McGraw Hill Geometry Chapter 2

Inductive Reasoning

(2-1) The kind of reasoning based on observed patterns rather than properties and definitions.

Conjecture

(2-1) A concluding statement based on inductive reasoning

Counterexample

(2-1) A false example used to disprove a conjecture or a statement. Can be a number, drawing, or another statement.

Statement

(2-2) A sentence that can be true or false.

Truth Value

(2-2) The truth or falsity of a statement (if a statement is true or false)

Negation

(2-2) The opposite meaning of a statement.
eg. the negation of "p: A rectangle is a quadrilateral" is
"~p: A rectangle is a quadrilateral"

Compound Statement

(2-2) Two or more statements joined by the words "and" or "or"

Conjunction

(2-2) A compound statement joined by the word "and"

Disjunction

(2-2) A compound statement joined by the word "or"

Truth Table

(2-2) a convenient method for organizing the truth values of statements.

Conditional Statement

(2-3) a statement that can be written in the "if-then form"
eg. If you would like to speak to a representive, then you will press zero now"

If-Then Statement

(2-3) A statement in the form of "p→q"

Hypothesis

(2-3) The part of the if-then statement directly following the word "if"

Conclusion

(2-3) The part of the if-then statement directly following the word "then"

Related Conditionals

(2-3)Conditionals that are based on a given conditional statement.

Converse

(2-3) to switch the hypothesis and conclusion in a conditional eg. "q→p" instead of "p→q"

Inverse

(2-3) To negate the hypothesis and conclusion of a conditional. eg. "~p→~q" instead of "p→q"

Contrapositive

(2-3) to negate the converse of a conditional statement.
"~q→~p" instead of "p→q"

Logically Equivalent

(2-3) two statements with the same truth value.

Deductive Reasoning

(2-4) using facts and rules instead of patterns to make a conclusion.

Valid

(2-4) If something is logically correct (true

Law of Detachment

(2-4) if "p→q" is true and if p is true, then q is also true.

Law of Syllogism

(2-4) if p→q is true and q→r is true, then p→r is also true

Postulate

(2-5) a statement that is accepted as true without proof. also known as an "axiom"

(2-5) through any two points, there is.....?

through any two points, there is exactly one line.

(2-5) Through any three non-collinear points, there is.....?

Through any three non-collinear points, there is exactly one plane.

(2-5) A line contains....?

A line contains at least two points.

(2-5) A plane contains at least.....?

A plane contains at least three non-collinear points

(2-5) If two points lie in a plane, then.....?

If two points lie in a plane, then the entire line containing those points lies in that plane.

(2-5) If two lines intersect, then......?

If two lines intersect, then their intersection is exactly one point

If two planes intersect, then.....?

If two planes intersect, then their intersection is a line

Proof

(2-5) a logical argument in which each statement you make is supported by a statement that is accepted as true.

Theorem

(2-5) Once a statement or conjecture is proven, then it's called a theorem and can be used to justify other statements in other proofs.

Paragraph Proof

(2-5) also called informal proof, is a written paragraph that proves a statement.

Midpoint Theorem

(2-5) if M is the midpoint of AB, then AM is congruent to MB

Algebraic Proof

(2-6) a proof that is made of algebraic statements

Two Column Proof

(2-6) a proof with two columns, one for the statement and one for the reasons/postulates/theorems

Reflexive Property

(2-6) a=a

Symmetric Property

(2-6) if a=b, then b=a

Transitive Property

(2-6) If a=b and b=c, then a=c

Ruler Postulate

(2-7) the points on an line or line segment can be put into one-to-one correspondence with real numbers.

Segment addition postulate

if a, b, and c are colluinear, then point B is between a and c if and only if AB+BC=AC

transversal

a line that intersects two coplanar lines at two different points; makes 8 angles

interior angle

if t is a transversal of q and r, the interior angles are in the region BETWEEN q and r (4)

exterior angle

if t is a transversal of q and r, the exterior angles are in the regions that are NOT between lines q and r (4)

consecutive interior angles

interior angles that lie on the same side of transversal t (will be two pairs)

alternate interior angles

Nonadjacent interior angles lying on opposite sides of a transversal t (2 pairs)

alternate exterior angle

nonadjacent exterior angles that lie on opposite sides of transversal t (2 pairs)

corresponding angle

lie on the same side of transversal t and and on the same side of lines q and r

parallel lines

two lines that are in the same plane, but do not intersect

skew lines

Noncoplanar, non-parallel lines that do not intersect

parallel planes

planes that do not intersect

Alternate Interior Angles Theorem

if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent

Consecutive Interior Angles Theorem

if two parallel lines are cut by a transversal, then each pair of consecutive interior lines is supplementary

alternate exterior angles theorem

if two parallel lines are intersected by a transversal, then alternate exterior angles are equal in measure

transitive property

if AB equals CD, and CD equals EF, then AB equals EF

Reflexive Property of Congruence

line AB is equal to line AB

Symmetric Property of Congruence

If line AB≅line CD, then line CD≅line AB

supplement theorem

If two angles form a linear pair, then they are supplementary
Linear Pair→Supplementary (equals 180)

complement theorem

if the noncommon sides of two adjacent angles form a right angle (angle 1 plus angle 2 equals 90)

vertical angles

two nonadjacent angles formed by interscting lines