditional statement is
true ONLY IF the conditional
and the converse are both
true.
Examples of Biconditionals
• Conditional statement – If
x² = 4, then x = 2 or x = -2.
• Is the statement true?
What is the biconditional of
that statement?
• Is the biconditional true?
2.3 Deductive Reasoning
• Conditional – If p, then q.
pq
• Converse – If q, then p.
q p
• Inverse – If p, then q.
p  q
• Contrapositive – If q, then p.
q  p
• Biconditional – p iff q.
p
q
Laws of Logic
• Law of Detachment –
If pq is a true conditional
statement and p is true,
then q is true.
• Law of Syllogism If pq and qr are true
conditional statements,
then pr is true.
Example – Law of Detachment
Is the argument valid?
Michael knows that if he does
not do his chores in the
morning, he will not be
allowed to play video games
later that same day.
Michael does not play video
games Friday afternoon. So
Michael did not do his chores
on Friday morning.
Example – Law of Detachment
• Is the argument valid?
If two angles are vertical
then they are congruent.
ABC and DBE are vertical.
So ABC and DBE are
congruent.
Example – Law of Syllogism
• What can you conclude?
• If a fish swims at 68 mi/h,
then it swims at 110 km/h.
• If a fish can swim at 110
km/h, then it is a sailfish.
• Therefore,
Example – Law of Syllogism
• What can you conclude?
• If the stereo is on, then the
volume is loud.
• If the volume is loud, then
the neighbors will
complain.
• Therefore,
2.4 Reasoning with Properties
from Algebra
• Algebraic Properties of
Equality
• Algebraic Properties of
Equality and Congruence
Properties Of Equality
• ADDITION
PROPERTY
• SUBTRACTION
PROPERTY
• MULTIPLICATION
PROPERTY
• DIVISION
PROPERTY
• SUBSTITUTION
PROPERTY
• If a = b, then
a+c = b +c.
• If a = b, then
a-c = b-c.
• If a = b, then
ac = bc.
• If a = b and c  0,
then a/c = b/c.
• If a = b, then a
can be
substituted for b
in any equation
or expression.
Properties of Equality and
Congruence
• REFLEXIVE
PROPERTY
• SYMMETRIC
PROPERTY
• TRANSITIVE
PROPERTY
• For any real
number a, a = a.
or AB  AB.
• If a = b, then, b = a.
If AB  CD, then
CD  AB.
• If a = b and b = c,
then a = c.
If AB  CD and
CD  EF, then
AB  EF.
2.5 – Proving Statements About
Segments
Q is midpoint PR
Q
PQ = ½PR and
i = ½PR
QR
s
t
h
e
M
i
d
2.6 – Proving Statements About
Angles
1 supp 2
3 supp 4
1  4
2  3
Angle Theorems
• If two angles are
supplementary to the same
angle or congruent angles,
then they are congruent.
(Supp of  s are .)
• If two angles are
complementary to the same
angle or congruent angles,
then they are congruent.
(Comp of  s are .)
More Theorems
• All right angles are
congruent.
• All vertical angles are
congruent.
• Linear Pair Postulate – If
two angles form a linear
pair, then they are
supplementary.