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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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warm up
Warm Up

Find the distance and midpoint between the two points below

distance
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
Midpoint: the value in the middle of a segment

On a # line:

On a coordinate plane:

In 3-d:

homework check
Homework Check

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

2 1 conditional statements
2-1 Conditional Statements
  • Objectives
    • To recognize conditional statements
    • To write converses of conditional statements
if then 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
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
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
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
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
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.”
example1
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
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
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
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 contrapositives1
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
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
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
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
2-2 Biconditionals and Definitions
  • Objectives
    • To write biconditionals
    • To recognize good definitons
2 2 biconditionals and definitions1
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
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.
example2
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
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
example3
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
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
Homework

Worksheet

Scrapbook Project due Friday

Distance/Midpoint Mini-Project due Sept 18

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