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Chapter 2 Section 2.3

Chapter 2 Section 2.3. Laws of Reasoning Deductive Reasoning. 1.) If a figure is a triangle, then it has three angles. Converse: If a figure has three angles, then it is a triangle. Inverse: If a figure is not a triangle, then it does not have three angles.

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Chapter 2 Section 2.3

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  1. Chapter 2 Section 2.3 Laws of Reasoning Deductive Reasoning

  2. 1.) If a figure is a triangle, then it has three angles. Converse: If a figure has three angles, then it is a triangle. Inverse: If a figure is not a triangle, then it does not have three angles. Contrapositive: If a figure does not have three angles, then it is not a triangle. 2.) If a number is divisible by 6, then it is divisible by 2. Converse: If a number is divisible by 2, then it is divisible by 6 Inverse: If a number is not divisible by 2, then it is not divisible by 6. Contrapositive: If a number is not divisible by 6, then it is not divisible by 2.

  3. Warming Up • If Chris goes to the basketball game, he will not go to the movies. Chris went to the basketball game. What conclusion can you draw? • What is the difference between the statement above and the following? • If Rachael goes to the basketball game, she will not go to the movies. • Rachael did not go to the movies. Can you conclude that Rachael went to the basket ball game?

  4. If-then Statement: If you have a high school diploma, then you graduated from high school. Is this true or false? True Statement about the hypothesis: A person has a high school diploma. True Conclusion: The person graduated from high school.

  5. If-then Statement: If you plan on cutting class, then you will be written up. True Statement: A person plans on cutting class. True Conclusion: The person will be written up.

  6. Law of Detachment The Law of detachment offers us a way to draw conclusions from if-then statements. Whenever a conditional (if-then statement) is true, and it’s hypothesis is true, we can assume that its conclusion is true.

  7. If-then Statement: If two numbers are odd then their sum is even. Is this statement true? Hypothesis: Two numbers are odd. 3 and 5 satisfy the given hypothesis. Can you conclude that the sum is even? Since the statement is true, and the hypothesis is true, we know that 3 + 5 must be even. Yes, it is true.

  8. Does statement 3 follow from 1 and 2 by the law of detachment? 1.) If you plan on attending Princeton, then you need to be in the top 5% of your class. 2.) Jaime plans on attending Princeton. 3.) Jaime needs to be in the top 5% of her class. Yes, this satisfies the law of detachment.

  9. Can you draw a valid conclusion from the two statements using the Law of Detachment? 1.) If the class is long and boring, then a student in the class is likely to day dream. 2.) John day dreams in science class. Can you draw a conclusion? 1.) If you want good health, then you should eat vegetables and fruits everyday. 2.) Jackie wants good health. Can you draw a conclusion? No Conclusion Yes. Jackie should eat vegetables and fruits every day.

  10. What can you conclude from the following. 1.) If Erin is the same age as Jackie, and Jackie is the same age as Jaime, then … 2.) If John makes the same amount as Jose, and Jose makes the same amount as Andrew, then… 3.) If AB = BC and BC = DE then …

  11. 2nd Law of Logic (2nd page) Law of Modus Tollens( yes more Latin) • This law states when given two premises: one a conditional and the other the negation of the conclusion, then the negation of the hypothesis is also true. • What does this mean? • p If Al does not study, then he will fail. • Al did not fail • Al studied

  12. More and More Examples If we do not have Geometry books, then Ms. Valentino has to copy her notes. Ms. Valentino does not have to copy her notes. If Ms. Valentino does not have her microphone on, then she got her voice back. Ms. Valentino did not get her voice back.  ~p:

  13. 3rd Law of Logic Law of Syllogism – law very similar to the Transitive Property of Equality from algebra.

  14. Law of Syllogism Example If the canal is open, then the ship can go through. If the ship can go through the canal, then the company can transport the goods. By the law of syllogism, the canal is open the ship can go through the company can transport the goods We can conclude then that

  15. Does statement 3 follow from statements 1 and 2 by the law of syllogism? 1.) If an angle has a measure less than 90, it is acute. 2.) If an angle is acute, then its supplement is obtuse. 3.) If an angle has a measure of less than 90, then its supplement is obtuse. Yes! p – if an angle has a measure less than 90 q – the angle is acute r – the angles supplement is obtuse.

  16. Can a valid conclusion be made from statements 1 and 2? Write statement three if it applies. 1.) If the team wins, then we will celebrate. 2.) If we celebrate, then we will be out late. 3.) If the team wins, then we will be out late. 1.) If two angles are vertical, then they do not form a linear pair. 2.) If two angles are vertical, then they are congruent. 3.) No, the law of syllogism does not apply.

  17. Determine if statement 3 follows from 1 and 2 and decide if the law of syllogism, law of detachment or none apply. 1.) If you drive safely, the life you save may be your own. 2.) Sean drives safely. 3.) The life he saves may be his own. 1.) Right angles are congruent. 2.) A ≡ B 3.) A and B are right angles. Law of Detachment Applies None Apply

  18. HOMEWORK!! Pg. 90 #2 – 14 even

  19. 2.4 Postulates

  20. 2.4 Postulates • Postulates are facts about geometry that are accepted as true.

  21. Postulate 1-Ruler Postulate The points on any line can be paired with real numbers so that, given any two points P and Q on the line, P corresponds to zero, and Q corresponds to a positive number. P Q 0 Since P is at -2 and Q is at 4, we can say the distance from X to Y or Y to X is: -2 – 4 = 6 or 4 – (-2) = 6

  22. Postulate 2 Segment Addition Postulate If Q is between P and R, then PQ + QR = PR. If PQ +QR = PR, then Q is between P and R. 2x 4x + 6 R P Q PQ = 2x QR = 4x + 6 PR = 60 Use the Segment Addition Postulate find the measure of PQ and QR.

  23. More Postulates: YIPPEE!! Postulate 3 (Angle Addition Postulate): If R is in the interior of PQS, then m PQR + m RQS = m PQS. If m PQR + m RQS = m PQS, then R is in the interior of PQS. P R Q S

  24. Two points determine a line. There is only one line that contains points P and Q. P Q Postulate 4 and 5

  25. If two distinct lines intersect, then their intersection is a point. Lines l and m intersect at point T. l m T Postulate 7

  26. Three noncollinear points determine a unique plane. There is only one plane that contains points A, B, and C. Postulate 8 and 9 A B C

  27. R l g A H T W M F V EXAMPLE 3 TRUE TRUE or FALSE: There is only one plane that contains all of the points F, T, & M

  28. Postulate 10 and 11 • If 2 points lie in a plane, then the line containing them lies in a plane • If two distinct planes intersect, then their intersection is a line.

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