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# Chapter 2

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## Josh Truax

on 8 September 2017

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#### Transcript of Chapter 2

Conditional -
an "if-then" statement

Hypothesis -
the portion of the sentence follow the word "if"

Conclusion -
the portion of the sentence following the word "then"
Inductive Reasoning -
reasoning based on observed patterns

Conjecture -
a conclusion reached by using inductive reasoning
Section 2.1 - Patterns and Inductive Reasoning
Section 2.2 - Conditional Statements
Biconditional -
a single, true statement that combines a true conditional and its true converse using "if and only if"

**It is only possible to write a biconditonal if both the conditional and converse are true**
Section 2.3 - Biconditionals and Definitions
Deductive Reasoning -
sometimes called "
logical reasoning
," is the process of reasoning logically from given statements or facts to a conclusion.
2.4 - Deductive Reasoning
Theorem -
a conjecture or statement that is proven to be true

Difference between Postulates and Theorems
Postulates are accepted facts
Theorems require proof
Section 2.6 - Proving Angles Congruent
Vertical angles are congruent.
Theorem 2.1 - Vertical Angles Theorem
Chapter 2
Reasoning and Proof

State the next two terms in each pattern by making a conjecture. Also, state the pattern you observed.

1, 3, 5, 7 ... 1, 4, 9, 16...

M, T, W... O, T, T, F, F, S, S...
It's important to understand that not all conjectures turn out to be true. Sometimes patterns can be observed incorrectly.

Counterexample -
an example that shows the conjecture is false.
Find a counterexample that proves the following conjectures to be false.

If the name of a month starts with the letter J, it is a summer month.

You can connect any three points to form a triangle.

When you multiply a number by 2, the product is greater than either number.
Homework #7 - Section 2.1

Complete the assignment on Math XL

Remember!!! You must earn 100% on the assignment in order to earn full credit on your homework assignment.
Identify the hypothesis and conclusion in each of the following conditional statements.

If an animal is an eagle, then the animal is a bird.

If an angle measures 90, then the angle is a right angle.

The window will break if the wind blows too hard.
Creating Conditional Statements

Convert the following sentences into conditional statements:

Vertical Angles share a vertex.

Dolphins are mammals.
Every conditional statement has a
truth value
associated with it. The conditional is either true or false.

True Conditional -
every time the hypothesis is true, the following conclusion is also true.

False Conditional -
if a counterexample can be found to show the conclusion is false.
Identify the truth value of the following conditional statements. If the conditional is false, give the counterexample.

If a woman is Hungarian, then she is European.

If a number is divisible by three, then it is odd.

If angles form a linear pair, then those angles are supplementary.
Negation -
stating the opposite of a statement
Ex: The sky is blue.
Negation: The sky is not blue.

Converse -
the hypothesis and conclusion of a statement switch spots (the words if and then
do not
move)

Inverse -
both the hypothesis and conclusion of a statement are negated (
)

Contrapositive -
write the converse then negate both the hypothesis and conclusion (
switch and negate
)
Write the converse, inverse, and contrapositive of each statement:

If the figure is a square, then the figure is a quadrilateral.

If a vegetable is a carrot, then it contains beta carotene.
Homework #8 - Section 2.2

Complete assignment on Math XL

Remember!!! You must earn 100% on the assignment in order to earn full credit on your homework assignment.

Steps to Writing a Biconditional
1. Check to see if the conditional statement is true
2. Write the converse and check to see that it is also true
3. If both statements are true, proceed to write the biconditional
4. Use the original conditional statement
5. Cross out the word "
if
"
6. Replace "
then
" with "
if and only if
"
Write the converse of each conditional statement. If both statements are true, write a biconditional.

If the sum of two angles is 180, then the angles are supplementary.

If a figure is a square, then the figure is a quadrilateral.
Take the following conditionals and write the conditional and converse that were used to create the biconditional.

A ray is an angle bisector if and only if it divides an angle into two congruent angles.

Two numbers are reciprocals if and only if their product is 1.
Good Definitions
Good Definition -
a statement that helps you to classify or identify an object.

Three Requirements
1. Uses clearly understood terms
2. Precise (no large/sort of/almost)
3. Reversible (can be written as a biconditional)
Are the following good definitions?

A fish is an animal that swims.

Rectangles have four corners.

Giraffes are animals with very long necks.

A penny is a coin worth one cent.
Homework #9 - Section 2.3

Complete assignment on Math XL

Remember!!! You must earn 100% on the assignment in order to earn full credit on your homework assignment.

Property: Law of Detachment
If the hypothesis of a true conditional is true, then the conditional is true.

Using the Law of Detachment
Identify the hypothesis of the given true conditional. If the second statement matches the hypothesis, then you can make a valid conclusion.
Draw a conclusion from the given statements:

If a student gets an A on a final exam, then the student passes the course.
Fred got an A on the final exam.

If there is lightning, then it is not safe to be outside.
Sylvia sees lightning from the soccer field.

If a figure is a square, then it has sides of equal length.
ABCD has sides of equal length.
Property: Law of Syllogism
Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement.

**Shortcut**
Cross out the matching portions of each conditional statement and combine what is left over into one conditional statement
What can you conclude from the given information?
If a figure is a square, then the figure is a rectangle.
If a figure is a rectangle, then the figure has four sides.

If a whole number ends in 0, then it is divisible by 10.
If a number is divisible by 10, then it is divisible by 5.

If you do gymnastics, then you are flexible.
If you do ballet, then you are flexible.
Homework #10 - Section 2.4

Complete assignment on Math XL (PDF also available)

Remember!!! You must earn 100% on the assignment in order to earn full credit on your homework assignment.
1
2
3
4
m<1 = m<2

m<3 = m<4
Notebook Question #1
Solve for x and y.
(2x + 20)
4x
2y
Theorem 2.2 - Congruent Supplements Theorem
If two angles are supplements of the same angle (or of congruent angles) then the two angles are congruent.
3
1
2
If m<3 + m<1 = 180 and m<3 + m<2 = 180, then m<1 = m<2.
Theorem 2.3 - Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.
2
1
3
If m<1 +m<2 = 90 and m<3 + m<2 = 90, then m<1 = m<3
Theorem 2.4 - Right Angles Theorem
All right angles are congruent.
Theorem 2.5 - Congruent Supplements Theorem
If two angles are congruent and supplementary, then each is a right angle.
Homework #11 - Section 2.6

Complete assignment on Math XL (PDF also available)

Remember!!! You must earn 100% on the assignment in order to earn full credit on your homework assignment.
Notebook Question #1
Notebook Question #2
Notebook Question #3
Notebook Question #1
Notebook Question #2
Notebook Question #3
Notebook Question #1
Notebook Question #2
Notebook Question #3
Notebook Question #1
Notebook Question #2
Notebook Question #3
Use the Law of Syllogism and the Law of Detachment to make conclusions from the following statements.
If you live in the Bronx then you live in New York.
Tracy lives in the Bronx.
If you live in New York, then you live in the 11th state to enter the Union.

If a river is longer than 4000 mi long, then it is longer than the Amazon.
If a river is longer than the Amazon, then it is the longest river in the world.
The Nile River is 4132 mi long.
Notebook Question #2
Solve for each variable.
Notebook Question #3
Solve for each variable.
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