1. Lesson: Pages: Objectives: 2.6 Using Proof 92 – 94  To Learn the Midpoint Theorem  To Describe Properties Used in Proofs  To Learn Required Elements of Proof  To Learn the Forward-Backward Approach to Proof  To PRACTICE

2. MIDPOINT THEOREM If M is the midpoint of, then

3. GEOMETRY 2.6 Using Proof Why are PROOFS Needed? Proofs Justify THEOREMS with GENERAL APPLICATION General Application Proofs avoid REPETITION SO

4. GEOMETRY 2.6 Using Proof Why are PROOFS Needed? Proofs BUILD Theorems Proofs are the PATH to Theorems

5. GEOMETRY 2.6 Using Proof Some EXAMPLES of BUILDING a PATH

6. GEOMETRY 2.6 Using Proof

14. The POINT of all these pics? Sometimes the PATH is OBVIOUS OTHER TIMES the PATH requires Looking FIRST at your DESTINATION!

15. GEOMETRY 2.6 Using Proof CONSIDER: How did they Find the PATH to Build the St. Louis ARCH?

16. GEOMETRY 2.6 Using Proof

17. The LESSON:  Some PROOFS are OBVIOUS PATHS from their Start  Some PROOFS are NOT OBVIOUS; you find the PATH by looking at the DESTINATION and plotting your path BACKWARDS!  Some PROOFS require BUILDING from BOTH ENDS to the MIDDLE!

18. GEOMETRY 2.6 Using Proof DEFINITION: A PROOF is a LOGIC PATH leading from a HYPOTHESIS to a CONCLUSION using AXIOMS, POSTULATES, DEFINITIONS, THEOREMS, and ALGEBRA

19. GEOMETRY 2.6 Using Proof DEFINITION: A PROOF is a LOGIC PATH EACH STEP on the LOGIC PATH is an IF – THEN LINK EACH STEP is composed of a STATEMENT, and a REASON

20. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof

21. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing

22. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing GIVEN Statement

23. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing GIVEN Statement PROVE Statement

24. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing GIVEN Statement PROVE Statement List of Deductive Logic Steps

25. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing GIVEN Statement PROVE Statement List of Deductive Logic Steps A REASON justifying each Logic Step

26. GEOMETRY 2.6 Using Proof The REQUIRED Elements of a 2-Column Proof A Drawing GIVEN Statement PROVE Statement List of Deductive Logic Steps A REASON justifying each Logic Step A CONCLUSION

27. GEOMETRY 2.6 Using Proof Required BRAIN Elements of a Proof: Knowledge of Linking Tools Intuition about the Best Path CREATIVITY

28. GEOMETRY 2.6 Using Proof ANOTHER LOOK AT THE LINKING TOOLS

29. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements:

30. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements: Undefined Terms (the Big 3)

31. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements: Undefined Terms (the Big 3) Postulates

32. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements: Undefined Terms (the Big 3) Postulates Definitions

33. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements: Undefined Terms (the Big 3) Postulates Definitions Theorems previously proved

34. GEOMETRY 2.6 Using Proof We can use several tools to support proof statements: Undefined Terms (the Big 3) Postulates Definitions Theorems previously proved It’s BEST to think of your Linking Tools By CATEGORY

35. GEOMETRY 2.6 Using Proof We can also use ALGEBRA!

36. GEOMETRY 2.6 Using Proof We can also use ALGEBRA! We measure angles and segments in numbers

37. GEOMETRY 2.6 Using Proof We can also use ALGEBRA! We measure angles and segments in numbers Frequently measures are part of Equations

38. GEOMETRY 2.6 Using Proof We can also use ALGEBRA! We measure angles and segments in numbers Frequently measures are part of Equations Use Algebra, to solve equations:

39. GEOMETRY 2.6 Using Proof We can also use ALGEBRA! We measure angles and segments in numbers Frequently measures are part of Equations Use Algebra, to solve equations:  Addition & Subtraction Properties  Multiplication & Division Properties

40. GEOMETRY 2.6 Using Proof In this class, when we use these properties of Algebra Addition Subtraction Multiplication Division we will state the REASON as simply “ALGEBRA”

41. GEOMETRY 2.6 Using Proof There are OTHER properties of Algebra used for proofs that are named SPECIFICALLY:

42. GEOMETRY 2.6 Using Proof

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