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WARM-UPS
Identify the hypothesis and conclusion of each statement.
1. If it rains on Monday, then I will stay home.
H: it rains on Monday; C: I will stay home
2. If x – 3 = 7, then x = 10.
H: x – 3 = 7; C: x = 10
3. If a polygon has six sides, then it is a hexagon.
H: a polygon has six sides; C: it is a hexagon
Lesson 2-1 Inductive Reasoning and Conjecture
Lesson 2-2 Logic
Lesson 2-3 Conditional Statements
Lesson 2-4 Deductive Reasoning
Lesson 2-5 Postulates and Paragraph Proofs
Lesson 2-6 Algebraic Proof
Lesson 2-7 Proving Segment Relationships
Lesson 2-8 Proving Angle Relationships
Example 4 Related Conditionals
OBJECTIVE: To write the converse, inverse, and
contrapositive of if-then statements (2.8.11Q) (M8.D.1.1)
KEY CONCEPTS:
Conditional (p -> q); Converse (q->p); Inverse (~p->~q);
Contra-positive (~q -> ~p)
Related Conditions
Conditional Converse Inverse Contrapositive
p q
p -> q
q -> p ~p -> ~q
~q -> ~p
T T
T
T
T
T
T F
F
T
T
F
F T
T
F
F
T
F F
T
T
T
T
Write the converse, inverse, and contrapositive of the
statement All squares are rectangles. Determine
whether each statement is true or false. If a statement
is false, give a counterexample.
First, write the conditional in if-then form.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.
Write the converse by switching the hypothesis and
conclusion of the conditional.
Converse: If a shape is a rectangle, then it is a square.
The converse is false. A rectangle with = 2
and w = 4 is not a square.
Inverse:
If a shape is not a square, then it is not a
rectangle. The inverse is false. A 4-sided
polygon with side lengths 2, 2, 4, and 4 is
not a square, but it is a rectangle.
The contrapositive is the negation of the hypothesis and
conclusion of the converse.
Contrapositive: If a shape is not a rectangle, then it is
not a square. The contrapositive is true.
Write the converse, inverse, and contrapositive of the
statement The sum of the measures of two
complementary angles is 90. Determine whether each
statement is true or false. If a statement is false, give
a counterexample.
Answer: Conditional: If two angles are complementary,
then the sum of their measures is 90; true.
Converse: If the sum of the measures of two
angles is 90, then they are complementary;
true.
Inverse: If two angles are not complementary,
then the sum of their measures is not 90; true.
Contrapositive: If the sum of the measures of
two angles is not 90, then they are not
complementary; true.
Write the conditional and converse of the statement
The sum of the measures of two complementary
angles is 90. Determine whether each statement is
true or false. If a statement is false, give a
counterexample.
CONDITIONAL: If two angles are complementary, then
the sum of their measures is 90; true.
CONVERSE: If the sum of the measures of two angles
is 90, then they are complementary; true.
Write the inverse and contrapositive of the
conditional statement: If two angles are complementary,
then the sum of their measures is 90. Determine
whether each statement is true or false. If a statement
is false, give a counterexample.
INVERSE: If two angles are not complementary,
then the sum of their measures is not 90;
true.
CONTRAPOSITIVE: If the sum of the measures of
two angles is not 90, then they are not
complementary; true.
REVIEW
Write the converse, inverse, and
contrapositve of the following statement,
then determine if each is true or false. If
false, find a counterexample.
If plants have water, then they will grow.
CONVERSE: If plants grow, then they have water; true.
INVERSE: If plants do not have water, then they will
not grow; true.
CONTRAPOSITIVE: If plants do not grow, then they do
not have water. False; they may have been killed by
overwatering.
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