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G E O M E T R Y
CHAPTER 2
REASONING AND PROOF
Notes & Study Guide
CHAPTER 2 NOTES
2
TABLE OF CONTENTS
CONDITIONAL STATEMENTS ............................................................................ 3
DEFINTIONS & BICONDITIONAL STATEMENTS .............................................. 6
DEDUCTIVE REASONING ................................................................................... 9
REASONING WITH PROPERTIES OF ALGEBRA ........................................... 12
HOMEWORK EXAMPLES ................................................................................. 15
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CHAPTER 2 NOTES
CONDITIONAL STATEMENTS
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INTRODUCTION
Chapter 2 may be the most difficult chapter in the course because it deals with logic and
reasoning. These topics can be difficult to describe to students who may not have welldeveloped logic/common sense skills.
What we will discuss are the different types of statements (conditionals) that are used in
Geometry that help establish the rules and laws we’ll use.
We will also examine the step-by-step process (deductive reasoning) that
mathematicians use to prove the statements that they make are true (proofs).
CONDITIONAL STATEMENTS
Almost all the rules and laws that we use in Geometry are written as conditional
statements.
In general, all conditional statements state that “if A happens, then B is true.”
This is why we often call them if–then statements.
In other words, if all the requirements in A are met, then we accept whatever B
says as true.
THE TWO PARTS
All conditional statements contain two parts…
Hypothesis  the first part of the statement with the “if”.
Conclusion  the second part of the statement with the “then”.
CONDITIONAL STATEMENT EXAMPLES
If an animal has stripes, then the animal is a tiger.
If the temperature drops below 70, then the pool will close.
If a trapezoid is isosceles, then each pair of base angles is congruent.
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
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CONDITIONAL STATEMENTS
HOW CONDITIONAL STATEMENTS WORK
When everything here happens…
…everything here can be accepted.
If a geometric shape has 3 sides, then the shape is a triangle.
If today is Wednesday, then tomorrow is Thursday.
If Tommy’s grade is higher than 90%, then he will receive an “A”.
Conclusions are not accepted unless all of
the hypothesis requirements are met!
TYPES OF CONDITIONAL STATEMENTS
There are four types of conditional statements. The conditional itself is the first
type. The other three are created directly from the conditional.
They are called converses, inverses and contrapositives.
IMPORTANT! When we write these statements, we DO NOT care if they are true
or if they even make sense. We’ll deal with that later.
 CONVERSE
The converse of a conditional statement is formed by simply switching the
hypothesis and the conclusion.
Conditional  If the wind is blowing, then the kite will fly.
Converse  If the kite will fly, then the wind is blowing.
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CONDITIONAL STATEMENTS
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 INVERSE
The inverse of a conditional statement is formed by writing its’ negation. This
means the negative (or opposite) of something. To do it, we use the word “not”.
Conditional  If the wind is blowing, then the kite will fly.
Inverse  If the wind is NOT blowing, then the kite will NOT fly.
 CONTRAPOSITIVE
The contrapositive of a conditional statement is formed by writing its’ negation
AND switching its hypothesis and conclusion. It is a combo of a converse and
an inverse.
Conditional  If the wind is blowing, then the kite will fly.
Contrapositive  If the kite will NOT fly, then the wind is NOT blowing.
 SUMMARY OF CONDITIONAL STATEMENTS
Standard Notation
(written or spoken)
Symbol
Notation
CONDITIONAL
“If p, then q.”
p q
CONVERSE
“If q, then p.”
q p
INVERSE
“If not p, then not q.”
~p  ~q
CONTRAPOSITIVE
“If not q, then not p.”
~q  ~p
The symbol notation is used as a more convenient way to write conditional
statements. Assign a variable (p and q) to each part and use a squiggle.
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CHAPTER 2 NOTES
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DEFINITIONS AND BICONDITIONALS
 BICONDITIONAL STATEMENTS
A conditional statement and its’ converse can be combined into one larger
statement that is known as a biconditional.
Biconditionals can be identified by their use of the phrase “if and only if”.
Conditional  If the wind is blowing, then the kite will fly.
Converse  If the kite will fly, then the wind is blowing.
Biconditional  The wind is blowing if and only if the kite will fly.
