Chapter Two

1. Chapter Two Emma Risa Haley Kaitlin

2. 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

3. 2.2 Conditional statement: a logical statement with two parts- hypothesis and conclusion If then form: if contains hypothesis then contains conclusion Example: If it is raining, then there are clouds in the sky. Negation: a statement opposite the original statement Statement: The wall is purple. Negation: The wall is not purple. Converse: exchange the hypothesis and conclusion of a conditional statement Example: If there are clouds in the sky, then it is raining Inverse: negate both the hypothesis and conclusion Example: If it is not raining, then there are no clouds in the sky. Contrapositive: write converse then negate both the hypothesis and conclusion Example: If there are no clouds in the sky, then it is not raining.

4. 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.

5. 2.3 Deductive Reasoning: The process of using logic to draw conclusions Inductive Reasoning: Reasonings from examples Law 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

6. 2.4 Postulates: Rules that are accepted without proof Theorems: Rules that are proved Point, Line, and Plane Postulates 5. Through any two points there exists exactly one line 6. A line contains at least two points 7. 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.

7. 2.5 Algebraic Properties of Equality Addition Property Ex: If a=b, then a-c=b-c Subtraction Property Ex: If a=b, then a-c=b-c Multiplication Property Ex: If a=b, then ac=bc Division Property Ex: If a=band cdoes not =0, then a/b=b/c Substitution Property Ex: If a=b, then a can be substituted for b in any equation or expression.

8. 2.5 Continued… Distributive Property of Equality Ex: a(b+c)=ab+ac Reflexive Property of Equality Ex: For any segment AB, AB=AB Symmetric Property of Equality Ex: For any segments AB and CD, if AB=CD, then CD=AB Transitive Property of Equality Ex: For any segments AB, CD, and EF, if AB=CD and CD=EF, then AB=EF

9. 2.6 Proof: A logical argument that shows a statement true Two-Column Proof: Numbered statements and corresponding reasons that show an argument in a logical order. Theorem: A statement that can be proven.

10. 2.7 Right Angles Congruence Theorem: All right angles are congruent Congruent 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 supplementary Vertical Angles Congruence Theorem: Vertical angles are congruent.