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Deductive Reasoning

“Indubitably.”. “The proof is in the pudding.”. Le pompt de pompt le solve de crime!". Deductive Reasoning. Je solve le crime. Pompt de pompt pompt.". Proving Theorems. The Midpoint Theorem. If m is the midpoint of , then. Written form. A. M. B. Simplistic form. Hypothesis.

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Deductive Reasoning

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  1. “Indubitably.” “The proof is in the pudding.” Le pompt de pompt le solve de crime!" Deductive Reasoning Je solve le crime. Pompt de pompt pompt."

  2. Proving Theorems The Midpoint Theorem If m is the midpoint of , then Written form A M B Simplistic form Hypothesis Given: Conclusion Prove:

  3. Proving Theorems The Midpoint Theorem Given: Prove: A M B Def of mp Def of mp Statements Reasons Given AM = MB Definition of midpoint AM + MB = AB Segment Addition Post. AM + AM = AB Substitution 2 AM = AB Combine like terms __ __ 2 2 Mental step Division Prop. of Equality

  4. As you continue to learn to prove theorems, it is important to get accustomed to the various forms of stating a theorems. You need to get accustomed to using the hypothesis and the conclusion. This will take time. You will be given enough time. You are learning a new way of thinking. Understanding the importance of this type of thinking will take even more time. Please be patient!!! Let’s begin.

  5. These proofs are using deductive reasoning. Deductive reasoning is a linking of cause and effect statements to reach a conclusion. It is like a chain that starts and moves to other the end. Beginning Hypothesis Given End Conclusion To Prove

  6. It is also like a row of dominos. Like dominos which have two blocks on each tile, each step of the proof also has two parts – statements and reasons. Statements without reasons are not valid or substantiated. Each line of the proof must contain a statement supported by a reason or justification. When this happens, the hypothesis logical forces you to accept the conclusion just like a row of dominos being toppled.

  7. How are we going to learn how to do proofs? 1] Examine the structure of completed proofs. 2] Fill in the reasons for steps of partially completed proofs. 3] Fill in proofs that have blanks on either the statement or the reason side. 4] Do our own proofs with simple geometry and simple algebra.

  8. This is a slow very process. However, it is relatively painless. It leads to success and understanding. How many proofs will we do? How many swings at a baseball before you are a good hitter?

  9. Given: Prove: ? ? Label diagram to help visualize. Statements Reasons Given Angle Add. Postulate Angle Add. Postulate Substitution Prop. Of Equality Reflexive Prop. Of Equality Subtr. Prop. Of Equality

  10. g Given: Prove: ? ? g g g Label diagram to help visualize. Statements Reasons Given Given Addition Prop. Of Equality Angle Add. Postulate Angle Add. Postulate Substitution Prop. Of Equality

  11. W L g K ? WK = RN LK = RU Given: Prove: g g ? WL = UN R U N g Label diagram to help visualize. Statements Reasons Given WK = RN WK ____ = WL + LK Segment Add. Postulate RN Segment Add. Postulate ____ = RU + UN Substitution Prop. Of Equality WL + LK = RU + UN Given LK = RU Subtr. Prop. Of Equality WL = UN

  12. W L g K ? Notice that the given doesn’t have to come first. You can put it in when it is needed. g g ? R U N g Statements Reasons Given WK = RN WK ____ = WL + LK Segment Add. Postulate RN Segment Add. Postulate ____ = RU + UN Substitution Prop. Of Equality WL + LK = RU + UN Given LK = RU Subtr. Prop. Of Equality WL = UN

  13. Let’s try an alternate approach to the same problem.

  14. W L g K ? WK = RN LK = RU Given: Prove: g g ? WL = UN R U N g Label diagram to help visualize. Statements Reasons Given WK = RN LK = RU Given WK ____ = WL + LK Segment Add. Postulate RN ____ = RU + UN Segment Add. Postulate WL + LK = RU + UN Substitution Prop. Of Equality Subtr. Prop. Of Equality WL = UN

  15. Did you notice that it was not as easy to follow? It is not wrong. But understanding is more difficult. In geometry there are often many correct approaches to the same problem. If you find it easier to put the given down first. Do it. Eventually, you will learn easier ways to do your proofs. It is not uncommon to be able to complete only 50% of yourproofs

  16. It is not uncommon to be able to complete only 50% of your homework. This is the nature of geometry in the beginning. Do not feel bad. Many students experience frustration in the beginning. Persevere and the frustration will disappear.

  17. C’est fini. Good day and good luck.

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