Download Math definitions and theories and postulates

Definitions and Descriptions
Synthetic Geometry:
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Description of a point: A point marks an exact location
Description of a line: A set of two points that continue in both directions, and which
contains the shortest route between any of the two points on the line
Distance on a Number Line:
● Distance: The value of the difference of two coordinates in absolute its absolute value, is
the distance between two points on a number line
Plane Coordinate Geometry:
● Description of a Point: A point contains an ordered pair of real numbers that mark its
location
● Description of a line: In Plane Coordinate Geometry, A line contains a set of ordered
pairs which must satisfy the standard form of a line, Ax + By = C
Undefined Terms and definitions of those terms:
● Figure: A figure is a set of points
● Space: The space of a geometry contains the set all points in the geometry given
● Collinear: In order for points to be collinear, there must be three or more points on the
same line
● Coplanar: In order for something to be coplanar, there must be four or more points in the
same space
Euclidian Geometry:
● Parallel lines: Two coplanar lines are parallel lines IFF they are identical, or have
absolutely no points in common
Betweenness and Distance:
● Segment or line segment: A segment is a piece (or segment) of a line that has two
endpoints
● Ray: a ray is the part of a line that has one endpoint, and continues in the other direction
● Opposite rays: ray CA and ray CB are opposite rays IFF C is between
Convex Sets:
● Convex set: A convex set is a set in which every segment can connect to any two any
two points that lie inside the set
If-Then Statements:
● Instance of a conditional: An instance of a conditional is when if (the antecedent) and
then (the consequent) are both true
● Counterexample to a conditional: A counterexample to a conditional is when the
antecedent is true, and the consequent is false
Converses:
● Converse: The converse of A ---> B is B ---> A
Midpoints and circles:
● Midpoint: The midpoint of a segment is the point that is directly in the middle of it (the
midpoint of segment BC is the point Y on BC with BY = YC
Unions and Intersections of Figures:
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Union of two sets: The union of two sets C and D, is the things that are in C and B, or
either one of them
● Intersection of two sets: The intersection of two sets, C and D, is the set of elements that
are shared or that are found in both of them
Polygons:
● Polygon: A polygon is a shape that is formed by three or more segments
Angles and Their Measures:
● Angle: An angle is two rays that come together in order to share same endpoint
● Bisector: A bisector of an angle is a ray that directly cuts that angle in half, so that the
two angles it creates are equal to each other
Arcs and Rotations:
● Degree measure of a minor arc or semicircle: The degree measure of a minor arc or
semicircle is the measurement of its central angle
● Degree measure of a major arc: The degree measure of a major arc of a circle, is the
measurement of the circle (360 degrees), minus the measurement of the arc
Properties of Angles:
● Measurement of angles: a.) An angle is zero, IFF the measurement (n) equals zero
b.) an angle is acute IFF zero is greater than n, but less than ninety degrees
c.) The angle is a right angle IFF n equals ninety degrees exactly
d.) The angle is obtuse IFF it is greater than ninety degrees but less than one hundred eighty
degrees
e.) The angle is straight IFF n equals one hundred eighty degrees exactly
● Complementary angles: An angle is complementary IFF the measurement of that angle
equals ninety degrees
● Supplementary angles: An angle is supplementary IFF the measurement of that angle
equals one hundred eighty degrees
● Adjacent angles: two angles having the same vertex and having a common side
between them
● Linear pair: A linear pair is formed by two adjacent angles, IFF their sides that are not
common are opposite rays
● Vertical angles: Vertical angles are angles that are equal and opposite, formed by two
intersecting lines
One-Step Proofs and Arguments:
● Proof argument: A proof argument for a conditional, is a justified conclusion, that starts
with the antecedent and ends with the consequent
Parallel Lines:
● Slope: The slope of a line equals y2 minus y1 divided by x2 minus x1
Perpendicular Lines:
● Two lines, segments or rays are perpendicular IFF the angles that they form are ninety
degrees
Reflecting Points:
● Reflecting image P over line m: When point P is not on line m, the reflection image of P
over line m is the point Q IFF m is the perpendicular bisector of segment PQ
● If point P is on line m, then the reflection image of P over line m is P
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Transformation: Transformation is a similarity between two sets of points so that every
point in the preimage set has a unique image, and the points within the image sets has
exactly one image
Composing Reflections Over Parallel Lines:
● Composite: The composite of a first transformation and a second transformation would
be the transformation that maps each point onto the image
● Translation/slide: A translation or slide is the composite of two reflections over lines that
