“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 A B , then A M A M Hypothesis Conclusion B 1 AB 2 Written form Simplistic form Given: m is the m idpt . of A B Prove: A M 1 2 AB Proving Theorems The Midpoint Theorem Given: A M Def of mp B m is the m idpt . of A B AM = MB AM + MB = AB AM + AM = AB 2__ AM = __ AB AM 1 2 Prove: A M Def of mp Statements 2 m is the m idpt . of A B 2 AB 1 AB 2 Reasons Given Definition of midpoint Segment Addition Post. Substitution Combine like terms Mental step Division Prop. of Equality 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. 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 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. 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. 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? B Given: m A E C m B E D A ? 1 C 2 3 ? D E Statements m AEC m BED m AEC m 1 m 2 m BED m 2 m 3 m 1 m 2 m 2 m 3 m2m2 m 1 m 3 Prove: m 1 m 2 Label diagram to help visualize. Reasons Given Angle Add. Postulate Angle Add. Postulate Substitution Prop. Of Equality Reflexive Prop. Of Equality Subtr. Prop. Of Equality B g ? C g D ? 4 1 m2m3 Prove: m A E C m C E F 2 3 g Given: m 1 m 4 g Label diagram to help visualize. A E F Statements Reasons m 1 m 4 Given m2m3 Given m 1 m 2 m 3 m 4 m AEC m 1 m 2 m CEF m 3 m 4 m AEC m CEF Addition Prop. Of Equality Angle Add. Postulate Angle Add. Postulate Substitution Prop. Of Equality W ? g R L gK Given: WK = RN LK = RU g ? U Prove: N g WL = UN Label diagram to help visualize. Statements Reasons Given WK = RN WK = WL + LK ____ Segment Add. Postulate RN = RU + UN ____ Segment Add. Postulate WL + LK = RU + UN LK = RU WL = UN Substitution Prop. Of Equality Given Subtr. Prop. Of Equality W ? g R L gK g ? U g Notice that the given doesn’t have to come first. You can put it in when it is N needed. Statements Reasons Given WK = RN WK = WL + LK ____ Segment Add. Postulate RN = RU + UN ____ Segment Add. Postulate WL + LK = RU + UN LK = RU WL = UN Substitution Prop. Of Equality Given Subtr. Prop. Of Equality Let’s try an alternate approach to the same problem. W g R L gK ? Given: WK = RN LK = RU g ? U Prove: N g WL = UN Label diagram to help visualize. Statements Reasons Given Given WK = RN LK = RU WK = WL + LK ____ Segment Add. Postulate RN = RU + UN ____ Segment Add. Postulate WL + LK = RU + UN Substitution Prop. Of Equality WL = UN Subtr. Prop. Of Equality 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 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. C’est fini. Good day and good luck.