2.  Explanation of Content  Deductive Structure : A system of thought in which conclusions are justified by means of previously proved or assumed statements. ▪ Every deductive structure contains the following four elements. ▪ Undefined Terms ▪ Assumptions known as postulates ▪ Definitions ▪ Theorems and other conclusions

3.  Definitions of the elements.  Undefined terms : Terms that are described, but not precisely defined. (Because it is impossible) ▪ Examples : Point Line Plane  Postulates: An unproved assumption ▪ Example: You can always assume a straight line.

4.  Definitions cont’d.  Definition: States the meaning of a term or idea. ▪ ***DEFINITIONS ARE REVERSIBLE *** ▪ Example: A ray that divides an angle into two congruent angles is called a bisector.  Theorems: A proven fact through logic and using definitions, undefined terms, postulates, and other theorems. ▪ Examples: Theorem 2- If two angles are straight angles, then they are congruent.

5.  Sample Problems  State the converse of each of the following statements. ▪ A. If a ray divides an angle into two congruent angles, then it bisects the angle. ▪ B. If two angles have the same measure, then they are congruent.  Answers ▪ A. If a ray bisects an angle, it divides the angle into two congruent angles. ▪ B. If two angles are congruent, then they have the same measure.

6.  Practice Problems 1. What are the four elements of deductive structure? 2. Tell the converse of the following statements and identify them as True or False. a)If an angle is a 115° angle, then the angle is obtuse. b)If two angles are congruent, then they are straight angles. 3. Comment on the reasoning used in this statement: There is a pencil and a pen in a bag. I picked once and got the pen, picked again, and get the pen again. Therefore, the odds are greater than 50-50 that I will pick the pen again.

7.  Explanation of Content  Conditional Form: ▪ A sentence that is written in “If Then” Form.  The clause following the if is called the hypothesis.  The clause following the then is called the conclusion.  The statement “If p, then q” can be written as “p  q”  The negation of any statement p is “not p” ▪ The symbol for “not p” is ~p and ~~p = p.

8.  Converse, Inverse, and Contrapositives  Converse of “If p, then q” would be “If q, then p”  Inverse of “If p, then q” would be “If ~p, then ~q”  Contrapositive of “If p, then q” would be “If ~q, then ~p”  Statements where the converse does not make sense  If Levy lives in Pittsburgh, then he must live in Pennsylvania. ▪ Converse- If Levy lives in Pennsylvania, then he must live in Pittsburgh.  Theorem 3 :  If a conditional statement is true, then the contrapositive of the statement is also true. (If p, then q  If ~q, then p)

9.  Chains of Reasoning  If p  q and q  n, then p  n. ▪ Called the chain rule, and a series of conditional statements connected is called the chain of reasoning.

10.  Venn Diagram: A useful tool for determining if a statement is true or false. ▪ Example: “If Richard lives in Boise, then Richard must live in Idaho.” ▪ (Use the diagram below) All the people live in Idaho are represented by the large circle (I) ▪ All the people who live in Boise are represented by the smaller circle (B). ▪ Notice that all the people in set B, including Richard (R), is also in set I. I R B

11.  Sample Problem  Write the converse, inverse, and contrapositive of the following statement: ▪ “If two angles have the same measure, then they are congruent.”  Answers ▪ Converse- “If two angles are congruent, then they have the same measure.” ▪ Inverse- “If two angles do not have the same measure, then they are not congruent.” ▪ Contrapositive- “If two angles are not congruent, then they do not have the same measure.”  Is the statement true or false? ▪ 1. If Richard lives in Idaho, then he must live in Boise. ▪ 2.If Richard does not live in Idaho, then he does not live in Boise.

12.  Answers  1. This statement is not necessarily true. Richard could live in Idaho and yet not live in Boise.  2. This statement is true. If he does not live in the state of the city then he obviously does not live in the city. I R B R I B

13.  Practice Problems 1. Tell which one of these statements is “biconditional” ( The statement and its converse are true) a)If two angles are congruent, then they have the same measure. b)If two angles are right angles, then they are congruent. 2. Write “Two consecutive angles of a rectangle are supplementary” in “If then” form. 3. Write a concluding statement for the following chain of reasoning n  m r  ~p ~p  n m  s

14.  Section 1.7-  1) Undefined terms, Assumptions or Postulates, Definitions, and Theorems and Other Conclusions  2) a. If an angle is obtuse, then the angle is a 115° angle. False. b. If two angles are straight angles, then they are congruent. True.  3) Not true, probabilities do not “grow”  Section 1.8-  1) a.  2) If two angles are consecutive angles of a rectangle, then the two angles are supplementary.  3) r  s

15.  Bowman, Dave. "Geometry Notes — Conditionals and Other Statements « Bowman’s Website." Bowman’s Website. Word Press Classic, 28 Aug. 2009. Web. 18 Jan. 2011..  Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Evanston, IL: McDougal, Littell, 1991. Print.  Thorne, Lisa. "Geometry Definition: Undefined Terms." FREE Animated and Narrated Step-by-Step Tutorials in Math and Computer Applications for High School and Middle School Students. University of California Las Angeles, 2004. Web. 18 Jan. 2011.