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Section 2.1 – Conditional Statements
Conditional Statement – A logical statement containing 2 parts, a hypothesis and
a conclusion.
If – Then Form
If Then -
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Example 1- Write the following as a conditional statement in if-then form.
a. A number divisible by 9 is also divisible by 3.
b. All sharks have a boneless skeleton.
c. Two points are collinear if they lie on the same line.
Recall:
Counterexample
– an example showing a statement is false.
Converse- switching the hypothesis (if) and conclusion (then).
Example 2 – Find a counterexample and then write the converse
a. If x2 = 16, then x = 4
b. If a number is odd, then it is divisible by 3
Negation – the negative of the original conditional statement
*** adding “not” to the
statement
Inverse – Negating the hypothesis (if) and conclusion (then)
Contrapositive – negating the converse of a conditional statement (flip and negate)
Example 3 – Write the inverse, converse, and contrapositive of the statement.
If an animal is a fish, then it can swim.
Inverse .
Converse -
.
Contrapositive-
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Equivalent Statements- When two statements are both true or false so, original =
contrapositive and inverse = converse.
Point-Line-Plane Postulates
 Through any 2 points there is exactly 1 line

A line consists of at least 2 points

Through any 3 noncollinear points, there is a plane

A Plane consists of at least 3 noncollinear points

If 2 points are in a plane, then the line containing them are in the same plane

If 2 planes intersect, their intersection is a line
Section 2.2 – Definitions & Biconditional Statements
Perpendicular lines- 2 lines that form a right angle
*Every definition can be written as 2 different conditional statements, If… Then…
1. The original statement and 2. Its converse.
Biconditional Statement – a conditional statement that contains “if and only if” (Can be
denoted as iff).
Ex. 1- Rewrite the biconditional statement as a conditional statement and then write its
converse. Decide if it is a true biconditional statement. (Meaning both are true.)
Two lines intersect if and only if their intersection is exactly one point.
X=3 if and only if x2 = 9
Ex. 2- The following statements are true. Write the converse and decide whether it is
true of false. If the converse is true, combine it with the original to form a
biconditional.
If x2 = 4 , then x = 2 or –2
If a number ends is 0, then the number is divisible by 5
Section 2.3 – Deductive Reasoning
Symbol NotationP
Q
is the same as
P
is the same as
Converse:
Inverse:
Contrapositive:
Biconditional:
.
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Ex. 1- p = a quadrilateral has 4 right angles
q = the quadrilateral is a square
a. Write the converse and then the contrapositive
Ex. 2- p = the value of x is –4
q = the square of x is 16
a. Write the inverse, converse, and the decide if the biconditional is true.
Inductive Reasoning – a process that relies on previous examples and patterns for making
a conjecture
Deductive Reasoning – a process based on logic and order
Two Laws of Deductive Reasoning
1. Law of Detachment-
2. Law of Syllogism-
Ex. 3 – Using the Law of Detachment, state whether the argument is valid.
a. Mike knows that if he does not do his chores in the morning, then he will not be
allowed to play video games later that day. Mike does not play video games on
Friday afternoon. So Mike did not do his chores on Friday morning.
b. If 2 angles are vertical, then they are congruent. Angle ABC and Angle DBE are
vertical. Conclusion: Angle DBE and Angle ABC are congruent.
c. Sarah knows that all sophomores take driver education in her school. Hank
takes driver education. So Hank is a sophomore.
Ex. 4 – Using the Law of Syllogism, write a conditional statement that can be written
from the following statements.
a. If it’s Wednesday, then it’s late start. If it’s late start, then school starts at
9:40.
b. If a fish can swim at 110 km/h, then it’s a sailfish. If a fish swims at 68 mi/h,
then it swims at 110 km/h.
Section 2.4 – Reasoning with Properties from Algebra
Algebraic Properties of Equality (Used in equations)
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 -
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Ex. 1 – Solve for the variable and write a reason for each step.
a. 3x + 12 = 8x – 18
b. 55z – 3(9z + 12) = -64
Geometric Properties of Equality
Property
Segment Length
Reflexive P of E
Symmetric P of E
Transitive P of E
Angle Measure
Section 2.5 – Proving Statements about Segments
Postulate- Rules that are accepted without proof. Also called axioms
Theorem- A true statement that follows as a result of other true statements
To prove something is true
To prove something is false
Paragraph proof- a proof written in paragraph form
Properties of Congruence
Reflexive Property of CongruenceSymmetric Property of CongruenceTransitive Property of Congruence-
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1. Given: EF = GH
Prove: EG  FH
E
F
G
H
Section 2.6 – Proving Statements about Angles
The Reflexive, Symmetric, and Transitive Properties also apply to Angle Congruence.
Right Angle Congruence Theorem – all right angles are congruent
Congruent Supplements Theorem – If two angles are supplementary to the same angle, then
they are congruent
Congruent Complements Theorem – If two angles are complementary to the same angle, then
they are congruent
Linear Pair Postulate – If two angles form a linear pair, then they are supplementary
Vertical Angles Theorem – Vertical angles are congruent