McGraw Hill Reveal Geometry Chapter 3

Inductive Reasoning

the process of reaching a conclusion based on a pattern of examples

conjecture

an educated guess based on known information and specific examples.

counterexample

one example that contradicts the conjecture, however, to show that a conjecture is not always true

statement

any sentence that is either true or false

Truth Value

the truth or falsity of a statement

negation

The opposite of a statement

compound statement

two or more statements joined by the word "and" or "or"

conjunction

is true only when both statements that form it are true

disjunction

a compound statement that uses the word or

conditional statement

A compound statement that consists of a premise, or hypothesis, and a conclusion, which is false only when its premise is true and its conclusion is false.

converse

the statement formed by exchanging the hypothesis and conclusion of a conditional statement

inverse

the statement formed by negating both the hypothesis and conclusion of a conditional statement

contrapositive

the statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement

biconditional statements

the conjunction of a conditional and its converse. Often phrase "if and only if..."

deductive reasoning

uses general facts, rules, definitions, or properties to reach specific valid conclusions from given statements

valid

it is impossible for all the premises, or supporting statements, of the argument to be true and for its conclusion to be false.

Law of Detachment

If a conditional statement ("if P, then Q") is true, and its hypothesis (P) is true, then its conclusion (Q) must also be true

Law of Syllogism

allows you to draw conclusions from two true conditional statements when the conclusion of one statement is the hypothesis of the other.

3.1 Postulate Points, Lines, & Planes

Through any two points, there is exactly one line.

3.2 Postulate Points, Lines, & Planes

Through any three noncollinear points, there is exactly one plane.

3.3 Postulate Points, Lines, & Planes

A line contains at least two points.

3.4 Postulate Points, Lines, & Planes

A plane contains at least three noncollinear points.

3.5 Postulate Points, Lines, & Planes

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

3.6 Postulate Points, Lines, & Planes

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

3.7 Postulate Points, Lines, & Planes

If two planes intersect, then their intersection is a line.

proof

a logical argument in which each statement is supported by a statement that is accepted as true.

two-column proof

proof that contains statements and reasons that are organized in a two-column format.

Deductive Argument

a proof formed by a group of algebraic steps used to solve a problem

flow proof

uses boxes and arrows to show the logical progression of an argument.

paragraph proof

prove a conjecture is to write a paragraph that explains why the conjecture for a given situation is true.

Ruler Postulate

When you use a ruler to measure the length of an object, you match the mark for zero at one endpoint of the object. Then you look for the ruler mark that corresponds to the other endpoint.

Postulate 3.8 Ruler Postulate

The points on any line or line segment can be put into one-to-one correspondence with real numbers.

Postulate 3.9: Segment Addition Postulate

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

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

Postulate 3.10: Protractor Postulate

Every angle has a measure that is between 0 and 180.

Postulate 3.11: Angle Addition Postulate

D is in the interior of

Theorem 3.3: Supplement Theorem

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

Theorem 3.4: Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.

Congruent Angles

The properties of algebra that apply to the congruence of segments and the equality of their measures also hold true for the congruence of angles and the equality of their measures.

Theorem 3.8: Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.

Theorem 3.9 Right Angle Theorems

Perpendicular lines intersect to form four right angles.

Theorem 3.10 Right Angle Theorems

All right angles are congruent.

Theorem 3.11 Right Angle Theorems

Perpendicular lines form congruent adjacent angles.

Theorem 3.12 Right Angle Theorems

If two angles are congruent and supplementary, then each angle is a right angle.

Theorem 3.13 Right Angle Theorems

If two congruent angles form a linear pair, then they are right angles.

Skew lines

are lines that do not intersect and are not coplanar.

Parallel planes

are planes that do not intersect.

Transversal

A line that intersects two or more lines in a plane at different points

Same-side Interior Angles

Consecutive interior angles are also called

Theorem 3.14: Corresponding Angles Theorem

If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

Theorem 3.15: Alternate Interior Angles Theorem

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

Theorem 3.16: Consecutive Interior Angles Theorem

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

Theorem 3.17: Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

Theorem 3.18: Perpendicular Transversal Theorem

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Slope

the ratio in the change in the y-coordinate (rise) to the corresponding change in the x coordinate (run) as you move along a line

Slope Criteria

outlines a method for proving the relationship between lines based on a comparison of the slopes of the lines

Slopes of Parallel Lines

Two distinct nonvertical lines have the same slope if and only if they are parallel. All vertical lines are parallel.

Slopes of Perpendicular Lines

Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.

slope intercept form of a line

y=mx+b

Point slope form of a line

y-y1=m(x-x1)

Theorem 3.19 Converse of Corresponding Angles Theorem

If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel

Postulate 3.13 Parallel Postulate

If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

Theorem 3.20 Alternate Exterior Angles Converse

If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.

Theorem 3.21: Consecutive Interior Angles Converse

If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.

Theorem 3.22: Alternate Interior Angles Conserve

If two lines in a plane re cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel

Theorem 3.23: Perpendicular Transversal Converse

If two lines in a plane are perpendicular to the same line, then the lines are parallel.

Postulate 3.14 (Perpendicular Postulate)

If given a line and a point not on the line, then there exists exactly one line through the point that is perpendicular to the given line.

Theorem 3.24: Two Lines Equidistant from a Third

In a plane, if two lines are each equidistant from a third line, then the two lines are parallel to each other.