1. Conditional Statements
Section 2–1
Geometry PreAP, Revised ©2013
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2. Conditionals Chart – Page 2
Symbol
Concept p
Original Hypothesis q
Original Conclusion →
“Implies” ~
“Not”
p → q
“p implies q” “if p, then q” ~p
“not p”
Statement
Symbols
Conditional
p → q
Converse
q → p
Inverse
~p → ~q
Contrapositive
~q → ~p
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3. 2-1: Logic with Inductive Reasoning
Conditionals – Page 3
If p, then q
Notation: p → q (if p then q)
HYPOTHESIS – all words after “If” and before comma
CONCLUSION – all words after “Then”
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4. Example 1
In this statement, “If a polygon is a hexagon, then it has exactly six sides” state the hypothesis and conclusion
Subject
Predicate
A polygon is a hexagon is a polygon with six sides.
If it is a hexagon, then it is a polygon with six sides.
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5. 2-1: Logic with Inductive Reasoning
Example 2
In this statement, “a number is a rational number if it is an integer.” State the hypothesis and conclusion.
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6. 2-1: Logic with Inductive Reasoning
Example 3
Rewrite this conditional statement in if-then form, “All 90° angles are right angles.”
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Your Turn
Write an example of a conditional statement (if  then) about a topic in geometry.
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Truth Values – Page 5
Determining whether a statement is TRUE or FALSE.
True: To prove that the conclusion is true every time the hypothesis is satisfied.
False: To prove if it is false, one has to find an example in which the conclusion is not true when the hypothesis is satisfied.
A conditional is FALSE ONLY when hypothesis is true and conclusion is false.
To show false you need a COUNTEREXAMPLE.
Counterexample – example that proves a statement is false.
Related conditional statements that have the same TRUTH VALUE are logically equivalent.
Create a table (can all be T or all F)
Conditional and Contrapositive always the same
Converse and Inverse always the same
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Example 4
In this statement, “If this month is August, then next month is September” state whether the conditional statement is true. If it is false, provide a counterexample.
When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true.
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Example 5
In this statement, “If n2 = 144, then n = 12” state whether the conditional statement is true. If it is false, provide a counterexample.
When the hypothesis is true, the conclusion is also false because if n = –12, n2 also equals 144. So the conditional is false.
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Your Turn
In this statement, “If it is February, then there are only 28 days in the month” state whether the conditional statement is true. If it is false, provide a counterexample.
When the hypothesis is true, the conclusion is also false because if February can have 29 days in a leap year. So the conditional is false.
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12. Conditionals Chart – Page 2
Symbol
Concept p
Original Hypothesis q
Original Conclusion →
“Implies” ~
“Not”
p → q
“p implies q” “if p, then q” ~p
“not p”
Statement
Symbols
Conditional
p → q
Converse
q → p
Inverse
~p → ~q
Contrapositive
~q → ~p
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13. 2-1: Logic with Inductive Reasoning
Conditionals – Page 3
If p, then q
Notation: p → q (if p then q)
HYPOTHESIS – all words after “If” and before comma
CONCLUSION – all words after “Then”
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14. Example 6
In this statement, “If a polygon is a hexagon, then it has exactly six sides” state the hypothesis and conclusion
Subject
Predicate
A polygon is a hexagon is a polygon with six sides.
If it is a hexagon, then it is a polygon with six sides.
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Example 7
In this statement, “a number is a rational number if it is an integer.” State the hypothesis and conclusion.
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Example 8
Rewrite this conditional statement in if-then form, “All 90° angles are right angles.”
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Your Turn
Write an example of a conditional statement (if  then) about a topic in geometry.
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18. Converse, Inverse, and Contrapositive – Page 4
A. Converse
The CONVERSE is formed by REVERSING the hypotheses and conclusion.
Notation: q → p (if q then p)
B. Inverse
The INVERSE is formed by NEGATING the hypotheses and conclusion.
Notation: ~p → ~q (if not p then not q)
C. Contrapositive
The CONTRAPOSITIVE is formed by REVERSING and NEGATING the hypotheses and conclusion.
Notation: ~q → ~p (if not q then not p)
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19. Example 9
In this statement, “If a polygon is a hexagon, then it has six sides” state the converse and determine whether it is true of false.
If it is a hexagon, then it is a polygon with six sides.
If a polygon has exactly six sides, then it is a hexagon.
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Example 10
In this statement, “All squares are rectangles” state the converse and determine whether it is true or false.
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Your Turn
In this statement, “If an animal is an adult insect, then it has six legs” state the converse and determine whether it is true or false.
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22. Example 11
In this statement, “If a polygon is a hexagon, then it has six sides” state the inverse and determine whether it is true of false.
If it is a hexagon, then it is a polygon with six sides.
If the polygon is NOT a hexagon, then it is NOT a polygon with six sides.
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Example 12
In this statement, “If a number is even then it is divisible by two” state the converse and determine whether it is true or false.
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Your Turn
In this statement, “If an animal is an adult insect, then it has six legs” state the inverse and determine whether it is true or false.
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25. Example 13
In this statement, “If a polygon is a hexagon, then it has six sides” state the contrapositive and determine whether it is true of false.
If it is a hexagon, then it is a polygon with six sides.
If a polygon does NOT have exactly six sides, then it is NOT a hexagon.
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Example 14
In this statement, “If you are a guitar player, then you are a musician,” state the contrapositive and determine whether it is true or false.
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Your Turn
In this statement, “If mA = 99°, then A is obtuse” state the contrapositive and determine whether it is true or false.
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Example 15
Fill out these statements below and identify whether it is true or false.
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Your Turn
In this statement, “If I measure two angles which are acute, then they are congruent” state whether the conditional statement is true. If it is false, provide a counterexample.
You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false.
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Assignment
Page 77 8, all, 16, 20-23, 25, 28, 31, 37, 38
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