geometry midterm definitions

(1.2) undefined terms

intuitive ideas, basis of ALL geometry (point, line, and plane)

(1.2) point

undefined; a location is space that has no thickness, all figures are made up of points

(1.2) line

undefined; extends in opposite directions without ending and has no thickness and cannot be measured. made up of infinite number of points

(1.2) plane

undefined; a flat surface that goes on forever and has no thickness

(1.2) space

the set of ALL points, represented by 4 non-coplanar points

(1.2) collinear points

points all in ONE line

(1.2) coplanar points

points all in ONE plane

(1.2) intersection

the set of all points in both (all) figures

(1.3) between

for N to be "between" M and P, all 3 points must be collinear and N must be in the middle

(1.3) segment

2 points and all points BETWEEN them

(1.3) ray

segment XY and all points Z, such that Y is between X and Z

(1.3) opposite rays

collinear rays that share EXACTLY ONE point

(1.3) postulates

statements that are accepted without proof

(1.3) segment addition postulate

if B is between A and C, then AB + BC \= AC

(1.3) midpoint

point that divides a segment into two CONGRUENT segments

(1.3) congruent

objects (or figures) that have the same EXACT size and shape

(1.3) congruent segments

segments are congruent if and only if their measurements are EQUAL

(1.3) segment bisector

a line, segment, ray, or plane that intersects a segment at its midpoint

(1.4) angle

figure formed by two rays that have the endpoint

(1.4) point on interior

a point is on the interior of an angle if and only if it lies on a segment whose endpoints are on the angle, but the point is not

(1.4) acute angle

angle whose measure is between 0 and 90

(1.4) right angle

angle whose measure is 90

(1.4) obtuse angle

angle whose measure is between 90 and 180

(1.4) straight angle

angle whose measure is 180

(1.4) protractor postulate

(NOT WORD FOR WORD) says that every straight line is 180

(1.4) angle addition postulate

if point K lines in the interior of angle JGH, the the measure of angle JGK + the measure of angle KGH \= the measure of angle JGH

(1.4) congruent angles

angles that have EQUAL measures

(1.4) adjacent angles

TWO angles IN A PLANE that have a common vertex and a common side but no common interior points

(1.4) linear pair

adjacent angles whose non-common sides form opposite rays

(1.4) angle bisector

the ray that divides and angle into CONGRUENT, adjacent angles

(1.5) 1. a line contains at least ________

2. a plane contains at least __________

3. space contains at least ______________

1. 2 points

2. 3 noncollinear points

3. 4 non-coplanar points

(1.5) through any 2 points,

there is EXACTLY ONE line

(1.5) 1. through any three points, there is at least _________


2. through any three noncollinear points, there is EXACTLY ONE _____

1. one plane

2. plane

(1.5) if 2 points are in a plane,

then the line that contains the points, is in that plane

(1.5) if 2 planes intersect,

then their intersection is a line

(1.5) if 2 lines intersect,

then they intersect in EXACTLY ONE point

(1.5) through a line and a point not on that line,

there is EXACTLY ONE plane

(1.5) if 2 lines intersect,

then EXACTLY ONE plane contains the lines

(1.5) existence

"at least one"; the situation exists, however there could be more than one example

(1.5) uniqueness

"exactly one"; there is only one example that satisfies a given condition

logic

study of methods and principles that allows you to classify arguments

(logic) simple statements

statements that are either true or false

(logic) compound statements

the joining of 2 or more simple statements

(logic) conjunction

p ^ q; the joining of 2 simple statements with the word "and"

(logic) disjunction

p ⌄ q; joining of 2 simple statements with the word "or"

(logic) truth tables

used to tell under what conditions a compound statement is true or false

(logic) inclusive "OR"

TRUE if the first, second, or both statements are true (USED IN OUR CLASS)

(logic) exclusive "OR"

implies that the first or second statement are true, but not both

(logic) negation

p: Mr. Mann loves football

\~p: Mr. Mann does not love football

(logic) conditionals

if p, then q

ex. if you study hard, then you pass the test

expressed as p-> q

(logic) hypothesis

you study hard (p)

(logic) conclusion

you pass the test (q)

