Day 3

1. Day 3

2. Warm Up Find the distance and midpoint between the two points below

3. 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:

4. Midpoint: the value in the middle of a segment On a # line: On a coordinate plane: In 3-d:

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

8. 2-1 Conditional Statements • Objectives • To recognize conditional statements • To write converses of conditional statements

9. 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)

10. 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

11. 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.”

12. 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

13. 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

14. 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.”

15. 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.”

16. 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!

17. 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

18. 5-4 Inverses and Contrapositives • Objectives • To write the negation of a statement • To write the inverse and contrapositive of a conditional statement

19. 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

20. 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

21. 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!

22. 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

23. Summary

24. 2-2 Biconditionals and Definitions • Objectives • To write biconditionals • To recognize good definitons

25. 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.

26. 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.

27. 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.

28. 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

29. 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.

30. 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!

31. Homework Worksheet Scrapbook Project due Friday Distance/Midpoint Mini-Project due Sept 18