Starter:

1. Starter: Write a conjecture based on the given information. Draw a picture if necessary. • Given: <1 and <2 form a right angle. Conjecture: 2. Given: Today is Thursday. Conjecture: Determine if the conjecture is true or false. If false, give a counterexample. 3. Given: <1 and <2 are supplementary Conjecture: <1 and <2 are not congruent 4. Given: <1 and <2 are vertical angles Conjecture: <1 and <2 are congruent

2. Starter: Fill in the blank with the correct term: A is an educated guess. Inductive/deductive reasoning is when you make a conjecture based on a set of observations (or a pattern). In order for a conjecture to be true it must be true for cases. It only takes 1 to prove a conjecture is false.

3. Questions on Homework?

4. Lesson 2.2 Notes If-Then Statements and Postulates

5. Conditional Statement: any statement written in if-thenform or where a condition is given for a conclusion to be true. Conditional Words: if, when, provided Example: If today is Monday, then Mrs. Reese is wearing ! blue

6. Hypothesis and Conclusion Hypothesis: the part of a conditional statement immediately following “if”. Conclusion: the part of a conditional statement immediately following “then”. Example: If , then today is Monday Mrs. Reese is wearing blue!

7. Symbolically: p = hypothesis q = conclusion • if p, then q is written as: p  q

8. Example 1: If two angles are vertical, then they are congruent. State the hypothesis and conclusion. Hypothesis: Conclusion: Two angles are vertical They are congruent

9. Example 2: If 3 points lie on the same line, then they are collinear. State the hypothesis and conclusion. Hypothesis: Conclusion: 3 points lie on the same line They are collinear

10. Example 3: If you pack up before Mrs. Reese tells you to, then… State the hypothesis and conclusion. Hypothesis: Conclusion: You pack up b4 Mrs. Reese tells you to

11. How to write a Conditional (If-Then) Statement • Divide the subject and the verb • Remove words like “All”, “Every”, “Each”. • Identify a basic noun which describes the subject. • Make sure your pronouns agree with the subject.

12. Write the following as a conditional statement: All girls like chocolate. • All girls / like chocolate • Girls / like chocolate • If a person is a girl, then girls like chocolate. • If a person is a girl, then SHE likes chocolate.

13. Write the following as a conditional statement: All right angles are congruent. • All right angles / are congruent • Right angles / are congruent • If 2 angles are right angles, then THEY are congruent.

14. Sometimes you may need to rearrange a statement before you rewrite it in if-then form: Maya can go to the party on Friday, as long as she doesn’t get in trouble this week. • THINK: What is the CONDITION? IF Maya doesn’t get in trouble this week, THEN she can go to the party on Friday.

15. Two more examples: • Mauldin Fans love orange and brown! • If a person is a Mauldin fan, then they love orange and brown. • All linear pairs are supplementary. • If 2 angles form a linear pair, then they are supplementary. Can you identify the hypothesis and conclusion for each statement?

16. Two more examples (IF NEEDED) • Every acute angle does not equal 90°! • If an angle is acute, then it does not equal 90°. • Angles with the same measure are congruent. • If 2 angles have the same measure, then they are congruent. Can you identify the hypothesis and conclusion for each statement?

17. PRACTICE: Write the answers to the following and turn it in before you leave: • Identify the hypothesis and conclusion: • If 3x – 7 = 5, then x = 2. • If Carl scores 85%, then he passes. • Write each conditional statement in if-then form. • All students like vacations. • The game will be played provided it doesn’t rain.

18. HOMEWORK • Textbook page 80-81 #6-9 and 19-24

19. Converse Inverse Contrapositive Symbolically q p ~ p ~ q ~ q ~ p There are 3 additional types of conditional statements that are used in Geometry.

20. A conditional statement being true, does not guarantee these additional statements will be true - even if we start with the same hypothesis and conclusion.

21. Converse Statement • Formed by interchanging (switching) the hypothesis and conclusion of a conditional statement. • Example: If today is Friday, then we will have a football game. • CONVERSE: If we have a football game, then today is Friday.

22. Inverse Statement • Formed by negating the hypothesis and conclusion of a conditional statement. • Example: If today is Friday, then we will have a football game. • INVERSE: If today is NOT Friday, then we will NOT have a football game.

23. Contrapositive Statement • Formed by negating AND interchanging the hypothesis and conclusion of a conditional statement. (OR by negating the converse) • Example: If today is Friday, then we will have a football game. • CONTRAPOSITIVE: If we do not have a football game, then today is not Friday.

24. Breaking Down Conditionals • Example: If 2 angles are vertical, then they are congruent. • Hypothesis: • Conclusion: • Converse: • Inverse: • Contrapositive: 2 angles are vertical they are congruent If 2 angles are congruent, then they are vertical. If 2 angles are not vertical, then they are not congruent. If 2 angles are not congruent, then they are not vertical.

25. Practice: Write the Converse of each conditional statement and determine if the converse is true or false. If false, give a counterexample. 1. If it rains, then it is cloudy. 2. If a number is even, then it is divisible by 2. Next, write the Inverse and Contrapositive for each statement.

26. 1. Converse: If it is cloudy, then it rains. False: It can be cloudy without raining. • Converse: If a number is divisible by 2, then it is an even number. True 1. Inverse: If it does not rain, then it is not cloudy. Contrapositive: If it is not cloudy, then it does not rain. 2. Inverse: If a number is not even, then it is not divisible by 2. Contrapositive: If a number is not divisible by 2, then it is not an even number.

27. Starter 10/4/2010 Determine if conjecture is true based on the given information. If false, give a counterexample. • Given: AB and AC are collinear • Conjecture: AB and AC are opposite rays. • Given: <1 and <2 are complementary • Conjecture: m<1 + m<2 = 180°

28. All students love math. • Rewrite the statement as a conditional. • Identify the hypothesis and conclusion of the conditional you wrote for #1. • Write the converse of the conditional you wrote in #1. • Is the converse true or false? If false, give a counterexample. • Write the inverse and contrapositve of the conditional statement you wrote in #1.

29. CLASSWORK: • PAGE 81 #25-30 and 36-41

30. Biconditional Statement • When you combine a conditional statement and its converse, you create a biconditional statement. • A biconditional statement is written in the form “p if and only if q” • Symbolically: • p q OR • p iff q

31. Biconditional Statement • In order for a biconditional statement to be true, the conditional statement AND the converse must both be true! • A true biconditonal statement is considered to be a definition. Any definition can be written as a true biconditional statement.

32. Are the following biconditonal statements true? • Two angles are complementary if and only if the sum of their measures is 90°. • Think about the conditional and converse statements… The biconditonal is TRUE; this IS a definition! • If 2 angles are complementary then the sum of their measures is 90°. • If the sum of the measures of 2 angles is 90°, then the angles are complementary. true true

33. Are the following biconditonal statements true? • Two angles are vertical if and only if they are congruent. • Think about the conditional and converse statements… • If 2 angles are vertical then they are congruent. • If 2 angles are congruent, then they are vertical. true FALSE The biconditional is false; this is NOT a definition!