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```Warm-Up:
EOC Prep
What is RD if RL=54 cm?
A) 81
B) 108
C) 162
What is x if WL=15x and
D) 216
LJ=5x+3?
A) 0.3
B) 0.4
C) 0.6
D) 1.2
Inverses,
Contrapositives, and
Indirect Reasoning
5.4
Today’s Goals
By the end of class today, YOU should be able to…
1. Write the negation of a statement and the
inverse and contrapositive of a conditional
statement.
2. Use indirect reasoning to solve a problem.
Negations

A negation of a statement has the
opposite meaning of the original
statement.
Ex.1: Negations
Write the negation of the following
statement…
Statement: ABC is obtuse.
Ex.1: Solution
Negation: ABC is not obtuse.
You Try…
Write the negation of the following
statement…
Statement: Lines m and n are not
Inverses and contrapositives

A conditional statement is a statement that tells if
one thing happens, another will follow
It is written as p  q
The inverse of a conditional statement negates
both the hypothesis and the conclusion.
The contrapositive of a conditional statement
switches both the hypothesis and conclusion and
negates both.



Ex.2:
Write the inverse and contrapositive of
the following statement:
Statement: If a polygon has six sides,
then it is a hexagon.
Ex.2: Solution
Conditional: If a polygon has six sides, then it is a hexagon.

Negate both.

Inverse: If a polygon does not have six sides, then it is not a hexagon.
Conditional: If a polygon has six sides, then it is a hexagon.
 Switch and negate both. 
Contrapositive: If a polygon is not a hexagon, then it does not have six sides.
You Try…
Write the inverse and contrapositive of the
following statement:
If a polygon is a triangle, then the sum of
the measures of its angles is 180.
A summary of negation,
inverse, and contrapositive
statements
Statement
Example
Symbolic form
Conditional
If an angle is a
straight angle, then
its measure is 180.
pq
If p, then q
Negation (of p)
An angle is not a
straight angle
~p
Not p
Inverse
If an angle is not a
straight angle, then
its measure is 180
~p  ~q
If not p, then not q
Contrapositive
If an angle’s measure
is not 180, then it is
not a straight angle
~q  ~p
If not q, then not p
Indirect reasoning
Indirect reasoning is when all
possibilities are considered and then all
but one are proved false. Then the
remaining possibility must be true.
 A proof involving indirect reasoning is an
indirect proof.

An example of indirect
reasoning…
Suppose your brother tells you, "Susan
called a few minutes ago." You think
through these three steps.
1. You have two friends named Susan.
2. You know that one of them is at band
practice.
3. You conclude that the other Susan must
have been the caller.
Writing an indirect proof
1.
2.
3.
State as an assumption the opposite
(negation) of what you want to prove.
Show that this assumption leads to a
Conclude that the assumption must be
false and that what you want to prove
must be true.
Ex.3: Indirect proofs
Write the first step of an indirect proof:
Quadrilateral QRWX does not have four
acute angles.
Ex.3: Solution
Quadrilateral QRWX does not have four
acute angles.
Assume that quadrilateral QRWX has four
acute angles.
You Try…
Write the first step of an indirect proof:
An integer n is divisible by 5.
Ex.4: Indirect proofs
what is given and what you are to prove.
Then give the steps of an indirect proof.
If Jaeleen spends more than \$50 to buy two
items at a bicycle shop, then at least one
of the items costs more than \$25.
Ex.4: Solution
Given: The cost of two items is more than \$50. This means
that each item costs \$25 or less.
Prove: At least one of the items costs more than \$25.
Step 1: Assume as true the opposite of what you want to
prove. That is, assume that neither item costs more than \$25.
Step 2: This, in turn, means that the two items together cost
\$50 or less. This contradicts the given information that the
amount spent is more than \$50.
Step 3: Conclude that the assumption is false. One item
must cost more than \$25.
Homework
Page 267 #s 8, 13, 15, 16
 Page 268 #s 21, 24-27, 31
 The assignment can also be found
online at:

• http://www.pearsonsuccessnet.com/snpap
p/iText/products/0-13-037878-X/Ch05/0501/PH_Geom_ch05-01_Ex.pdf
```
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