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undefined terms
can not be defined using other figures.

Point
names a location and has no size. it is represented by a dot.
A capital letter

Line
a straight path that has no thickness and extends forever
A lower case letter or two points on the line

Plane
 a flat surface that has no thickness and extends forever.
 a script capital letter
 or
 three points not on a line.

collinear
points that lie on the same line

noncollinear
points that do not lie on the same line

line segment
the part of a line consisting of two points and all points between them

endpoint
a point at one end of a segment or the starting point of a ray
A capital letter

ray
part of a line that starts at an endpoint and extends forever in one direction
It's endpoint and any other point

Opposite Rays
two rays that have a common endpoint and form a line

postulate
a statement that is accepted as true without proof

intersection
the set of all points the two or more figures have in common

system of equations
a set of two or more equations containing two or more of the same variables

coordinate
a number used to identify the location of a point

distance between any two points
the absolute value of the difference of the coordinates

length of AB
the distance between A and B

Finding the Length of a Segment
 ab  or  ba 

congruent segments
segments that have the same length
uses tick marks

midpoint
the point that bisects, or divides the segment into two congruent segments

segment bisector
any ray, segment, or line that intersects a segment at its midpoint

angle
a figure formed by two rays, or sides, with a common endpoint


How to name an angle
 Vertex
 a point on each ray and the vertex
 number

interior of an angle
the set of all points between the sides of the angle

exterior of an angle
the set of all points outside the angle

the measure of an angle
the absolute value of the difference of the real numbers that the rays correspond with on a protractor

Acute Angle
measures greater than 0 degrees and less than 90 degrees

Right Angle
 Measures 90 degrees
 must have box

obtuse angle
measures greater than 90 degrees and less than 180 degrees

straight angle
 formed by two opposite rays
 measures 180 degrees

congruent angles
angles that have the same measure

angle bisector
a ray that divides an angle into two congruent angles
the (particular points) of all points in the interior of the angle that are equidistant from the sides of the angle

Adjacent Angles
two angles in the same plane with a common vertex and a common side, but no common interior points

linear pair of angles
a pair of adjacent angles whose noncommon sides are opposite rays

complementary angles
two angles whose measures have a sum of 90 degrees

supplementary angles
two angles whose measures have a sum of 180 degrees

Finding the Complementary angle
90x

finding the supplementary angke
180x

vertical angles
two nonadjacent angles formed by two intersecting lines

perimeter
the sum of the side lengths of the figure
 rectangle 2l+2w
 square 4s
 triangle a+b+c

area
the number of non overlapping square units of a given size that cover the figure
 rectangle l*w
 Square side squared
 Triangle 1/2bh
 circle pi(r) squared

base of a triangle
any side of a triangle

height
a segment from a vertex that forms a right angle with a line containing the base
 may be:
 inside the triangle (interior)
 outside (exterior)
 a side of the triangle

diameter
a segment that passes through the center of the circle and whose endpoints are on the circle

radius
a segment whose endpoints are the center of the circle and a point on the circle

circumference
the distance around a circle
c=pi(d) or 2pi(r)

pi
irrational
3.14 or 22/7

coordinate plane
a plane that is divided into 4 regions by the x and y axis

Midpoint formula
m= (x_{1}+x_{2} / 2, y_{1}+y_{2} /2)

distance formula
d= square root of (x_{2}x_{1})^{2} + (y_{2}y_{1})^{2}

pythagorean theorem
 ONLY in a right triangle
 used to find distance
 a^{2}+b^{2}=c^{2}

legs of a right triangle
the 2 sides that form the right angle

hypotenuse
the side across from the right triangle that stretches from one leg to the other

transformation
a change in the position, size, or shape of a figure

preimage
the original figure

image
the resulting figure

> arrow notation
used to describe a transformation

' primes
used to label the image

reflection (flip)
a transformation across a line, called the line of reflection.
each points and its image are the same distance from the line of reflection

Rotation (turn)
a transformation about a point P, called the center of rotation.
Each point and its image are the same distance from P

Translation (slide)
a transformation in which all the points of a figure move the same distance in the same direction

inductive reasoning
the process of reasoning that a rule or statement is true because specific cases are true
used to draw a conclusion from a pattern
 1) look for a pattern
 2) Make a conjecture
 3) prove the conjecture or find a counterexample

conjecture
a statement believed to be true based on inductive reasoning

counterexample
an example that proves a conjecture false

conditional statement
a statement that can be written in the form "if p, then q"
p> q

hypothesis
the part of p of a conditional statement following the word "if"

conclusion
the part q of a conditional statement following the word, "then"

truth value
true or false of a conditional statement
false when the hypothesis is true and the conclusion if false

negation of statement p (opposite)
~p
the negation of a true statement is false, the negation of a false statement is true

converse
statement formed by exchanging the hypothesis and conclusion
q>p

inverse
statement formed by negating the hypothesis and the conclusion
~p> ~q

contrapositive
statement formed by both exchanging and negating the hypothesis and conclusion
~q > ~p

logically equivalent statements
related conditional statements that have the same truth value

deductive reasoning
process of using logic to draw conclusions from given facts, definitions, and properties

biconditional statement
a statement that can be written in the form, "p if and only if q" (iff)
if p, then q
if q, then p
used to write defintions

definition
a statement that describes a mathematical object and can be written as a true biconditional
iff

polygon
closed plane figure formed by 3 or more line segments
each segment intersects exactly two other segments only at their endpoints
no two segments with a common endpoint are collinear

triangle
a three sided polygon

quadrilateral
a 4 sided polygon

proof
an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true

theorem
any statement that you can prove

two column proof
proof that lists the steps of the proof in the left column and the matching reason in the right column.

