Download 2.1 Conditional Statements

Deductive Reasoning
Algebraic Properties of Equality
• Addition property: If a=b, then a+c = b+c.
– Allowed to add same number on both sides
• Subtraction property: If a=b, then a-c = b-c.
– Allowed to add same number on both sides
• Multiplication property: If a=b, then ac = bc.
– Allowed to add same number on both sides
• Division property: If a=b, and c≠0, then a/c = b/c.
– Allowed to add same number on both sides
Solve
• Solve 5x – 18 = 3x + 2 and explain each step
in writing.
5x – 18 = 3x + 2
2x – 18 = 2
2x = 20
x = 10
Subtraction prop. of =.
Addition prop. of =.
Division prop. of =.
More properties of equality
• Reflexive property: For any real number a,
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, then a can be
substituted for b in any equation or
expression.
Properties of Equality
Segment Length
Angle Measure
Reflexive
AB = AB
m<A = m<A
Symmetric
If AB = CD, then
CD = AB.
If m<A = m<B, then
m<B=m<A.
Transitive
If AB = CD and
CD = EF, then
AB=EF.
If m<A = m<B and
m<B=m<C, then
m<A=m<C.
Challenge
• How is the product 4 · 6 related to 52?
• How is the product 5 · 7 related to 62?
• Make a conjecture about how the product of
two positive inters n and n + 2 is related to the
square of the integer between them.
• Write a convincing argument to justify your
conjecture.
Conditional Statements
Warmup
• State whether each sentence is true or false.
– If you live in Los Angeles, then you live in
California. True
– If you live in California, then you live in Los
Angeles. False
– If today is Wednesday, then tomorrow is Thursday.
True
– If tomorrow is Thursday, then today is Wednesday.
True
Conditional Statement
• Conditional statement has two parts,
hypothesis and a conclusion.
• If _____________, then____________.
hypothesis
conclusion
– Hypothesis is after “if” and the conclusion is after
“then”
Rewrite in If-Then form
• A number divisible by 9 is also divisible by 3.
– If a number is divisible by 9, then it is divisible by
3.
• Two points are collinear if they lie on the same
line.
– If two points lie on the same line, then they are
collinear.
Writing a Counterexample
• Write a counterexample to show that the
following conditional statement is false.
– If x2 = 16, then x = 4.
– False, x could be negative four; x = (-4)
Converse Example
• Two points are collinear if they lie on the same
line.
– If two points are collinear, then they lie on the
Conditional Statement
same line.
– If two points lie
on
the
same
line,
then
they
are
Converse
collinear.
• A statement can be altered by negation, that
is, by writing the negative of the statement.
– Statement: m<A = 30°
– Negation: m < A ≠ 30°
– Statement: <A is acute.
– Negation: <A is not acute.
Inverse Example
• If two points lie on the same line, then they
Conditional
are collinear.
• If two points do not lie on the same line, then
Inverse
they are not collinear.
Contrapositive Example
• If two points lie on the same line, then they
Conditional
are collinear.
• If two points are not collinear, then they do
Contrapositive
not lie on the same line.
• When two statements are both true or both
false, they are called logically equivalent
(have same truth value) statements.
– A conditional statement is logically equivalent to
its contrapositive.
– The inverse and converse of any conditional
statement are logically equivalent.
•
Write the
– a) inverse
– b) converse
– c) contrapositive
If there is snow on the ground, then flowers are
not in bloom.
a) If there is no snow on the ground, then flowers
are in bloom.
b) If flowers are not in bloom, then there is snow
on the ground.
c) If flowers are in bloom, then there is no snow
on the ground.
Definitions and Biconditional Statements
Definition
• Two lines are called perpendicular lines if and
only if they intersect to form right angles.
• A line perpendicular to a plane is a line that
intersects the plane in a point and is
perpendicular to every line in the plane that
intersects it.
Exercise
•
Decide whether each statement about the
diagram is true. Explain your answer using the
definitions you have learned.
a. Points D, X, and B are collinear.
b. AC is perpendicular to DB.
c. <AXB is adjacent to <CXD.
.A
.D
X
.C
.B
Biconditional Statement
• When a conditional statement and its
converse are both true, you can write them as
a single biconditional statement.
• A biconditional statement is a statement that
contains the phrase “if and only if. ”
• Any valid definition can be written as a
biconditional statement.
• Rewrite the biconditional as conditional
statement and its converse.\
– Biconditional :
Two angles are supplementary if and only if the sum
of their measures is 180°.
– Conditional:
If two angles are supplementary, then the sum of their
measures is 180°.
– Converse:
If the sum of two angles measure 180°, then they are
supplementary.
• State a counterexample that demonstrates
that the converse of the statement is false.
– If three points are collinear, then they are
coplanar.
– If an angle measures 48°, then it is acute.
Warmup
•
Which statement about the diagram is not true?
.
.
A
B
G
H
C
.
.
D
a.
b.
c.
d.
e.
<GHE is adjacent to <CHD.
.F
BF is perpendicular to AH.
<BGH and <BGA are supplementary.
.GHC  EHD
m<BGH = 90
.E