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Warm Up

Find the distance and midpoint between the two points below

Distance:

**Remember: AB = distance between A and B**

AB = length of

= segment between A and B (Notation)

Distance: on a # line:

on a coordinate plane:

Pythagorean Theorem or

in 3-d:

Homework Check

1. sqrt(41) = 6.4 2. (6.5, 6)

2-1 Conditional Statements

- Objectives
- To recognize conditional statements
- To write converses of conditional statements

If-Then Statements

- Real World Example:
- “If you are not completely satisfied, then your money will be refunded.”
- Another name of an if-then statement is a conditional.
- Parts of a Conditional:
- Hypothesis (after “If”)
- Conclusion (after “Then”)

“If you are not completely satisfied, then your money will be refunded.”

(hypothesis) (conclusion)

Identifying the Parts

- Identify the hypothesis and the conclusion of this conditional statement:
- If it is Halloween, then it is October
- Hypothesis: It is Halloween
- Conclusion: It is October

Writing a Conditional

- Write each sentence as a conditional:
- A rectangle has four right angles

“If a figure is a rectangle, then it has four right angles.”

- An integer that ends with 0 is divisible by 5

“If an integer ends with 0, then it is divisible by 5.”

Truth Value

A conditional can have a truth value of true or false.

To show that a conditional is true, you must show that every time the hypothesis is true, the conclusion is also true.

To show that a conditional is false, you need to only find one counterexample

Example

- Show that this conditional is false by finding a counterexample
- “If it is February, then there are only 28 days in the month”
- Finding one counterexample will show that this conditional is false
- February 2012 is a counterexample because 2012 was a leap year and there were 29 days in February

Converses

- The converse of a conditional switches the hypothesis and the conclusion
- Example
- Conditional: “If two lines intersect to form right angles, then they are perpendicular.”
- Converse: “If two lines are perpendicular, then they intersect to form right angles.”

Example

- Write the converse of the following conditional:
- “If two lines are not parallel and do not intersect, then they are skew”
- “If two lines are skew, then they are not parallel and do not intersect.”

Are all converses true?

- Write the converse of the following true conditional statement. Then, determine its truth value.
- Conditional: “If a figure is a square, then it has four sides”
- Converse: “If a figure has four sides, then it is a square”
- Is the converse true?
- NO! A rectangle that is not a square is a counterexample!

Assessment Prompt

- Write the converse of each conditional statement. Determine the truth value of the conditional and its converse.
- If two lines do not intersect, then they are parallel
- Converse: “If two lines are parallel, then they do not intersect.”
- Conditional is false
- Converse is true
- If x = 2, then |x| = 2
- Converse: “If |x| = 2, then x = 2”
- Conditional is true
- Converse if false

5-4 Inverses and Contrapositives

- Objectives
- To write the negation of a statement
- To write the inverse and contrapositive of a conditional statement

5-4 Inverses and Contrapositives

- Is the statement, “Knightdale is the capital of North Carolina,” true or false?
- False!
- The negationof a statement is a new statement with the opposite truth value
- The negation, “Knightdale is not the capital of North Carolina” is true

Examples

- Write the negation of each statement.
- Statement: ABC is obtuse

Negation: ABC is not obtuse

- Statement: mXYZ > 70

Negation: mXYZ is not more than 70

Inverse versus Contrapositive

Conditional: If a figure is a square, then it is a rectangle.

Definition: The inverse of a conditional statement negates both the hypothesis and the conclusion

Inverse: If , then

Definition: The contrapositive of a conditional statement switches the hypothesis and the conclusion and negates both.

a figure is not a square

it is not a rectangle

NEGATION!

NEGATION!

- Contrapositive: If , then

it is not a square

a figure is not a rectangle

NEGATION!

NEGATION!

Equivalent Statements

A conditional statement and its converse may or may not have the same truth values.

A conditional statement and its inverse may or may not have the same truth values

HOWEVER, a conditional statement and its contrapositive will ALWAYS have the same truth value. They are equivalent statements.

Equivalent Statementshave the same truth value

2-2 Biconditionals and Definitions

- Objectives
- To write biconditionals
- To recognize good definitons

2-2 Biconditionals and Definitions

When a conditional and its converse are true, you can combine them as a true biconditional. This is a statement you get by connecting the conditional and its converse with the word and.

You can also write a biconditional by joining the two parts of each conditional with the phrase if and only if

A biconditional combines p → q and q → p as p ↔ q.

Example of a Biconditional

- Conditional
- If two angles have the same measure, then the angles are congruent.
- True
- Converse
- If two angles are congruent, then the angles have the same measure.
- True
- Biconditional
- Two angles have the same measure if and only if the angles are congruent.

Example

- Consider this true conditional statement. Write its converse. If the converse is also true, combine them as a biconditional
- If three points are collinear, then they lie on the same line.
- If three points lie on the same line, then they are collinear.
- Three points are collinear if and only if they lie on the same line.

Definitions

- A good definition is a statement that can help you identify or classify an object.
- A good definition has several important components:
- …Uses clearly understood terms. The terms should be commonly understood or already defined.
- …Is precise. Good definitions avoid words such as large, sort of, and some.
- …is reversible. That means that you can write a good definition as a true biconditional

Example

- Show that this definition of perpendicular lines is reversible. Then write it as a true biconditional
- Definition: Perpendicular lines are two lines that intersect to form right angles.
- Conditional: If two lines are perpendicular, then they intersect to form right angles.
- Converse: If two lines intersect to form right angles, then they are perpendicular.
- Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

Real World Examples

- Are the following statements good definitions? Explain
- An airplane is a vehicle that flies.
- Is it reversible?
- NO! A helicopter is a counterexample because it also flies!
- A triangle has sharp corners.
- Is it precise?
- NO! Sharp is an imprecise word!

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