Day 3

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Day 3. 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

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### Day 3

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:

Midpoint: the value in the middle of a segment

On a # line:

On a coordinate plane:

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: mXYZ > 70

Negation: mXYZ 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!
Homework

Worksheet

Scrapbook Project due Friday

Distance/Midpoint Mini-Project due Sept 18