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Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
(For help, go to Lesson 2-1 and 2-2.)
Write the converse of each statement.
1. If it snows tomorrow, then we will go skiing.
2. If two lines are parallel, then they do not intersect.
3. If x = –1, then x2 = 1.
Write two conditional statements that make up each biconditional.
4. A point is on the bisector of an angle if and only if it is equidistant from
the sides of the angle.
5. A point is on the perpendicular bisector of a segment if and
only if it is equidistant from the endpoints of the segment.
6. You will pass a geometry course if and only if you are successful
with your homework.
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Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
Solutions
1. Switch the hypothesis and conclusion: If we go skiing tomorrow, then it snows.
2. Switch the hypothesis and conclusion: If two lines do not intersect, then they
are parallel.
3. If x2 = 1, then x = –1.
4. Rewrite the statement as an if-then statement; then rewrite it by writing
its converse: If a point is on the bisector of an angle, then it is equidistant from
the sides of the angle. If a point is equidistant from the sides of an angle,
then it is on the bisector of the angle.
5. Rewrite the statement as an if-then statement; then rewrite it by writing
its converse: If a point is on the perpendicular bisector of a segment, then
it is equidistant from the endpoints of the segment. If a point is equidistant
from the endpoints of a segment, then it is on the perpendicular bisector
of the segment.
6. Rewrite the statement as an if-then statement; then rewrite it by writing its
converse: If you pass a geometry course, then you are successful with your
geometry homework. If you are successful with your geometry homework,
then you will pass the geometry course.
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5-4: Inverses, Contrapositives, and Indirect Reasoning
Objective:
• Learn to write the negation of a statement and the inverse and
contrapositive.
• To use indirect reasoning.
Vocabulary:
• Negation: Has opposite truth value of a given statement.
• Inverse: Negates both hypothesis and conclusion of a conditional
• Contrapositive: The inverse of the converse of a conditional.
• Equivalent Statements: Statements with the same truth value.
Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
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.”
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Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
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.
Converse: If
Contrapositive: If
ABC is isosceles,
ABC is not isosceles,
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then it is equilateral.
then it is not equilateral.
Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
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.”
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Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
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
PQR==60.
60.
Three points that lie
on the same line are
both coplanar and
collinear, so these
two statements do
not contradict each
other.
Three points that lie
on an angle are
coplanar, so these
two statements do
not contradict each
other.
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If three distinct
points are collinear,
they form a straight
angle, so m PQR
cannot equal 60.
Statements II and III
contradict each
other.
Inverses, Contrapositives, and Indirect Reasoning
GEOMETRY LESSON 5-4
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.
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ABC cannot