Download Izzy Kurkulis September 28, 2013 EXTRA CREDIT GEOMETRY

Izzy Kurkulis September 28, 2013
EXTRA CREDIT GEOMETRY
POSTULATES, THEOREMS AND DEFINITIONS
1-1
 Discrete lines: Lines made from dots.
 Discrete Geometry: The study of points as dots and discrete lines.
 Description of a point: A point is a dot.
 Description of a line: A line is a set of dots in a row.
 Every line is either horizontal, vertical or oblique.
1-2
 Synthetic geometry: The study of the description of points.
 Description of a point: A point is an exact location.
 Description of a line: A line is a set of points extending in both directions containing the
shortest path between any two points on it.
 Number Line: Unique distance between two locations on a line with a number assigned to
each location.
 Coordinate: The number or numbers associated with the location of a point on a line, a
plane, or in space. (Lines are said to be coordinatized.)
 Number lines are dense.
 Distance between two points: the absolute value of the difference of their coordinates on
a coordinatized line.
1-3
 Ordered pair: The pair of numbers (a,b) identifying a point in a two dimensional
coordinate system.
 Plan coordinate geometry: The study of points as ordered pairs of numbers.
 Coordinate plane (a.k.a Cartesian Plane): A plane containing points.
 Coordinates of a point are determined by the y axis and the y axis.
 Description of a point: An ordered pair of real numbers.
 Descripton of a Line: set of ordered pairs which satisfy the equation Ax+By= C, where A
and B are not both zero.
1-4 POINTS IN NETWORKS
 Network: A union of points. (Networks are also sometimes called graphs.)
 Graph theory: the geometry of networks.
 Vertices (Nodes): Endpoints with no size.
 Description of a point: A node of a network.
 Description of a line: An arc connecting either two nodes or one node to itself.
 Traversable network: A network in which all the arcs may be traced exactly once
wihtout picking up the tracing instrument.
 Even Node: A node which is the endpoint of an even number of arcs in a network.
 Odd Node: A node which is the endpoint of an odd number odd arcs in a network.
1-5 DRAWING IN PERSPECTIVE
N/A
1-6 THE NEED FOR UNDEFINED TERMS
 Circularity: You have circled back to any word previously defined, not necessarily the
original one.
-To avoid circularity, certain basic geometry terms must be undefined.
 Undefined terms= point, line and plane.
 Figure: a set of points
 Space: set of all points in that geometry.
 Coplanar: four or more points ar this if and only if they are in the same plane.
 One-dimensional: a space in which all points are coplanar.
 Two-dimensional: pertaining to figures that lie in a single plane, or to their geometry.
 Three dimensional: Real objects that do not lie in a single pane.
1-7 POSTULATES FOR EUCLIDEAN GEOMETRY
 Postulates: assumptions of a description of a point or a line.
 Point line plane postulate
o Unique line assumption: Through any two points, there is exactly one line.
o Number line assumption: Every line is a set of points that can be put into a one-toone correspondence with the real numbers, with any point on it corresponding
to 0 and any other point corresponding to 1.
o Dimension assumption: Given a line in a plane, there is at least one point in the
plane that is not in the line.
 -Given a plane in space, there is at least one point in space that is not in
the plane.
 Line intersection theorem: Two different lines intersect in at most one point.
 Parallel lines: two coplanar lines can be this if and only if the have no points in common
or they are identical.
1-8 BETWEENNESS AND DISTANCE
 Segment (line segment): The set consisting of the distinct points A and B (Its end points)
and all points between A and B.
 Ray: The ray with endpoint A containing B, denoted AB(arrow is on top), is the union of
AB (Segment symbol on top) and the set of all points for which B is between each of th
and A.
 Opposite rays: When ray AB and ray AC iif A is between B and C
 Distance Postulate:
o Uniqueness property: On a line, there is a unique distance between two points.
o Distance formula: If two points on a line have coordinates x and y, the distance between
them is |x - y|.
o Additive property: If B is on line AC then AB+BC=AC
2-1 THE NEED FOR DEFINITIONS
 Convex
Set: A set in which every segment that connects points of the set lies entirely in
the set lies entirely in the set.
