1. Bell Work Conditional: If the car is running, then it has fuel 1) Write the converse 2) Write the “opposite” statement of the conditional 3) Write the “opposite” statement of the converse

2. Chapter 5.4 Inverses, Contrapositives, and Indirect Reasoning 2.0 Students write geometric proofs, including proofs by contradiction

3. Review Conditional – If p, then q Conditional: – If it snows tomorrow, then we will go skiing Converse – If q, then p Converse: – If we go skiing, then it snows tomorrow

4. Review Biconditional – When both the conditional and converse are both true statements – Joins the hypothesis and conclusion with if and only if – Example: A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment

5. Vocabulary Negation has the opposite meaning of the original statement Example: Statement: “An angle is a straight angle” Negation: “An angle is NOT a straight angle”

6. Try this one Write the negation of a statement Statement: “Angle ABC is obtuse” Negation: “Angle ABC is NOT obtuse”

7. One more!! Write the negation of the statement Statement: “Lines m and n are not perpendicular” Negation: “Line m and n ARE perpendicular”

8. Vocabulary Inverse: negates both the hypothesis and the conclusion – If ~p, then ~q (if not p, then not q) Conditional: – If it snows tomorrow, then we will go skiing Inverse: – If it does not snow tomorrow, then we will not go skiing

9. Example! Write the inverse of the conditional statement Conditional: “If a figure is a square, then it is a rectangle” Inverse: “If a figure is NOT a square, then it is NOT a rectangle”

10. Woot! More! Write an inverse for the conditional statement Conditional: “If two angles add up to 180, then they are supplementary” Inverse: “If two angles do not add up to 180, then they are not supplementary”

11. Vocabulary Contrapositive: Switches the hypothesis and the conclusion and negates both – If not q, then not p – Conditional: “If it snows tomorrow, then we will go skiing” – Contrapositive: – “If we do not go skiing, then it does not snows tomorrow”

12. Example! Find the Contrapositives Conditional Statement: “If an angle is a straight angle, then its measure is 180” Contrapositives: “If an angle’s measure is not 180, then it is not a straight angle”

13. One more! Find the contrapositive Conditional: “If two lines are parallel, then they do not intersect” Contrapositive: “If they do intersect, then two lines are not parallel”

14. Random practices Statement: The angle is obtuse Negation: The angle is not obtuse

15. Vocabulary Equivalent statements = statements with the same truth value Example: – “If a figure is a square, then it is a rectangle” (Conditional) – “If a figure is not a rectangle, then it is not a square” (Contrapositive) – These two statements are the same

16. Vocabulary Indirect reasoning = type of reasoning in which all possibilities are considered and then the unwanted ones are proved false. The remaining possibilities must be true.

17. Vocabularies Indirect proof = see indirect reasoning

18. More random practice Write the negation – Statement: “Today is not Tuesday” – Negation:

19. Indirect Proof Step 1) State as an assumption the opposite (negation) of what you want to prove Step 2) Show that this assumption leads to a contradiction Step 3) Conclude that the assumption must be false and that what you want to prove must be true

20. Example of indirect proof If Jaeleen spends more than $50 to buy two items at a clothing store, then at least one of the items costs more than $25 Dollars – Given: The cost of two items is more than $50 – Prove: At least one of the items costs more than $25 Step 1) Assume negation of what you are trying to prove is true. “Neither item costs more than $25” Step 2) This means that each item costs $25 or less, which lead to that two items together cost $50 or less. Which contradict the given Step 3) So the negation is false. So one item must cost more than $25

21. Identify Contradiction Identify the two statements that contradict each other – I. ABC is acute – II. ABC is scalene – III. ABC is equiangular

22. Homework Pgs 283-284 #2-20 even, 21