Chapter Two

Emma Risa Haley Kaitlin

2.1

Inductive reasoning: find a pattern in specific cases and then write a conjecture

Conjecture: unproven statement based on observations

Example: the sum of two numbers is always greater than the larger number (2+3=5)

Counterexample: a specific case where the conjecture is false

Conjecture counter example: -2+-3=-5

2.2Conditional statement: a logical statement with two parts- hypothesis and

conclusion If then form: if contains hypothesis then contains conclusionExample: If it is raining, then there are clouds in the sky.Negation: a statement opposite the original statementStatement: The wall is purple.Negation: The wall is not purple.Converse: exchange the hypothesis and conclusion of a conditional statementExample: If there are clouds in the sky, then it is rainingInverse: negate both the hypothesis and conclusionExample: If it is not raining, then there are no clouds in the sky.Contrapositive: write converse then negate both the hypothesis and conclusionExample: If there are no clouds in the sky, then it is not raining.

2.2 continued…Biconditional statement: This is a statement that contains the phrase “if and only

if”. Example: It is raining if and only if there are clouds in the sky.

2.3Deductive Reasoning: The process of using logic to draw conclusionsInductive Reasoning: Reasonings from examplesLaw of Detachment: If the hypothesis of a true conditional statement is true, then

the conclusion is also true.Example: If it is Monday, then I will go to school. Today is Monday.Law of Syllogism:Example: If 5+b= 10, then b+7=12

If b+7=12, then b=5 If b=5, then 6+b=11

2.4Postulates: Rules that are accepted without proofTheorems: Rules that are proved

Point, Line, and Plane Postulates5. Through any two points there exists exactly one line6. A line contains at least two points7. If two lines intersect then their intersection is exactly one point.8. Through any three noncollinear points there exists exactly one plane.9. A plane contains at least three noncollinear points.10. If two points lie in a plane, then the line containing them lies in the plane.11. If two planes intersect, then their intersection is a line.

2.5Algebraic Properties of Equality

Addition PropertyEx: If a=b, then a-c=b-cSubtraction PropertyEx: If a=b, then a-c=b-cMultiplication PropertyEx: If a=b, then ac=bcDivision PropertyEx: If a=b and c does not =0, then a/b=b/cSubstitution PropertyEx: If a=b, then a can be substituted for b in any equation or expression.

2.5 Continued…Distributive Property of EqualityEx: a(b+c)=ab+acReflexive Property of EqualityEx: For any segment AB, AB=ABSymmetric Property of EqualityEx: For any segments AB and CD, if AB=CD, then CD=ABTransitive Property of EqualityEx: For any segments AB, CD, and EF, if AB=CD and CD=EF, then AB=EF

2.6Proof: A logical argument that shows a statement trueTwo-Column Proof: Numbered statements and corresponding reasons that show

an argument in a logical order.Theorem: A statement that can be proven.

2.7Right Angles Congruence Theorem: All right angles are congruentCongruent Supplements Theorem: 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.Linear Pair Postulate: If two angles form a linear pair, then they are

supplementaryVertical Angles Congruence Theorem: Vertical angles are congruent.