Reasoning and Proof Chapter 2

2.1 Use Inductive Reasoning Conjecture- an unproven statement based on an observation Inductive reasoning- finding a pattern in a specific case and then writing a conjecture for the general case (like a theory) Counterexample- a specific case which makes a conjecture false

2.4 Using Postulates and Diagrams Ruler Postulate Segment Addition Postulate Protractor Postulate Angle Addition Postulate

Through any two points there exists exactly one line. A line contains at least two points. If two lines intersect, then their intersection is exactly one point. Through any three noncollinear points there exists exactly one plane. A plane contains at least three noncollinear points. If two points lie in a plane, then the line containing them lies in the plane. If two planes intersect, then their intersection is a line.

Diagrams

2.2 Analyze Conditional Statements Conditional statement- a logical statement that has 2 parts, “If p, then q.” hypothesis (p) and conclusion (q) If the animal is a poodle, then it is a dog. If the quadrilateral has 4 right angles, then it is a rectangle.

Negation- the opposite of the original statement, not p or ~p The quadrilateral does not have 4 right angles.

Conditionals have truth values. To be true, the conclusion must be true every time the hypothesis is true. To be false, you need only one counter example.

Related conditionals Conditional - p → q Converse- q → p Inverse- ~p → ~q Contrapositive- ~q → ~p

Equivalent statements- same truth value conditional and contrapositive converse and inverse Definition can be written in if-then form. In this case, all four statements are true.

Biconditional- a statement with the phrase “if and only if” used when both conditional and converse are true p if and only if q, p iff q, p ↔ q Perpendicular lines- two lines that intersect to form a right angle Two lines are perpendicular if and only if they intersect to from a right angle.

2.3 Apply Deductive Reasoning Deductive Reasoning- using facts, definitions, accepted properties and laws of logic to form a logical argument.

Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Example: If 2 angles are right angles, then they are congruent. <C and <D are right angles. Conclusion ?

Law of Syllogism If p then q These statements are true. If q then r You can conclude: If p then _____

Truth Tables- conditional, converse, conjunction, disjunction TT p TT TF FT FF

More truth tables- inverse, contrapositive TT TF FT FF

And more truth tables TTT TTF TFT TFF FTT FTF FFT FFF

2.5 Reasons Using Properties for Algebra See handout

2.6 Prove Statements about Segments and Angles Proof- a logical argument that shows a statement is true Two column proof- numbered statements and corresponding reasons in logical order

2.7 Prove Angle Pair Relationships Right Angle Congruence Theorem- All right angles are congruent. Given: Prove:

Linear Pair Postulate If two angles form a linear pair, then they are supplementary. <1 and <2 form a linear pairdef. of linear pair <1 and <2 are supplementaryLinear Pair Post. m<1 + m<2 = 180 o def. of supplementary

Vertical Angles Theorem Vertical angles are congruent.

Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent.