Chapter 2 – Reasoning and Proof
2.1 Conditional, Converse, Biconditional
Conditional Statements
A conditional statement is a ____________ statement that people can use to ____________ something.
It has two parts, a ________________and a _________________.
When it is written in if-then form, the “________” part contains the hypothesis and the “________” part
contains the conclusion.
example:
If you play basketball at James Kenan, then you have a game on Friday.
Circle the hypothesis and underline the conclusion.
MY example:
Circle the hypothesis and underline the conclusion.
Rewrite the statements in if-then form:
Two points are collinear if they lie on the same
line.
All sharks have a boneless skeleton.
A number divisible by 9 is also divisible by 3.
Conditional statements can be ________ or _________.
If a conditional statement is true, it has to be true in ______ cases, not just sometimes.
Counterexamples
If a conditional statement is ____________, then it should be able to be proven wrong by providing a
_________________________.
Chapter 2 – Reasoning and Proof
Counterexamples in action…
conditional statement – If x2=16, then x=4.
conditional statement – If a number is odd, then it
is divisible by 3.
The conditional statement is false.
The conditional statement is _________.
counterexample – x = -4
counterexample – _________
Converse
The converse of a conditional statement is formed by switching the ______________ and
______________. You leave “if” at the beginning, and “then” in the middle to separate the two parts of
the statement.
Write the converse of this conditional statement:
conditional: If you see lightning, then you hear thunder.
converse:
Converse statements can be _________ or __________.
If a converse is true, it has to be true in _________ cases, not just sometimes.
If a converse is false, you then it should be able to be proven wrong by providing a
_________________.
Biconditional Statements
A biconditional statement is a statement that contains the phrase “___________________.” Writing a
biconditional statement is equivalent to writing a ____________________ AND its ________________.
biconditional statement: We will have practice if and only if it doesn’t rain.
Write the biconditional statement as a conditional statements and its converse.
conditional statement:
converse:
Chapter 2 – Reasoning and Proof
A biconditional statement can be ________ or __________.
If a biconditional statement is true, then the _____________________________ AND
the _______________ must be true.
If a biconditional statement is false, then either the _________________________ or
the _______________ must be false.
2.1 Negation, Inverse, and Contrapositive
Negation
A statement can be altered by _______________. Negation is the negative or _________________ of a
statement.
STATEMENT
NEGATION
m A = 30o
A is acute
Inverse
When you negate the hypothesis and conclusion of a _______________________________, you form
the inverse.
Write the inverse of each conditional statement.
If there is no snow on the ground, then the flowers
are in bloom.
If m GED = 130o, then it is obtuse.
If you are a doctor, then you love science.
Contrapositive
When you negate the hypothesis and conclusion of a ___________________, you form the
contrapositive.
Chapter 2 – Reasoning and Proof
Write the converse, then contrapositive of each conditional statement.
CONDITIONAL STATEMENT
CONVERSE
CONTRAPOSITIVE
If you eat lettuce, then you are
skinny.
If √100 is z, then z = 10.
If an animal is a monkey, then it
has a tail.
Summarizing Logical Statements
conditional statement – ____________
Under these conditions, this will
happen.
converse – ___________
Converse tried to switch it up
by coming out with their
“different” looking sneakers.
inverse – ____________
The prefix in- means not.
contrapositive – _________ AND ___________ Put it all together.
equivalent statements – If one of them is true, the other one is __________. If one of them is false,
then the other one is _____________.
2.4 Algebraic Proofs
Algebraic Properties of Equality
ADDITION PROPERTY OF EQUALITY
SUBTRACTION PROPERTY OF EQUALITY
MULTIPLICATION PROPERTY OF EQUALITY
DIVISION PROPERTY OF EQUALITY
REFLEXIVE PROPERTY OF EQUALITY
SYMMETRIC PROPERTY OF EQUALITY
TRANSITIVE PROPERTY OF EQUALITY
SUBSTITUTION PROPERTY OF EQUALITY
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 and c ≠ 0, then a ÷ c = b ÷ c.
For any real number a, a = a.
If a = b, then b = a.
If a = b and b = c, then a = c.
If a = b, then a can be substituted for b in any
equation or expression.
Chapter 2 – Reasoning and Proof
Properties of equality along with other properties from algebra, such as the distributive property, can be
used to solve equations. For instance, you can use the subtraction property of equality to solve the
equation x + 3 = 7. By subtracting 3 from each side of the equation, you obtain x = 4.
Writing Algebraic Proofs
The following equation is solved for you, step by step. Use the algebraic properties of equality to write
the reasons for step below.
STATEMENTS
5x – 18 = 3x + 2
REASONS
Given
2x – 18 = 2
2x = 20
x = 10
Try one more…
STATEMENTS
55z – 3(9z + 12) = -64
REASONS
Given
55z
2x = 20
x = 10
2.3/2.5/2.6 Geometric Proofs
Properties of Segment Congruence
Chapter 2 – Reasoning and Proof
Writing Geometric Proofs
Use the diagram and the given information to complete the missing steps and reasons in the proof.
given: LK = 5, JK = 5, JK
prove: LK
JL
K
JK
STATEMENTS
REASONS
J
Given
Given
LK = JK
LK
JK
JK
JL
Transitive property of equality
L
Given
Transitive Property of Congruence