Inductive Reasoning
the process of reasoning that a rule or a statement is true because specific cases are true
Conjecture
a statement that is believed to be true
Counterexample
an example that proves a conjecture or statement is false
Using Inductive reasoning...
1. Look for a pattern2. Make a conjecture3. Prove the conjecture or find a counterexample
Conditional Statement
a statement that can be written in the form "If p, then q" where p is the hypothesis and q is the conclusion
Hypothesis
the part of a conditional statement following the word "if
Conclusion
the part of a conditional statement following the word "then
Truth Value
a statement that can have a truth value of true or false
Negation
the negation of statement p is "not p" written as ~p
Converse
the statement formed by exchanging the hypothesis and conclusion of a conditional statement
Inverse
the statement formed by negating the hypothesis and conclusion of a conditional statement
Contrapositive
the statement formed by both exchanging and negating the hypothesis and conclusion of a conditional statement
Logically Equivalent Statements
statements that have the same truth value
Deductive Reasoning
the process of using logic to draw conclusions
Law of Detachment
If p --> q is a true statement and p is true, then q is true.
Law of Syllogism
If p --> q and q --> r are true statements, then p --> r is a true statement
Biconditional Statement
a statement that can be written in the form "p if and only if q
Definition
a statement that describes a mathematic object and can be written as a true biconditional statement
Proof
an argument that uses logic to show that a conclusion is true
Addition Property of Equality
If a=b, then a+c=b+c
Subtraction Property of Equality
If a=b, then a-c=b-c
Multiplication Property of Equality
If a=b, then ac=bc
Division Property of Equality
If a=b and c does not equal 0, then a/c = b/c
Reflexive Property of Equality
a=a
Symmetric Property of Equality
If a=b, then b=a
Transitive Property of Equality
If a=b and b=c, then a=c
Substitution Property of Equality
If a=b, then b can be substituted for a in any situation
Reflexive Property of Congruence
Figure A is congruent to Figure A
Symmetric Property of Congruence
If Figure A is congruent to Figure B, then Figure B is congruent to Figure A
Transitive Property of Congruence
If Figure A is congruent to Figure B and Figure B is congruent to Figure C, then Figure A is congruent to Figure C.
Theorem
any statement that you can prove
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or two congruent angles), then the two angles are congruent.
Two-Column Proof
a style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column
Right Angle Congruence Theorem
All right angles are congruent
Congruent Complements Theorem
If two angles are complementary to the same angle (or two congruent angles), then the two angles are congruent.
The Proof Process...
1. Write the conjecture to be proven2. Draw a diagram to represent the hypothesis of the conjecture3. State the given information and mark it on the diagram4. State the conclusion of the conjecture in terms of the diagram5. Plan your argument and prove the conjecture
Flowchart Proof
a style of proof that uses boxes and arrows to show the structure of the proof
Common Segments Theorem
Given collinear points, A, B, C, and D arranged as shown, if (segment) AB is congruent to (segment) CD then (segment) AC is congruent to (segment) BD.(AB is shown as being congruent to CD)
Vertical Angles Theorem
Vertical angles are congruent.
Theorem...If two congruent angles are supplementary...
then each angle is a right angle.