are parallel (reflects the preimage over a line l, and reflects that image over another line
n)
Composing Reflections over Intersecting Lines:
● Rotation: A rotation is the location of two reflections of an image over two intersecting
lines
Translations and Vectors:
● Vector: A set of numbers that can be characterized by its magnitude or direction
Isometries:
● Glide reflection: A glide reflection is a preimage that is reflected over a line, and then this
image then “slides” further up the line (preimage P reflects over line m (P’) and slides up
to P”)
Postulates, Theorems, Assumptions, Properties, and Formulas
Point-Line-Plane Postulate (Euclidian Geometry):
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Unique Line Assumption: There is one line that goes through any two points
Number Line Assumption: Every line is a set of set of points that can be put in bijective
(one-to-one) correspondence with the real numbers, with any point on it corresponding
to 0 and any other corresponding to 1
● Dimension Assumption: 1.) When there is a line in a plane, there is at least one point
inside the plane that is not on the line 2.) If there is a plane in space, there should be at
least one point in space that is not inside the plane
● Line Intersection Theorem (Euclidian Geometry): When two lines intersect, they should
intersect in at most one point
Distance Postulate:
● Uniqueness Property: On every line, there is a unique distance between the two points
● Distance Formula: If one of the points on a line is A, and the other point is B, then the
distance between them would equal the absolute value of A-B
● Additive Property: If C is the midpoint of segment BD, then BC + CD = BD
Triangle Inequality Postulate: If you add two sides of any triangle, than the sum of those two
sides will be greater than the third side
Angle Measure Postulate:
● Unique Measure Assumption: Every angle, from 0 degrees to 180 degrees has a unique
measure
● Unique Angle Assumption: There is a unique angle in every half-plane, when any real
number or ray is given
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Zero Angle Assumption: If ray BC and BD are the same exact ray, then the
measurement of angle CBD equals zero
● Straight Angle Assumption: if ray BC and ray BD are opposite rays, then the
measurement of angle CBD equals zero
● Angle Addition Property: If ray BM is in the middle of angle ABC, then ABM + MBC
equals ABC
Linear Pair Theorem: If two angles form a linear pair, then the measurement of both angles
must be supplementary
Vertical Angles Theorem: If two angles are vertical, then they both have the same measure
Postulates of Equality:
● Reflective Property of Equality: Reflective Property of Equality is when something equals
equals itself (b=b)
● Symmetric Property of Equality: If B=C then C=B
● Transitive Property of Equality: If B=C and B=D then A=C
Postulates of Equality and Operations:
● Addition Property of Equality: If b=c then a + d = b + d
● Multiplication Property of Equality: If b = c, then ad + bd
Postulates of Inequality and Operations:
● Transitive Property of Inequality: If b < c, and c < d then b < d
● Addition Property of Inequality: if b < c, then b + d < c + d
● Multiplication Property of Inequality: If a < b and c > 0 (or c < 0), then ac < bc
Postulates of Equality and Inequality:
● Equation to Inequality Property: If numbers b and c are positive, and b + c + d, then d >
b and d > c
● Substitution Property: If b is equal to c, then b can be substituted for c in any expression
Corresponding Angles Postulate:
a.) Two lines are parallel if the two corresponding angles have the same measure
b.) Corresponding angles have the same measure, if the lines are parallel
Parallel Lines and Slopes Theorem: If two nonvertical lines have the same slope, then they are
parallel
Transitivity of Parallelism Theorem: If line n is parallel to line m, and line m is parallel to line l,
then line n is parallel to line l
Two Perpendiculars Theorem: If line l is perpendicular to line m and n, then line n and m are
parallel
Perpendicular to Parallels Theorem: If a line is perpendicular to one of the two parallel lines,
then it is automatically perpendicular to the other
Perpendicular Lines and Slopes Theorem: If the product of two nonvertical lines are is -1, then
the two nonvertical lines are parallel
Reflection Postulate: (under reflection)
a.) There is a one-to-one similarity between the points and their images
b.) Collinearity is preserved (If three points, B, C, and D lie on the same line, then their images,
B’, C’, and D’ are collinear)
c.) Betweenness is preserved (if point C is between B and D, C’ is between B’ and D’)
d.) Distance is preserved (is segment B’C’ is the image of segment BC, then B’C’ equals BC
e. Angle measure is preserved (If angle B’D’F’ is the image of angle BDF, then the
measurement of angle B’D’F’ equals BDF
Two-Reflection Theorem of Translation: If line n is parallel to line m, then the translation (Rm °
Rl) then the magnitude should be twice the distance between n and m
Two-Reflection Theorem for Reflections: If line n intersects line m, the rotation Rn ° Rm has the
center of intersection of n and m and has the magnitude two times the measure of the angle
(non-obtuse) formed by these lines in the direction from n to m