Reasoning and Proof

Chapter 2

2.1 Use Inductive ReasoningConjecture- an unproven statement based

on an observationInductive 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 DiagramsRuler PostulateSegment Addition PostulateProtractor PostulateAngle 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 StatementsConditional 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 conditionalsConditional - 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 ReasoningDeductive Reasoning- using facts,

definitions, accepted properties and laws of logic to form a logical argument.

Law of DetachmentIf 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 SyllogismIf p then q

These statements are

true.If q then rYou can conclude: If p then _____

Truth Tables-conditional, converse, conjunction, disjunction

T T p

T TT FF TF F

More truth tables-inverse, contrapositive

T TT FF TF F

And more truth tables

T T TT T FT F TT F FF T TF T FF F TF F F

2.5 Reasons Using Properties for AlgebraSee handout

2.6 Prove Statements about Segments and AnglesProof- a logical argument that shows a

statement is trueTwo column proof- numbered statements

and corresponding reasons in logical order

2.7 Prove Angle Pair RelationshipsRight Angle Congruence Theorem- All

right angles are congruent.Given: Prove:

Linear Pair PostulateIf two angles form a linear pair, then they

are supplementary.

<1 and <2 form a linear pair def. of linear pair

<1 and <2 are supplementary Linear Pair Post.

m<1 + m<2 = 180o def. of supplementary

2 1

Vertical Angles TheoremVertical angles are congruent.

432

1

Congruent Supplements TheoremIf two angles are supplementary to the

same angle (or to congruent angles), then they are congruent.

3

21

34

21

Congruent Complements TheoremIf two angles are complementary to the

same angle (or to congruent angles), then they are congruent.

3

4

21

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