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# Day 3 - PowerPoint PPT Presentation

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

Find the distance and midpoint between the two points below

**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:

On a # line:

On a coordinate plane:

In 3-d:

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

• Objectives

• To recognize conditional statements

• To write converses of conditional 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)

• 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

• 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.”

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

• 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

• 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.”

• 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.”

• 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!

• 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

• Objectives

• To write the negation of a statement

• To write the inverse and contrapositive of a conditional statement

• 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

• 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

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!

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.

• 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.

• 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

• 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.

• 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!

Worksheet

Scrapbook Project due Friday

Distance/Midpoint Mini-Project due Sept 18