STEPS TO CREATE A BICONDITIONAL
 Start with the original statement
 Drop the “IF”
 Replace the “THEN” with “IF AND ONLY IF”
EXAMPLES
If you see lightning, then you hear thunder.
You see lightning if and only if you hear thunder.
If today is Saturday, then Troy is working at the restaurant.
If the measure of angle A is 30°, then angle A is acute.
If two lines intersect, then their intersection is exactly one point.
The phrase “if and only if” can be shortened using IFF
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DEFINITIONS AND BICONDITIONALS
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 PROVING CONDITIONAL STATEMENTS TRUE
Conditional statements can either be true or false.
To prove one is true, you must present an argument that shows the conclusion is
true for ALL possible hypotheses (proof).
To prove one is false, you must come up with a single example hat shows the
statement is not true (counterexample).
This process is the same as the counterexample definition in Chapter 1.
EXAMPLES
Show that each statement is false by providing a counterexample.
If Calvin’s hair is wet, then he went swimming in the pool
If a geometric shape has four sides, then it is a rectangle.
If Tracy’s car won’t start, then the car is out of gas.
 DEFINITIONS
A biconditional statement that is true in both directions (conditional and
converse) is classified as a definition.
If either direction is false, then it cannot be a definition. It is still a biconditional
though.
Definitions are important. Since they are true forwards and backwards, they can
be counted on to work all the time. There is no chance of finding a
counterexample that makes them fail.
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CHAPTER 2 NOTES
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DEFINITIONS AND BICONDITIONALS
EXAMPLES
An animal is a tiger IFF it has stripes.
COND: If an animal is a tiger, then it has stripes. TRUE
CONV: If an animal has stripes, then it is a tiger. FALSE
The converse of this example is false, so the statement is NOT a definition.
Two lines are perpendicular IFF they intersect at 90°.
COND: If 2 lines are perpendicular, then they intersect at 90°. TRUE
CONV: If 2 lines intersect at 90°, then they are perpendicular. TRUE
Both statements in this example as true, so the statement IS a definition.
Three points are collinear IFF they lie on the same line.
You can become President of the U.S. IFF you are 35 years old.
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CHAPTER 2 NOTES
DEDUCTIVE REASONING
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INTRODUCTION
In this section we cover the concept of deductive reasoning and how we use it to reach
conclusions when presented with a collection of statements.
We also will cover the symbol notation used for conditional statements.
 SYMBOL NOTATION
In symbol notation for conditionals, the letters p and q are used to represent the
hypothesis and the conclusion in a conditional statement.
An arrow represents the “if-then”. A double arrow means “if and only if”.
Conditional
Converse
Inverse
Contrapositive
Biconditional
pq
qp
~p  ~q
~q  ~p
p  q
 DEDUCTIVE REASONING
Deductive reasoning is the process of using facts, definitions and accepted
truths, written in a logical order (argument), to reach a conclusion.
Inductive: conclusions made by observing patterns
Deductive: conclusions made by using facts or definitions
There are two methods (laws) of applying deductive reasoning.
 The Law of Detachment
 The Law of Syllogism
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DEDUCTIVE REASONING
 THE LAW OF DETACHMENT
The Law of Detachment occurs when the hypothesis of a given conditional
statement has been confirmed by a second statement (fact/definition).
When this confirmation happens, the conclusion is accepted to be true.
EXAMPLES: What can be concluded from each pair of statements?
(1) If Mike visits New York, then he will spend a day with his aunt & uncle.
(2) Mike visited New York last month.
(C) Mike spent a day with his aunt & uncle.
(1) If Bill gets a “C” on his test, then he will get an “A” for the quarter.
(2) Bill got a 68% on his test.
(C)
 THE LAW OF SYLLOGISM
The Law of Syllogism uses two or more conditional statements at once so that
the conclusion of one statement matches the hypothesis of the other.
These matching pieces “cancel” each other out and create one final conditional.
EXAMPLES: What can be concluded from each pair of statements?
(1) If Mike visits New York, then he will spend a day with his aunt & uncle.
(2) If Mike spends a day with his aunt & uncle, then he will go to bed early.
(C) If Mike visits New York, then he will go to bed early.
(1) If the cat catches the mouse, then the dog will bark loudly.
