1 2.1-2.3: Reasoning in Geometry Helena Seminati Stephanie Weinstein.

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Presentation transcript:

: Reasoning in Geometry Helena Seminati Stephanie Weinstein

2 2.1: An Intro to Proofs  A proof is a convincing argument that something is true.  Start with givens: postulates or axioms.  Can be formal or informal.

3 Types of Proofs m<1m<2m<3m<420°??? 30°??? 40°??? x°???

4 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.

5 Reversing Conditionals  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. If a car is a Corvette, then it is a Cheverolet.

6 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.

7 Conditionals from Logical Chains  If cats freak, then mice frisk.  If sirens shriek, then dogs howl.  If dogs howl, then cats freak. First, identify the hypothesis and conclusions. Strike out any repeats. String them together to form a conditional.

8 If-Then Transitive Property  An extension of logical chains, the If-Then  Transitive Property is:  Given: One can conclude:  “If A then B, and“If A then C.”  if B then C.”

9 2.3 Definitions AAAA definition is a type of conditional, written in aaaa different form. AAAA definition can apply to made-up polygons oooor traditional ones. AAAA definition has a property that the original cccconditional and the converse are both true.

10 Definition of a Vehicle  Vehicles  Planes  Cars  Wheelbarrows  Bicycle  Roller-coaster Not vehicles BooksComputersDSL “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!

11 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.

12 Helpful Websites  An introduction to proofs:   Conditional statements and their converses:  conclusion-geometry-14 conclusion-geometry-14 conclusion-geometry-14  More on conditionals:  d.htm d.htm d.htm

13 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.