1. Welcome to Proofs!

2. The Basics Structure: Given information (sometimes assumption) Step-by-step reasoning Conclusion Types of reasoning: Inductive-use a number of specific examples to arrive at conclusion (called a “conjecture”) Deductive- use facts, rules, definitions, or properties to reach a conclusion

3. Example of inductive reasoning Write a conjecture that describes the pattern. Movie show times: 8:30am, 9:45am, 11:00am, 12:15am… The show time is 1 hour and 15 minutes greater than the previous show. The next show time will be 12:15am + 1:15= 1:30pm. The examples of movie times showed us a pattern, and we were able to come to a conclusion based on that pattern.

4. Example of deductive reasoning At Fumio’s school if you are late 5 times, you will receive a detention. Fumio has been late to school 5 times; therefore he will receive a detention.

5. Are the following examples of inductive or deductive reasoning? Discuss with your partner. 1.) Every Wednesday Lucy’s mother calls. Today is Wednesday, so Lucy concludes that her mother will call. 2.) A person must have a membership to work out at a gym. Jesse is working out at a gym. I conclude that Jesse has a membership to the gym. 3.) A dental assistant notices a patient has never been on time for an appt. She concludes the patient will be late for her next appt. 4.) If Eduardo decides to go to a concert tonight, he will miss football practice. Tonight, Eduardo went to a concert, so he missed football practice.

6. Conditional Statements (“If-Then” statements) Uses a hypothesis-conclusion format Example 1: If the forecast is rain (hypothesis), then I will take an umbrella (conclusion) Example 2: A number is divisible by 10 (conclusion) if its last digit is 0 (hypothesis) If p q If q p (converse) If –p -q (inverse) If –q -p (contrapositive)

7. If-Then Statements With your partner, determine what the hypothesis and conclusion are of each statement. Also, determine if the converse is true. If a polygon has 6 sides, then it is a hexagon. Tamika will advance to the next level of play if she completes the maze in her computer game. A five-sided polygon is a pentagon. An angle that measures 45° is an acute angle.

8. Who remembers the book If You Give a Mouse a Cookie? 2.5 POSTULATES AND PARAGRAPH PROOFS

9.  Objective: I will be able to… -Prove relationships between points, lines, and planes using deductive reasoning and paragraph proofs 2.5 POSTULATES AND PARAGRAPH PROOFS

12. The proof process

14. We just proved this!

16. Try one with your partner!

17. 2.6 Algebraic proof A proof that is made up of a series of algebraic statements. Reasoning uses properties of real numbers (addition, subtraction, multiplication, division, reflexive, distributive, substitution, etc.)

21. 2.7 Proving segment relationships C F G E D