(2) If the dog barks loudly, then the parrot will sneeze.
(C)
Think of the matching pieces as the links that chain the statements together.
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MORE EXAMPLES
The Shady Oaks apartments do not allow dogs.
Gloria lives in a Shady Oaks apartment.
If Jack’s wife cheats on him, then he will step in front of a bus.
If Jack steps in front of a bus, then Bonnie will be late for work.
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If Alberto finds a summer job, then he will buy a car.
If Alberto buys a car, then his girlfriend will dump him.
 LOGIC CHAINS
When the Law of Syllogism involves 3 or more statements it creates what is
called a logic chain.
Each matching “if-then” pair will cancel each other out.
In the example shown, the B’s, C’s and D’s do this.
After all the canceling happens, whatever’s left is the final
conclusion. In this case that’s “If A, then E”.
If A, then B.
If B, then C.
If C, then D.
If D, then E.
EXAMPLE
If Charlie finishes his homework, then he can play video games.
If Charlie can play video games, then he will stare at the TV too long.
If he stares at the TV too long, then his eyeballs will fall out.
If his eyeballs fall out, then Charlie will fail his driver’s test.
If Charlie finishes his homework, then he will fail his driver’s test.
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REASONING WITH PROPERTIES FROM ALGEBRA
ALGEBRAIC PROPERTIES OF EQUALITY
Surprisingly, geometry makes regular use of the standard properties of Algebra.
They are useful in building logical arguments (proofs) to back up our statements.
The properties of Equality all do the same thing…they say “whatever you do to
one side, you do to the other”. There is a property for each operation.
Addition:
Subtraction:
Multiplication:
Division:
THE FOUR PROPERTIES OF ALGEBRA
If A = B, then A + C = B + C
If A = B, then A – C = B – C
If A = B, then AC = BC
If A = B, then A/C = B/C
There are three other important properties that are used in Geometry also…
REFLEXIVE
A=A
This one just means that all values are equal to themselves.
SYMMETRIC
If A = B, then B = A
This one says that any equation is reversible.
TRANSITIVE
If A = B and B = C, then A = C
This one says that if two things are equal to the same third thing, then those two
things are equal to each other.
EXAMPLES: Identify the Property shown.




If x = 5, then x + 4 = 5 + 4.
If AB = BC, then BC = AB.
If x = 12, then 5x = 60.
If x – 3 = 12 and x – 3 = c + 6, then x – 3 = c + 6.
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 LOGICAL ARGUMENTS (PROOFS)
When you put a series of statements together, in a logical order, that’s called a
logical argument. In math, we call them proofs.
The final conclusion of the proof is the thing you’re trying to prove is true.
 TYPES OF PROOFS
Paragraph Proof – written using full sentences as if you were verbally trying to
prove something to someone. (Like a lawyer in a courtroom does)
Two column Proofs – written as a very organized small table that lists the
statements made and the reasons that support them
STATEMENTS
1. Given Info
2. Statement 2
3. Statement 3
4.
5.
6. Final Conclusion
REASONS
Given
Reason 2
Reason 3
Final Reason
 ALGEBRAIC PROOFS
Anytime you have solved an algebra equation and shown your work, you have
done a proof.
The steps of the work are the statements. The only thing you don’t usually do is
write down the reasons (the Properties that allow you to do your work).
EXAMPLE: Solve 5x – 18 = 3x + 2
STATEMENTS
1. 5x – 18 = 3x + 2
2. 2x – 18 = 2
3. 2x = 20
4. x = 10
REASONS
Given
Subtraction prop of =
Addition prop of =
Division prop of =
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REASONING WITH PROPERTIES FROM ALGEBRA
MORE EXAMPLES
Write an algebraic proof (in 2-column format) for each of the following…
Solve: 2(3x + 1) = 5x + 14
STATEMENTS
1.
2.
3.
4.
5.
6.
REASONS
Solve: 8x – 5 = -2x – 15
STATEMENTS
1.
2.
3.
4.
5.
6.
REASONS
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CHAPTER 2 NOTES
ADDITIONAL NOTES & EXAMPLES
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CHAPTER 2 NOTES
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ADDITIONAL NOTES & EXAMPLES
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CHAPTER 2 NOTES