 Nonconvex set: set that is not convex
2-2 IF-THEN STATEMENTS
 If-then statement: “If”_____ “then”
 Antecedent: the clause that follows the if in the statement
 Consequent: the clause that follows the then.
 Instance of a conditional: specific case in which both the antecedent (if part) and the
consequent (then part) of the conditional are true.
 Counterexample to a conditional: A specific case for which the antecedent of the
conditional is true and its consequent is false.
2-3 CONVERSES
 Converse: the converse of pqis qp
2-4 GOOD DEFINITIONS
 Midpoint: Segment Ab is the point M on segment AB with AM=MB
 Circle: set of all points in a plane at a certain distance
o radius: A segment connecting the center of a circle or a sphere with a point on that
circle or sphere
o center: the center point of the circle
2-5 UNIONS AND INTERSECTIONS OF FIGURES
 Union of two sets: A and B, written AB, is the set of elements which are in A, in B, or
both A and B.
 Intersection of two sets: A and B, written AB, is the set of elements which are in both A
and B.
 Null Set (Empty Set): Figures that have no points in common.
2-6 POLYGONS
 Polygon: The union of segments in the same plane such that each segment intersects
exactly two others, one at each of its endpoints.
 Sides: Segments which make up a polygon.
 Vertices: The endpoints of the segments.
 Vertex: Singular vertices.
 Consecutive (adjacent) vertices: endpoints of a side.
 Consecutive (adjacent) sides: Sides that share and endpoints.
 Diagonal: A segment connecting nonadjacent vertices.
 N-gons: polygons with n sides.
 When n is small, the polygon have the following special names:
o Triangle (3 sides)
o Quadrilateral (4 sides)
o Pentagon (5 sides)
o Hexagon (6 sides)
o Heptagon (7 sides)
o Octagon (8 sides)
o Nonagon (9 Sides)
o Decagon (10 sides)
 Polygonal Region: The union of polygon and its interior.
 A polygon is a convex iif its corresponding polygonal region is convex.
 Equilateral: A triangle with all equal sides.
 Isosceles: A triangle with at least two equal sides.
 Scalene: A triangle with no even sides.
 Hierarchy: Family tree of classification of triangles.
2-7 USING AN AUTOMATIC DRAWER: THE TRIANGLE INEQUALITY
 Static mode: point that is moved and changes its name
 Dynamic mode: some parts of a figure can be moved while other parts can be fixed.
 Triangle Inequality Postulate: The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
2-8 CONJECTURES
3-1 ANGLES AND THEIR MEASURES
 Angle: the union of two rays that have the same endpoint.
 Sides: The two rays that form an angle.
 Vertex: the common endpoint of the two rays.
 Interior set: convex set
 Exterior: nonconvex set
 Measure: indicates the amount of openness of the interior of the angle.
 Degree: unit of measure used on angles.
 Angle Measure Postulate
o Unique measure assumption: every angle has a unique measure from 0º to 180º
o Unique angle assumption: Given any ray VA and any real number r between 0
and 180 , there is a unique angle BVA in each half-plane of line VA such that
angle BVA=r.
o Zero angle assumption: if ray VA and VB are the same ray, then angle AVB=0
o Straight angle assumption: if ray VA and ray B are opposite rays, the angle
AVB=180
o Angle addition property: If ray VC (except for point V) is in the interior of angle
AVB, then angle AVC+ angle CVB= angle AVB.
 Bisector:
a point, line, ray or plane which divides a segment angle, or figure into two
parts of equal measure.
3-2 ARCS AND ROTATION
 Central Angle Of A Circle: angle whose vertex is the center of the circle.
 Minor arc: The points of Othat are on or in the interior of angle AOB.
 Major arc: The points of O that are on or in the exterior of angle AOB
 Degree measure of a minor arc or semicircle AB: the measure of its central angle.
 Degree measure of Major arc ACB: 360- arc ABO
 Concentric: circles can be concentric iif they lie in the same plane and have the same
center.
