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.1. 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

2. 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 negation of 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.1. Statement: ABC is obtuse

Negation: ABC is not obtuse

2. Statement: mXYZ > 70Negation: 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 rectangleNEGATION! NEGATION!

Contrapositive: If , then

a figure is not a rectangle it is not a squareNEGATION! 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 Statements have the same truth value

Summary

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

WorksheetScrapbook Project due FridayDistance/Midpoint Mini-Project due Sept 18