Chapter 3 Proving Statements in Geometry
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.
1 = 1
1+3 = 4
1+3+5 = 9
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 = aSymmetric 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.