Geometry2

Chapter 2 : Reasoning and Proof

Statements such as POSTULATES and CONJECTURES are written in Conditional if-then form. The hypothesis, the "if" part of the statement, contains the conditions that must be met in order for the conclusion, the "then" part of the statement, to be true.

Example:
IF K is the midpoint of LN, then LK=KN.

In Symbolic form it would look like
p -> q
where "p" represents the hypothesis
and "q" represents the conclusion.

The Converse of that statement
If LK = KN then K is the midpoint of LN
is not necessarily true.

In symbolic form the converse would be written as . . . q -> p

Conditional statements for which the converse is also true are called biconditional statements.

Properties of Equality that can be used to justify statements in a proof. Study these on page 96 of your text book.
Reflexive: a = a
Symmetric: if a = b then b = z
Transitive: if a = b and b = c then a = c
Substitution: replace something with its equivalent
Addition:           add, subtract, multiply, or divide
Subtraction       both sides of an equation
Multiplication     by the
Division             same number

Definitions that we will frequently use
Midpoint
Angle Bisector

Postulates (some are listed on pg. 73)
Angle Addition Postulate
Segment Addition Postulate
Corresponding Angles Postulate

We will usually use the T-Proof because it is easy to construct and easy to follow.

Statements

Reasons

How to construct a T-PROOF

1.Premise
2.Each statement
3.should be a
4.logical
5.conclusion
6.from the
7.previous
8.statements

1.Given
2.Each reason
3.should be a
4.definition, postulate
5.property, or
6.previously
7.proven
8.theorem

Some proofs start out as generalities like: Prove that all pairs of vertical angles are congruent.

This general statement is easier to prove with specific information, so draw a picture and restate the problem

INVALID mathematical proofs

Proof by obviousness: The proof is so clear I don't need to write it.

Proof by imagination; We'll pretend it's true

Proof by plausibility: It sounds good so it must be true

Proof by intimidation: Don't be stupid, of course it's true.

Proof by tautology: It's true because it's true.

Proof by plagarism: See page 289

Proof by hasty generalization: It works for 17 so it must be true for all Real Numbers.

Proof by intuition: I have this gut feeling that it must be true.

Proof by illegibility: If he can't read it he'll think that it's OK

Given: two intersecting lines,
Prove: <D is congruent to <B

Statements

Reasons