 Image: The result of applying a transformation to an original figure or preimage.
 Preimage: The original figure in a transformation.
 Magnitude: In rotation the amount that the preimage is turned about the center of rotation
measured in degrees from -180º (clockwise) to 180º (counterclockwise)
3-3 PROPERTIES OF ANGLES
 Zero: iif m= o
 Acute: iif 0<m<90
 Right: iif m=90
 Obtuse: iif 90<m<180
 Straight: iif m=180
 Complementary: iif two angles added together=180º
 Supplementary: iif two angles added together= 90º
 Adjacent angles: two non-straight and nonzero angle.
iif a common side is interior to the
angle formed by the noncommon sides.
 Linear pair: two adjacent angles form this iif their noncommon sides are opposite rays.
 Linear Pair theorem: If two angles form a linear pair, then they are supplementary.
 Vertical angles: iif the union of their sides is two lines.
 Vertical angles theorem: if two angles are certical angles, then they have equal measures.
3-4 ALGEBRA PROPERTIES USED IN GEOMETRY
 Postulates of Equality:
o Reflexive property: a=a
o Symmetric Property of Equality: if a=b then b=a
o Transitive Property of Equality: if a=b and b=c, then a=c
 Postulates of Equality and Operations:
o Addition property of equality: if a=b then a + c= b+ c
o Multiplication property of equality: if a=b then ac=bc
 Postulates of inequality and operations:
o Transitive property of inequality: if a <b and b<c, then a<c
o Addition property of inequality: If a<b, then a+c < b+c
o Multiplication properties of inequality: if a<b and c>0, then ac<bc
 If a<b and c<0, then ac>bc
 Postulates of Equality and Inequality:
o Equation to inequality property: if a and b are positive numbers and a+b-c then c
> a and c > b
o Substitution property: if a = b, then a may be substituted for b in any expression.
3-5 ONE STEP PROOF ARGUMENTS
 Proof argument: a sequence of justified conclusions starting with the antecedent and
ending with the consequent.
 Justification: a general property for which the step is a special case.
3-6 PARALLEL LINES
 Transversal: angles formed when two lines m and n are intersected by a third line t
 Corresponding angles: any pair of angles in similar location with respect to the
transversal and each line.
 Corresponding angles postulate:
o If two corresponding angles have the same measure, then the lines are parallel.
o If the lines are parallel, then corresponding angles have the same measure.
 Slope: In the coordinate plane, the change in y-values divided by the corresponding
changes in x-values.
 Parallel lines and slopes theorem:Two nonvertical lines are parallel iif they have the same
slope.
 Transitivity of parallelism theorem: In a plane, if line l is parallel to line m is parallel to
line n, then line l is parallel to line n.
3-7 PERPENDICULAR LINES
 Perpendicular: two segment, ray or lines can be this iif the lines ontaining them form a
90º angle.
 Two Perpendicular theorem: If two coplanar lines l and m are each perpendicular to the
same lines, then they are parallel to each other.
 Perpendicular to parallels theorem: In a plane if a line is perpendicular to to one of two
parallel lines, then it is also perpendicular to the other.
 Perpendicular lines and slopes theorem: Two nonvertical lines are perpendicular iif the
product of their slopes is -1.
3-8 DRAWING PARALLEL AND PERPENDICULAR LINES
 Bisector of a segment: midpoint, or any line, ray or segment which intersects the segment
only at its midpoint.
 Perpendicular bisector: " "only one line is a bisector and perpendicular to the segment.
 Construction: Precise way of drawing which uses specific tools and follows specific
rules.
 Drawing tools: unmarked straightedge and compass.
4-1 REFLECTING POINTS
 Preimage: original image before reflection.
 Reflecting line (line of reflection): The line the preimage reflects into the new image
 Reflect: a transformation in which each point is mapped onto its reflection image over a
line or plane.
 Transformation: A correspondence between two sets of points such that:
o each point in the preimage set has a unique image
o each point in the image set has exactly one preimage.
4-2 REFLECTION FIGURES
 Reflection postulate: under a reflection:
o There is a 1-1 correspondence between point and their images.