proof process
1) write the conjecture to be proven
2) draw a diagram to represent the hypothesis of the conjecture
3) state the given information and mark it on the diagram
4) state the conclusion of the conjecture in terms of the diagram
5) plan your argument and prove the conjecture

flow chart proof
uses boxes and arrows to show the structures of the proof
moves from left to right or top to bottom
justification written below the box

paragraph proof
presents the steps of the proof and their matching reasons as sentences in a paragraph

parallel lines ()
coplanar and do not intersect

perpendicular lines (upside down T)
intersect at 90 degree angles.

skew lines
are not coplanar, are not parallel, and do not intersect

parallel planes
planes that do not intersect

transversal
a line that intersects two coplanar lines at two different points
t
other 2 lines = r and s

corresponding angles
lie on the same side of the transversal, on the same sides of lines r and s.

alternate interior angles
nonadjacent angles
lie on opposite sides of the transversal t, between lines r and s

alternate exterior angles
lie on opposite sides of the transversal t
outside lines r and s

sameside interior angles (consecutive interior angles)
lie on the same side of the transversal
between lines r and s

perpendicular bisector of a segment
a line perpendicular to a segment at the segment's midpoint

distance from a point to a line
the length of the perpendicular segment from the point to the line

rise
the difference in the y values of the 2 points on a line

run
the difference in the x values of 2 points on a line

slope
the ratio of rise to run,
m= y_{2}y_{1} / x_{2}x_{1}





opposite reciprocals
a/b and b/a

point slope form of a line
yy_{1}=m(xx_{1})
x_{1}, y_{1} is a given point on the line

slope intercept form
y=mx+b
m=slope
b=y intercept

the equation of a vertical line
x=a
a= x intercept

equation of horizontal line
y=b
b= y intercept

y=5x+8
y=5x4
same slope, different y intercept

acute triangle
three acute sides

equiangular triangle
3 congruent acute angles

right triangle
one right angle

obtuse triangle
one obtuse angle

equilateral triangle
three congruent sides

isosceles triangle
at least 2 congruent sides

scalene triangle
no congruent sides

auxiliary line
a line that is added to a figure to aid in a proof


corollary
a theorem whose proof follows directly from another theorem

interior
set of all points inside the figure

exterior
the set of all points outside the figure

interior angle
formed by 2 sides of a triangle

exterior angle
formed by one side of the triangle and the extension of an adjacent side

remote interior angle
an interior angle that is not adjacent to the exterior angle

triangle rigidity
if the side lengths of a triangle are given, the triangle can have only one shape.

included angle
an angle formed by two adjacent sides of a polygon

included side
the common side of 2 consecutive angles in a polygon

coordinate proof
uses coordinate geometry and algebra

strategies for positioning figures in the coordinate plane
use the origin as a vertex, keeping the figure in Quad I
center the figure at the origin
center a side of the figure at the origin
use one or both axes as sides of the figure

vertex angle
angle formed by the legs

base
side opposite the vertex angle

base angles
2 angles that have the base as a side

equidistant
a point is the same distance from 2 or more objects

concurrent
three or more lines intersect at one point

point of concurrency
the point where the lines intersect

circumcenter of a triangle
the point of concurrency in a triangle

circumscribed
a circle that contains all the vertices of a polygon

incenter of the triangle
the point of concurrency where the angle bisectors meet? (unsure)

inscribed
intersects each line that contains a side of the polygon at exactly one point

incenter
the center of the triangle's inscribed triangle
always inside the triangle

median of a triangle
a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side
each triangle has 3 medians, which are concurrent

centroid of the triangle
the point of concurrency of the medians of a triangle
always inside the triangle
center of gravity

altitude of a triangle
perpendicular segment from a vertex to the line containing the opposite side
each triangle has 3 altitudes
can be inside, out, or on the triangle

the height of a triangle is the length of an altitude
miscellaneous shit

orthocenter of the triangle
the point of intersection of the 3 altitudes of a triangle

midsegment of a triangle
segment that joins the midpoints if two sides of the triangle
every triangle has 3 midsegments, qhich form the midsegment triangle

indirect proof
1) identify the conjecture
2) assume the opposite of the conclusion is true
3) use direct reasoning to show that the assumption leads to a contradiction
4) conclude that since the assumption is false, the original conjecture is true

