1. 2.1 conditionals, 2.2 Biconditionals, 5.4 inverse and contrapositive
Reasoning and Proofs
2.1 conditionals, 2.2 Biconditionals, 5.4 inverse and contrapositive

2. Agenda Warm-up Homework check
2.1 conditionals, 2.2 biconditionals, 5.4 inverse and contrapositive
Practice
Homework

3. Warm-Up:

4. 3-d Coordinate Plane!
Find the coordinates for: O (0,0,0) N (0, -4, 0) K (-3, 0, 0) J (0,0, 4) M (0, -4, 4) H (-3, -4, 4) L (-3, 0, 4) I (-3, -4, 0)

5. Homework Check

6. 2-1, 5-4, 2-2 Reasoning and Proof
Negate – to add or remove the word “Not” Ex. Negate: “<A is a right <.” Becomes: <A is not a right <. Ex. Negate: “I am not cold.” Becomes: I am cold.

7. Conditional Statement: an if-then statement.
(p  q)
All conditional statements have 2 parts.
A hypothesis
A conclusion.

8. Hypothesis: (p) Follows “IF” Conclusion: (q) Follows “THEN” Ex: If today is Valentines Day, then the month is February. **When stating these do NOT include the words “if” or “then” Hypothesis: Today is Valentines Day. Conclusion: The month is February.

9. EX: Three points are collinear, if they are all in a line
EX: Three points are collinear, if they are all in a line. Hypothesis: Three points are all in a line. Conclusion: The three points are collinear.

10. Converse: SWITCHES the hypothesis and the conclusion EX: Conditional: If two lines intersect to form right angles, then they are perpendicular Write the converse: If two lines are perpendicular then the two lines intersect to form right angles.

11. Inverse: NEGATE the hypothesis and the conclusion **If not p, then not q.** EX: Conditional: If two lines intersect to form right angles, then they are perpendicular Write the Inverse: If two lines do not intersect to form right angles, then they are not perpendicular.

12. Contrapositive: SWITCH AND NEGATE Conditional: If p, then q
Contrapositive: SWITCH AND NEGATE Conditional: If p, then q. Contrapositive: *If not q, then not p.* -Logically equivalent to the original conditional -If conditional is true, then the contrapositive is true.

13. EX: Conditional: If two lines intersect to form right angles, then they are perpendicular Contrapositive: If two lines are not perpendicular then they do not intersect to form right angles.

14. Biconditional: p if and only if q (p iff
Biconditional: p if and only if q (p iff. q)  occurs when a statement and its converse are true.

15. Counterexample: an example that shows the hypothesis can be true while the conclusion is false, thereby proving the statement is false. Ex. Conditional: If two angles are adjacent then they are a linear pair. Counterexample: <ABC and <CBD are Adjacent but not a linear Pair.

16. Examples
Ex1. An angle that measures 90 ̊ is a right <. Write As a conditional: If an angle measures 90 ̊, then it is a right <. Write the Converse: If an angle is a right <, then it measures 90 ̊

17. Ex1. An angle that measures 90 ̊ is a right <
Ex1. An angle that measures 90 ̊ is a right <. Write the Inverse: If an angle does not measure 90 ̊, then it is not a right <. Write the Contrapositive: If an angle is not a right <, then it does not measure 90 ̊

18. Ex. If possible, write the biconditional of each statement
Ex. If possible, write the biconditional of each statement. If not explain.
2. If an angle is a straight angle, then it measures 180 ̊ Converse: If an angle measures 180 ̊, then it is a straight angle. (True) Biconditional: An angle is a straight < if and only if it measures 180 ̊.

19. 3. If 2 angles are a linear pair, then their sum is 180 ̊
3. If 2 angles are a linear pair, then their sum is 180 ̊. Converse: If 2 angles’ sum is 180, then they are a linear pair. –False Counterexample: No biconditional because the converse is false!

20. Most good definitions can be written as biconditionals
Most good definitions can be written as biconditionals. **You can determine a definition is not good by finding a counterexample.

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