Unit 2 - Logic and Proofs Test

Inductive reasoning

making a conclusion based on observations and patterns

conjecture

a concluding statement reached using inductive reasoning

Counterexample

An example which proves a conjecture false

Statement

A sentence that is either true or false (truth value), represented using letters such as 'p' or 'q'. (ex. p: supplementary angles have a sum of 180 degrees - truth)

negation

A negation of a statement has the opposite truth value, shown by the symbol "~", or "not p" (ex. ~p: supplementary angles *do not* have a sum of 180 degrees - false)

compound statements

two or more statements joined by the words "and", or "or"

conjunction

statements joined together by the word "and", written as p ^ q. True when both statements are true.

disjunction

statements joined together by the word "or", written as p ˇ q. True when at least one statement is true.

conditional statement

statement that can be written in if-then form, or in symbolic form, p -> q. The hypothesis is the phrase that follows "if", and the conclusion is the phrase that follows "then"

inverse

formed by negating the hypothesis and conclusion (~p -> ~q)

converse

formed by switching the hypothesis and conclusion (q -> p)

contrapositive

formed by both negating and switching the hypothesis and conclusion (~q -> ~p)

biconditional statement

the conjunction of the conditional and its converse. Represented by p iff q, (p -> q) ^ (q -> p), or ( p <--> q) True when both the conditional and converse are both true or false.

deductive reasoning

the process of reasoning logically and drawing a conclusion from given facts and statements

law of detachment

Given a conditional statement, if the hypothesis is true, then the conclusion is true

law of syllogism

Allows you to draw a conclusion from two condition statements, in which the conclusion of the first one is the hypothesis of the second statement

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 axc=bxc

Division property of equality

If a=b, then a/c=b/c

Distributive property of equality

a(b+c), then a(b+c) = ab + ac

Substitution property of equality

If a=b, then a may be replaced by b in any expression or equation

reflexive property of equality

for any real number a, a=a

Symmetric property of equality

if a=b, b=a

transitive property

if a=b, and b=c, then a=c

reflexive property of congruence

For any segment AB, line AB is congruent to line AB

Symmetric property of congruence

If line AB is congruent to line CD, then line CD is congruent to line AB

transitive property of congruence

If line AB is congruent CD and line CD is congruent to line EF, then line AB is congruent to line EF

Definition of congruence

Segments are congruent if and only if they have the same measure:
If line AB is congruent to line CD, then AB = CD
If AB = CD, then line AB is congruent to line CD

Definition of midpoint

If M is the midpoint of line AB, then line AM is congruent to line MB

Segment Addition Postulate

If A, B, and C are collinear points and B is between A and C, then AB + BC = AC

Definition of congruence

The measures of two angles are equal if and only if the angles are congruent. If the measure of A = the measure of B, then angle A is congruent to angle B, vice versa.

Definition of a Right Angle

An angle measures 90 degrees if and only if it is a right angle. If the measure of A = 90 degrees, then angle A is a right angle

Definition of complementary angles

Two angles are complementary, if and only the sum of their measures is 90 degrees

Definition of supplementary angles

Two angles are supplementary, if and only the sum of their measures is 180 degrees

Definition of an angle bisector

An angle bisector divides an angle into two equal parts

Definition of perpendicular

perpendicular lines form right angles

Angle addition postulate

The measure of ABD and the measure of DBC = the measure of ABC

Vertical angles theorem

if two angles are vertical, then they are congruent.

complement theorem

If two angles form a right angle, then they are complementary

Linear pair theorem (supplement theorem)

If two angles form a linear pair, then they are supplementary

Congruent complements theorem

If two angles are complementary to the same angle, then they are congruent (if angle A is complementary to angle B, and angle C is complementary to angle B, then angle A is congruent to angle C)

Congruent supplements theorem

If two angles are supplementary to the same angle, then they are congruent (if angle A is supplementary to angle B, and angle C is supplementary to angle B, then angle A is congruent to angle C)