Chapter 3 Proving Statements in Geometry

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    Inductive reasoning:  reaching a conclusion based on recognizing patterns in data.   This does not necessarily constitute proof that your conclusion is correct.   Example from class: 

    1 dot on a circle & all possible segments drawn = 1 region

    2 dots (results in 1 segment) = 2 regions

    3 dots (results in 3 segments) = 4 regions

    4 dots (results in 6 segments) = 8 regions

    5 dots (results in 10 segments) = 16 regions

    See the pattern?   You think so, but 6 dots (results in 15 segments) results in 31 regions, not the expected 32. 

     

    Another example: 

    1 = 1

    1+3 = 4

    1+3+5 = 9

    1+3+5+7=16

    1+3+5+7+9=25

    1+3+5+7+9+11=36

    Notice that the sum of n-odd integers is equal to n²?

    The sum of the first 7 odd integers should equal to 7² = 49.   1+3+5+7+9+11+13 DOES equal 39.   This statement turns out to be true, but requires more advanced math to prove that it always holds true.


    Chapter 3 Postulates:

     

    Reflexive Property of Equality:   a = a

    Symmetric Property of Equality:   If a = b, then b = a
    Transitive Property of Equality:  If a = b and b = c, then a = c

    Substitution:  A quantity may be substituted for its equal in any statement of equality

    A whole is equal to the sum of all its parts

    A segment is congruent to the sum of all its parts

    An angle is congruent to the sum of all its parts

    Addition:  If equal quantities are added to equal quantities, the sums are equal

    If congruent segments are added to congruent segments, the sums are congruent

    If congruent angles are added to congruent angles, the sums are congruent

    Subtraction:  If equal quantities are subtracted from equal quantities, the differences are equal

    If congruent segments are subtracted from congruent segments, the differences are congruent

    If congruent angles are subtracted from congruent angles, the differences are congruent.

    Multiplication:  If equals are multiplied by equals, the products are equal.

    Doubles of equal quantities are equal

    Division:  If equals are divided by nonzero equals, the quotients are equal

    Halves of equal quantities are equal

    The squares of equal quantities are equal

    Positive square roots of equal quantities are equal.