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2.1-2.3: Reasoning in Geometry Helena Seminati Stephanie Weinstein 1 2.1: An Intro to Proofs A proof is a convincing argument that something is true. Start Can with givens: postulates or axioms. be formal or informal. 2 Types of Proofs m<1 m<2 m<3 m<4 20° ? ? ? 30° ? ? ? 40° ? ? ? x° ? ? ? 3 2.2 An Intro to Logic “If-then” statements are conditionals. Formed as “if p, then q” or “p implies q.” Conditionals are broken into two parts: Hypothesis is p. Conclusion is q. 4 Reversing Conditionals If a car is a Corvette, then it is a Cheverolet. A converse is created when you interchange p and q (hypothesis and conclusion). A counterexample proves a converse false. ex: If a car is a Cheverolet, then it is a Corvette. ex: A Silverado. 5 Logical Chains A logical chain is a set of linked conditionals. If cats freak, then mice frisk. If sirens shriek, then dogs howl. If dogs howl, then cats freak. 6 Conditionals from Logical Chains First, identify the hypothesis and If cats freak, then mice frisk. conclusions. If sirens shriek, then dogs Strike out any repeats. howl. If dogs howl, then cats freak. String them together to form a conditional. 7 If-Then Transitive Property An extension of logical chains, the If-Then Transitive Property is: Given: “If A then B, and if B then C.” One can conclude: “If A then C.” 8 2.3 Definitions A definition is a type of conditional, written in a different form. A definition can apply to made-up polygons or traditional ones. A definition has a property that the original conditional and the converse are both9 Definition of a Vehicle “Anything that has wheels and moves people from place to place.” Not all definitions may be precise, so when creating or following one, read carefully! Vehicles Planes Cars Wheelbarrows Bicycle Roller-coaster Not vehicles Books Computers DSL 10 Biconditionals Two true conditionals (of a definition) can be combined into a compact form by joining the hypothesis and the conclusion with the phrase “if and only if.” Statements using “if and only if” are biconditionals. 11 Helpful Websites An introduction to proofs: http://library.thinkquest.org/16284/g_intro_2.htm Conditional statements and their converses: http://www.slideshare.net/rfant/hypothesisconclusion-geometry-14 More on conditionals: http://library.thinkquest.org/2647/geometry/cond/con d.htm 12 A Quick Review What are some types of proofs? What two parts form a conditional statement? What is the If-Then Transitive Property What is the essential phrase in a biconditional? What is the converse of this statement: If bob is old, then his bones are frail. 13