(logic) converse

q -\> p

(logic) inverse

~p -\> ~q

(logic) contrapositive

~q -\> ~p

(logic) biconditional

(iff) can be used if the conditional and converse are both true

expressed as: p

(logic) tautology

LAST column of truth table is ALL TRUE

(logic) contradiction

LAST column of truth table is ALL FALSE

(logic) infer

to conclude from the given information

(logic) modus ponens

the way that affirms by affirming

p -> q

p

therefore, q

(logic) modus tollens

p -> q

\~q

therefore, \~p

(logic) simplification

p ^ q

therefore, p

(logic) disjunctive syllogism

p ⌄ q

\~p

therefore, q

(logic) contrapositive rule

p -\> q logically equivalent to ~q -\> ~p

(logic) double negation

~(~p) logically equivalent to p

(logic) commutative rules

p ^ q LE to q ^ p

p ⌄ q LE to q ⌄ p

(logic) associative rules

(p ^ q) ^ r LE to p ^ (q ^ r)

(p ⌄ q) v r LE to p v (q v r)

(logic) distributive rules

p ^ ( q ^ r) LE to (p ^ q) v (p ^ r)

p v (q ^ r) LE to (p v q) ^ (p v r)

(logic) DeMorgan's rules

\~ (p ^ q) LE to \~p v \~q

\~ (p v q) LE to \~p ^ \~q

(logic) venn diagram

picture that shows logical relationships between sets of data, used to determine whether an argument leads to a valid conclusion

(logic) logically equivalent statements

statements are either BOTH true or BOTH false

(2.2) properties of equality (POE)

addition, subtraction, multiplication, division, substitution, reflexive, symmetric, transitive

(2.2) properties of congruence (POC)

reflexive, symmetric, transitive

(2.2) midpoint theorem

if M is the midpoint of segment AB, then AM \= 1/2 AB and MB \= 1/2 AB

(2.2) angle bisector theorem

if ray BX is the bisector of angle ABC, then measure of angle ABX \= 1/2 measure of angle ABC and measure of angle XBC \= 1/2 measure of ABC

(2.4) complementary angles

2 angles whose measures have the sum of 90

(2.4) supplementary angles

2 angles whose measures have the sum of 180

(2.4) vertical angles

2 angles such that the sides of one angle are opposite rays to the sides of the other angle

(2.4) vertical angle theorem

"vertical angles are congruent"

(2.5) perpendicular lines

2 lines that intersect to form at least one RIGHT angle

(2.5) if 2 lines are perpendicular,

then they form congruent, adjacent angles

(2.5) if 2 lines form congruent, adjacent angles,

then the lines are perpendicular

(2.5) if the exterior sides of adjacent, acute angles are perpendicular,

then the angles are complementary

(2.6) if 2 angles are supplements of congruent angles (or the same angle),

then the 2 angles are congruent

(2.6) if 2 angles are complements of congruent angles (or the same angle),

then the 2 angles are congruent

(3.1) parallel lines

coplanar lines that DO NOT intersect

(3.1) skew lines (can be segments and rays too)

non coplanar lines

(3.1) parallel planes

planes that DO NOT intersect

(3.1) line and a plane are parallel if:

they DO NOT intersect

(3.1) if 2 parallel planes are cut by a third plane,

then the lines of intersection are parallel

(3.1) transversal

a line that intersects two or more coplanar lines in different points

(3.1) corresponding angles

2 angles in corresponding positions relative to the 2 lines

(3.1) alternate interior angles

2 non-adjacent interior angles on opposite sides of the transversal

(3.1) alternate exterior angles

2 non-adjacent exterior angles on opposite sides of the transversal

(3.1) same side interior angles

2 interior angles on the same side of the transversal

(3.1) same side exterior angles

2 exterior angles on the same side of the transversal

(3.2) corresponding angle postulate

if 2 parallel lines are cut by a transversal, then corresponding angles are congruent

(3.2) alternate interior angle theorem

if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent

(3.2) same side interior angle theorem

if 2 parallel lines are cut by a transversal, then same side interior angles are supplementary

(3.2) alternate exterior angle theorem

if 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent