# Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic - PowerPoint PPT Presentation Download Presentation Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic

Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic Download Presentation ## Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic

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1. Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic Learner Objective: Students will write the converse, inverse, and   contrapositive of a conditional statement, determine the truth   value of statements and will use the chain rule to make logical   conclusions.

2. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. 1 A B C D E

3. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Building blocks of Geometry: Undefined Terms - Terms which we can't officially   define. We describe their properties but don't   formally define them.  examples: point, line, plane Definitions - State the meaning of a term. There is   never a need to prove a definition. (It is true   because we say it is true.)  examples: Def. of a right angle, Def. of a    segment bisector...

4. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Building blocks of Geometry (cont.): Postulates - An unproved assumption. These are pretty   obvious statements which we are unable to prove but   which clearly must be true.  example: Two points determine a line. Theorems - A mathematical statement which can be   proved.  example: If two angles are both right angles,   then they are congruent.

5. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Definitions, Postulates and Theorems are all conditional   statements. They have some (sufficient) condition that   leads to a (necessary) conclusion. If two angles are both right angles, Having two right angles is sufficient evidence to conclude  that the angles are congruent. then they are congruent. When two angles are both right angles, it is necessary to  conclude that they are congruent (they have to be). What happens if we reverse the condition and conclusion?

6. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Here is the theorem with the condition and conclusion reversed: If two angles are congruent Is knowing that two angles are congruent sufficient  evidence to conclude that they must be right angles? then they are both right angles. If two angles are congruent, is it necessary to conclude  that they are right angles? Is this a true statement? For a statement to be considered true, it must ALWAYS be   true. NOT ALWAYS TRUE = FALSE

7. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Conditional statements can be written in "If p then q" form  where p is the condition or hypothesis of the statement  (trigger) and q is the conclusion (result). "If p then q" can be symbolized (also read "p implies q")

8. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Reversing the condition and the conclusion results in the  Converse of the original statement. Thus the convers is  symbolized If both the conditional and the converse are true, the  statement is said to be reversible. Definitions are always reversible. Postulates and Theorems are sometimes reversible.

9. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. The Negation (opposite) of the statement p is "not p" and  is symbolized The Inverse of the conditional statement is formed when  we negate both p and q. Thus the inverse is symbolized

10. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. The Contrapositive of the conditional statement is formed  by both reversing and negating p and q. Thus, the  contrapositive can be symbolized

11. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. The conditional statement and it's contrapositive have the  same "truth value". That is, they are either both true or  both false. The inverse and the converse have the same truth value.

12. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Example Conditional: If two angles are both right angles,  then they are congruent Converse: If two angles _______________ then ______________________ Inverse: If two angles _______________ then ______________________ Contrapositive: If two angles _______________ then ______________________

13. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Example If an angle is an acute angle, then its measure is greater than 0 and less than 90. Conditional: Converse: If an angle _____________________________, then __________________________________. Inverse: If an angle _____________________________, then __________________________________. Contrapositive: If an angle _____________________________, then __________________________________.

14. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. A series of conditional statements can be connected  together using the Chain Rule. If and then

15. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. Example: , , and ,

16. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting  must be given a three day suspension. Bill, Bob and Bo are all students  at Hilldale. Which of the following statements is/are true? Bill has been given a three day suspension so we know that he   must have been caught fighting. Bob has never been caught fighting, so we know that he has   never been given a three day suspension. Bo has never been given a three day suspension, so we know   that he has never been caught fighting.

17. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting  must be given a three day suspension. Bill, Bob and Bo are all students  at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bill has been given a three day suspension so we know that he   must have been caught fighting. Converse: If suspension then fighting False It is not necessary for someone to Fight in order to receive a  suspension. Bill could have been suspended for something else.  The converse is false.

18. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting  must be given a three day suspension. Bill, Bob and Bo are all students  at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bob has never been caught fighting, so we know that he has   never been given a three day suspension. Inverse: If NOT fighting then NO suspension. False Fighting is a sufficient condition to receive a suspension but not  a necessary one. Bob could have been suspended for something  else. The inverse is false.

19. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting  must be given a three day suspension. Bill, Bob and Bo are all students  at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bo has never been given a three day suspension, so we know   that he has never been caught fighting. If NO suspension then NOT fighting. Contrapositive: True Since suspension is a necessary result of fighting, NOT being  suspended is sufficient to conclude that there was NO fighting.  The contrapositive is true.

20. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

21. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

22. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

23. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

24. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

25. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

26. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

27. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

28. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

29. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

30. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

31. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

32. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

33. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

34. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

35. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

36. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.

37. Learner Objective: Students will write the converse, inverse, and contrapositive of a   conditional statement, determine the truth value of statements and will use the   chain rule to make logical conclusions.