Goal: To learn about and use deductive reasoning. Section 10.5 May 17/18 Goal: To learn about and use deductive reasoning.

Deductive Reasoning If you have no geometry homework, then a miracle happened. If a miracle happened, then there is a blue moon in China. If there is a blue moon in China, then you have geometry class today. If you have geometry class today, then you have geometry homework. If I tell you, “You have no geometry homework,” what can you conclude? You have geometry homework!

Vocab & Symbols Inductive reasoning uses observed patterns and examples to draw a conclusion. Deductive reasoning uses logic, facts, and definitions to draw a conclusion. “p” means hypothesis “q” means conclusion “→” means implies (or if- then) p → q means if p, then q p ↔ q means p if and only if q

Use Logic... Example 1 Given: 1) If 2 angles form a linear pair, then they are supplementary. 2) A and B form a linear pair. Valid conclusion:  A and  B are supplementary.

Use Logic... Example 2: Given: If the sun is shining, then it is a beautiful day. If it is a beautiful day, then I will have a picnic. Valid conclusion: If the sun is shining, then I will have a picnic.

You just used logic! 1) Law of Detachment 2) Law of Syllogism Given: 1) p  q 2) p is true Conclusion: q is true The HYPOTHESIS must be true, then you can conclude the conclusion! 2) Law of Syllogism Given: 1) p  q 2) q  r Conclusion: p  r

Examples: Make a conclusion if possible and tell which law allowed this. Example 3: Given: 1) Angles that are complementary have measures with a sum of 90 degrees. 2) A and B are complementary. Conclusion: m<A + m<B = 90 Law of Detachment Example 4: Given 1) If a number is a whole number, then it is an integer. 2) If a number is an integer, then it is a rational number. Conclusion: If a # is a whole, then it is rational. Law of Syllogism

Is this a valid conclusion? You Try! Is this a valid conclusion? Which law did you use? Given: All fish can swim. Fonzo can swim. Conclusion: Fonzo is a fish. Valid or not valid? Invalid (Fonzo MAY be a fish) Law of N/A

Practice Can a valid conclusion be made? If so, what Law did you use. Example 6 Given: If you live in Manhattan, then you live in New York. If you live in New York, then you live in the United States. Conclusion: If you live in Manhattan, then you live in the United States. Valid or not valid? Valid Law of Syllogism

More practice What can you conclude? Example 7 Given: If two angles are adjacent, then they share a common vertex. 1 and 2 share a common vertex. Conclusion: No conclusion Can’t go backwards! …remember… 1 2 They share a common vertex but are not adjacent. The HYPOTHESIS must be true, then you can conclude the conclusion!

Even more practice Determine if the conclusion is valid. Example 8 Given: Downloading pirated music is stealing. Isabella downloaded a pirated version of Iggy Azalea’s new song. Conclusion: Isabella doesn’t steal. Valid or Invalid? Invalid (Contradiction!)  Tip: The order in which they are given DOESN’T mater! Isabella downloaded. Downloading is stealing.

Enough with the practice! What can you conclude? Example 9 Given: If a ray divides an angle into two congruent angles, then the ray is an angle bisector. RS divides ARB so that  ARS ≅ SRB. Conclusion: RS bisects ARB

Really... More practice?!       Example 10 Assume that the following statements are true. If Maria is drinking juice, then it is breakfast time. If it is lunchtime, then Kira is drinking milk and nothing else. If it is mealtime, then Curtis is drinking water and nothing else. If it is breakfast time, then Julio is drinking juice and nothing else. Maria is drinking juice. Use only the information given above. Are the following definitely true, maybe true, or untrue? Julio is drinking water. Untrue Curtis is drinking water. Definitely true Kira is drinking milk. Maybe true      