Download Chapter Two - Clayton School District

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
2.2
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.
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.
2.3
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
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.
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=b and c does 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.
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
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.
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.