GPS Geometry: Conditional Statements Notes
In this lesson you will study a type of logical statement called a conditional statement. A
conditional statement has two parts, a hypothesis and a conclusion. When the statement is
written in if-then form, the "if" part contains the hypothesis and the "then" part contains the
conclusion. Here is an example:
If it is noon in Georgia, then it is 9 A.M. in California.
Hypothesis
Conclusion
I. Rewrite a statement in if-then form
a. Two points are collinear if they lie on the same line. If this statement is rewritten as a
conditional (if-then) statement, it would be:
If two points lie on the same line, then they are collinear.
b. All sharks have a boneless skeleton. If this statement is rewritten as a conditional (if then)
statement, it would be:
If a fish is a shark, then it has a boneless skeleton.
c. A number divisible by 9 is also divisible by 3. If this statement is rewritten as a conditional (ifthen) statement, it would be:
If a number is divisible by 9, then it is divisible by 3.
II. Evaluate a conditional statement for truth
Conditional statements can be either true or false. To show that a conditional statement is true,
you must present an argument that the conclusion follows for all cases that fulfill the hypothesis.
To show that a conditional statement is false, you only need to describe a single counterexample
that shows the statement is not always true.
a. Write a counterexample to show that the following conditional statement is false.
If x2 =16, then x = 4.
A counterexample is x = -4. The hypothesis is true, because (4)2 =16, but the conclusion is false in
this case. This proves the conditional statement is false.
III. Write the converse of a conditional statement
The converse of a conditional statement is formed by switching the hypothesis and the
conclusion. Here is an example:
Statement: If you see lightning, then you hear thunder.
Converse: If you hear thunder, then you see lightning.
IV: Write the inverse of a conditional statement
The inverse of a conditional statement is formed by negating the hypothesis and the conclusion.
Here are two examples:
Statement: If you see lightning, then you hear thunder.
Inverse: If you do not see lightning, then you do not hear thunder.
Statement: If you tell the truth, then you don't have to remember a lie.
Inverse: If you do not tell the truth, then you have to remember a lie.
V: Write the contrapositive of a conditional statement
The contrapositive of a conditional statement is formed by negating the hypothesis and
conclusion of the converse. In other words, both switching and negating the original conditional
statement. ***NOTE*** The contrapositive always follows the truth value of the original
conditional statement. If the statement was true, then the contrapositive is true. Here is an
example:
Statement: If two segments are congruent, then they have the same length.
Contrapositive: If two segments do not have the same length, then they are not congruent.
YOU TRY: Write the statement in if-then form, then write the converse, inverse, and
contrapositive.
Flowers do not bloom if there is snow on the ground.
GPS Geometry: Conditional Statements WS
Name ____________________
I. Rewrite the conditional statement in "if-then" form.
1. You have a fever if your body temperature is 103°F.
_________________________________________________________________________
2. A car with leaking antifreeze has a problem.
_________________________________________________________________________
II. Identify the hypothesis and conclusion of the following conditional statement. Then write the
converse, inverse, and contrapositive of the conditional statement.
3. "If you like soccer, then you go to soccer games.
(a) Identify the hypothesis: ___________________________________________________
(b) Identify the conclusion: ____________________________________________________
(c) Write the converse: _______________________________________________________
(d) Write the inverse: ________________________________________________________
(e) Write the contrapositive: ___________________________________________________
_________________________________________________________________________
4. "If an object weighs one ton, then it weighs 2000 pounds."
(a) Identify the hypothesis: ___________________________________________________
(b) Identify the conclusion: ____________________________________________________
(c) Write the converse: _______________________________________________________
(d) Write the inverse: ________________________________________________________
(e) Write the contrapositive: ___________________________________________________
_________________________________________________________________________
III. Decide whether the statement is true or false. If it is false, provide a counterexample.
5. A point may lie in more than one plane.
6. If it is snowing, the temperature is below freezing.
7. If x4 = 81. then x must equal 3.
8. If four points are collinear, then they are coplanar.
IV. Fill in the blank with the correct answer, then sketch a diagram to illustrate your answer.
9. If two lines intersect, then their intersection is ________ point.
10. Through any _________ points there is exactly one line.
11. If two points lie in a plane, then the __________ containing those points lies in the plane.
12. If two planes intersect, then their intersection is a ________ .