Geometry (H) Lesson 2.2
2.2 Notes: Conditional Statements
Lesson Objectives: In this lesson, we will be learning about conditional statements and their
parts. You should be able to do the following:
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


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
Identify the hypothesis and conclusion of a conditional statement
State the truth value of a conditional statement
Rewrite a statement into a conditional statement
Give the converse, inverse, and contrapositive of a conditional statement
Identify the relationships between the four related conditional statements
Use Venn Diagrams and symbols to represent conditional statements
Conditional Statements
“If you don’t clean your room, then you can’t go to the movies.” Perhaps you’ve heard statements
like these before.
A conditional is an if-then statement. Each conditional has two parts:
Hypothesis:
Conclusion:
Conditional statements are important in math because many of the postulates and theorems we
will learn are written as conditional statements. For instance, “If two angles form a linear pair, then
they are supplementary.”
Example 1: Identify the hypothesis and conclusion in the Linear Pair Postulate given above.
Hypothesis:
Conclusion:
Conditionals in Symbols and Diagrams
Consider the following example: If you live in Mercer Island, then you live in Washington.
hypothesis (p):
conclusion (q):
Symbols
Venn Diagram
Geometry (H) Lesson 2.2
TRY IT
What is the hypothesis and the conclusion of the conditional?
“If an angle measures 130 degrees, then the angle is obtuse.”
Writing a Conditional
We can re-write a statement so that it reads as a conditional.
Example 2: A rectangle has four right angles.
Conditional:
Example 3: A tiger is a mammal.
Conditional:
2. Re-write the following statements as conditional statements:
A: Squares have four sides of equal length.
Conditional:
B: Leprechauns always wear green.
Conditional:
Instances vs. Counterexamples
Consider the following conditional statement: If you have Mr. Shafer for Honors Geometry, then
you have math 2nd period.
Instance: A situation where both the hypothesis and conclusion of a conditional statement are
true.
3. Can you find an instance to the conditional statement given above?
Counterexample: A situation where the hypothesis of a conditional statement is true, but the
conclusion of the conditional statement is false.
4. Can you find a counterexample to the conditional statement give above?
5. Sam is a student in Mr. Shafer’s 6th period Algebra II class. Is Sam a counterexample to the
conditional statement above? Explain.
Geometry (H) Lesson 2.2
Truth Value of a Conditional
The truth value of a conditional statement is either true or false.
Identifying the Truth Value of a Conditional Statement
True: The truth value of a conditional statement is true if...
False: The truth value of a conditional statement is false if...
Example 4: State the truth value of the following conditional: If a number is divisible by 5, then it
ends in a 0.
Truth Value:__________________
5. State the truth value of the following conditional: If an angle’s measure is between 0 and 90
degrees, then the angle is acute.
Truth Value:__________________
Negation
The negation of a statement p is the opposite of the statement. The symbol is ~p and is read “not
p”.
Example 5: State the negation of the statement: The sky is green.
Negation:
Example 6: State the negation of the statement I am not going to the dance on Friday.
Negation:
Example 7: State the negation of the statement: x  3 .
Negation:
Geometry (H) Lesson 2.2
Converses, Inverses, and Contrapositives
Every conditional statement has three related conditional statements.
Statement
How to Write It
Example
Conditional
Use the given
hypothesis and
conclusion.
If mA  100 , then  A is obtuse.
Symbols
Converse
Inverse
Contrapositive
6. Write the converse, inverse, and contrapositive of the following conditional statement:
Conditional: If it is raining, then the ground is wet.
Converse:
Inverse:
Contrapositive:
7. Write the converse, inverse, and contrapositive of the following conditional statement:
Conditional: If you live in Mercer Island, then you live in Washington.
Converse:
Inverse:
Contrapositive:
Geometry (H) Lesson 2.2
8. State the truth value of each of the four related conditional statements in #7.
9. Which of the four related conditional statements is #7 have the same truth value?
10. Which of the other three related conditional statements must have the same truth value as the
original conditional statement? If necessary, use Venn diagrams.
11. Do any of the other related conditional statements always have the same truth value? Explain.
Equivalent Statements have the same truth value.
A conditional and its contrapositive are equivalent statements. They are either both true or both
false. The converse and inverse of a statement are also equivalent statements.
Geometry (H) Lesson 2.2
Assignment 2A
Conditional Statements (2 – 2)
At the end of this assignment, you should be able to do the following:
o
o
o
o
o
Identify the hpyothesis and conclusion of a conditional statement
Re-write a statement as a conditional statement
Identify the truth value of a conditional statement
Write the converse, inverse, and contrapositive of a conditional statement
Use a Venn diagram to write conditional statements
Part I: Practice
1. Identify the hypothesis and conclusion for each conditional.
a. If you want to be fit, then get plenty of exercise.
b. If x + 20 = 32, then x = 12.
c. You can fly if you have wings.
2. Write each sentence as a conditional.
a. Glass objects are fragile.
b. Good weather makes a picnic enjoyable.
c. Three non-collinear points determine a unique plane.
3. Write a conditional statement that each Venn diagram illustrates.
a.
b.
Geometry (H) Lesson 2.2
4. Determine if the conditional is true or false. If it is false, find a counterexample.
a. If you live in a country that borders the United States, then you live in Canada.
b. If an angle measures 80 degrees, then it is acute.
5. For each conditional statement given, write the converse, inverse, and contrapositive. Then
determine the truth value of each statement. If a statement is false, give a counterexample.
a. If a point is in the first quadrant, then both its coordinates are positive.
converse:
inverse:
contrapostive:
b. If you are in Chicago, then you are in Illinois.
converse:
inverse:
contrapostive:
c. If four points are collinear, then they are coplanar.
converse:
inverse:
contrapostive:
Geometry (H) Lesson 2.2
Part II: Application and Problem Solving
6. Write the following postulates as conditionals:
a. Two intersecting planes meet in exactly one line.
b. Two intersecting lines meet in exactly one point.
7. For each statement, write the converse. If the converse is true, write true; if the converse is
false, provide a counterexample.
a. If x  3  15, then x = 18.
b. If y is negative, then –y is positive.
c. If x  6, then x  6.
2
d. If x < 0, then x  0.
2
e. If x = 2, then x  4.
3
f. If x < 0, then x  0.
8. Alf sees an ad that states, “You want to look good at the beach this summer. Join 24 Hour
Fitness.” Alf figures, “I am going to join 24 Hour Fitness so I can look good at the beach.”
a. Write the ad as a conditional statement.
b. Write Alf’s statement as a conditional statement.
c. Explain why the two statements do not have the same meaning.
9. Angie saw an ad that stated “If you don’t drink Muscle Rex, then you won’t build muscles.” Angie
bought Muscle Rex and drank it, and nothing happened. She sent an e-mail to the company asking
for her money back. The company would not refund her money. They claimed that her reasoning
was faulty. Using one or more of the terms converse, inverse, or contrapositive, decide whether
Angie deserves a refund.
Geometry (H) Lesson 2.2
10. Draw a Venn diagram to illustrate the following statement:
If an angle measures 100 degrees, then it is obtuse.
11. Write three separate conditional statements that the Venn diagram illustrates.
12. Write a statement beginning with all, some, or no to match each Venn diagram.
a.
b.
c.
13. State the truth values of the following conditional statement. Justify your answer!
If the moon is made of cheese, then the sun is made of crackers.
14. Fill in the blank with always, sometimes, or never to make the following statement true.
If a conditional is true, then its converse is ______________________ true.