Transcript 5-4 Inverses Contrapositives and Indirect Reasoning

Inverses, Contrapositives, and Indirect Reasoning
LESSON 5-4
Additional Examples
Write the negation of “ABCD is not a convex polygon.”
The negation of a statement has the opposite truth value. The
negation of is not in the original statement removes the word not.
The negation of “ABCD is not a convex polygon” is “ABCD is a
convex polygon.”
Quick Check
HELP
GEOMETRY
Inverses, Contrapositives, and Indirect Reasoning
LESSON 5-4
Additional Examples
Quick Check
Write the inverse and contrapositive of the conditional
statement “If ABC is equilateral, then it is isosceles.”
To write the inverse of a conditional, negate both the hypothesis and
the conclusion.
Hypothesis
Conclusion
Conditional: If
Inverse: If
ABC is equilateral,
then it is isosceles.
Negate both.
ABC is not equilateral,
then it is not isosceles.
To write the contrapositive of a conditional, switch the hypothesis and
conclusion, then negate both.
Conditional: If
ABC is equilateral,
then it is isosceles.
Switch and negate both.
Contrapositive: If
HELP
ABC is not isosceles, then it is not equilateral.
GEOMETRY
Inverses, Contrapositives, and Indirect Reasoning
LESSON 5-4
Additional Examples
Write the first step of an indirect proof.
Prove: A triangle cannot contain two right angles.
In the first step of an indirect proof, you assume as true the negation
of what you want to prove.
Because you want to prove that a triangle cannot contain two right
angles, you assume that a triangle can contain two right angles.
The first step is “Assume that a triangle contains two right angles.”
Quick Check
HELP
GEOMETRY
Inverses, Contrapositives, and Indirect Reasoning
LESSON 5-4
Additional Examples
Quick Check
Identify the two statements that contradict each other.
I. P, Q, and R are coplanar.
Two statements contradict each other
II. P, Q, and R are collinear.
when they cannot both be true
III. m PQR = 60
at the same time.
Examine each pair of statements to see whether they contradict each other.
II and III
I and II
I and III
P, Q, and R are
P, Q, and R are
P, Q, and R are
collinear, and
coplanar and
coplanar, and
m PQR = 60.
collinear.
m PQR = 60.
Three points that lie
on the same line are
both coplanar and
collinear, so these
two statements do
not contradict each
other.
HELP
Three points that lie
on an angle are
coplanar, so these
two statements do
not contradict each
other.
If three distinct
points are collinear,
they form a straight
angle, so m PQR
cannot equal 60.
Statements II and III
contradict each
other.
GEOMETRY
Inverses, Contrapositives, and Indirect Reasoning
LESSON 5-4
Additional Examples
Write an indirect proof.
Prove:
ABC cannot contain 2 obtuse angles.
Step 1: Assume as true the opposite of what you want to prove. That
is, assume that ABC contains two obtuse angles. Let A and B
be obtuse.
Step 2: If A and B are obtuse, m
so m A + m B > 180.
A > 90 and m
B > 90,
Because m C > 0, this means that m A + m B + m C > 180.
This contradicts the Triangle Angle-Sum Theorem, which states
that m A + m B + m C = 180.
Step 3: The assumption in Step 1 must be false.
contain 2 obtuse angles.
HELP
ABC cannot
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GEOMETRY