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) = (6.5, 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: 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! Contrapositive: If, then a figure is not a rectangle it is not a square 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 Worksheet Scrapbook Project due Friday Distance/Midpoint Mini-Project due Sept 18