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

MIDPOINT THEOREM If M is the midpoint of, then

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

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

GEOMETRY 2.6 Using Proof Some EXAMPLES of BUILDING a PATH

GEOMETRY 2.6 Using Proof

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

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

GEOMETRY 2.6 Using Proof

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!

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

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

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

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

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

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

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

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

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

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

GEOMETRY 2.6 Using Proof ANOTHER LOOK AT THE LINKING TOOLS

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

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

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

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

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

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

GEOMETRY 2.6 Using Proof We can also use ALGEBRA!

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

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

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:

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

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”

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

GEOMETRY 2.6 Using Proof

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