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Logical Equivalence in Geometric Proofs

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Understand the concepts of contrapositives, inverses, and logical implications in geometric proofs. Learn how to apply these principles to prove statements in geometry effectively.

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Logical Equivalence in Geometric Proofs

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  1. Even Answers 2)(a)No (b)Yes (c)Yes (d)Yes (e)No (f)No 4)(a)Yes (b)Yes (c)No (d)No 6)(1)Given (2)CPCTC (3)Segment Addition Prop (4)Prop of Inequality (5)Substitution 8)(2)Segment Addition (3)Property of Inequality (4) Substitution 10)(2)Defintion ┴ lines (3)Exterior Angle Inequality Theorem (4)Substitution (5) Definition of Obtuse Angle

  2. 6-2 Inverses and Contrapositives

  3. P: HYPOTHESIS Q: CONCLUSION • Statement If p, then q • Converse If q, then p • (con-artist—does a switch) • Inverse If not p, then not q • Add a word In---not • Contrapositive If not q, then not p • Weirdest word—so do both, add NOT and Switch

  4. EXAMPLES: Give the Inverse and Contrapos. State Tor F 1. If a parallelogram is a square, then it is a rectangle. (T) I: If a parallelogram is not a square, then it is not a rectangle (F) C+: If a parallelogram is not a rectangle, then it is not a square (T) 2. If it is snowing, then the game is canceled. I: If it is not snowing, then the game is not canceled (F) C+: If the game is not canceled, then it is not snowing (T)

  5. TRY ON OWN: 1. If I can sing, then you can dance. I: If I can’t sing, then you can’t dance (F) C+: If you can’t dance, then I can’t sing (T) 2. If Taylor is not here, then he is not well. I: If Taylor is here, then he is well (F) C+: If Taylor is well, then he is here (T)

  6. Use a VENN DIAGRAM to tell if an assumption is True or False Example: All marathoners have stamina Statement: If you are a marathoner, then you have stamina • Nick is a marathoner • Heidi has stamina • Mimi does not have stamina • Arlo is not a marathoner He has stamina No conclusion She is not a marathoner No conclusion Marathoner Stamina

  7. Try On Own: All Squares Are Rhombuses If it is a square, then it is a rhombus • ABCD is a Rhombus • PQRS is a square • LAST is not a rhombus • GHIJ is not a square What do we notice from this example & the last? 2 No Conclusions!! No Conclusion PQRS is a rhombus LAST is not a square No Conclusion Square Rhombus

  8. RULE: The Statement and Contrapositive are logically equivalent! That means if the statement is True, the Contra+ is also True. The 2 others will be false.

  9. HOMEWORK • Pg 210 #1-15 all

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