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POINTS, LINES AND PLANES-UNDEFINED TERMS.



POINTS, LINES AND PLANES--UNDEFINED TERMS





POINTS, LINES AND PLANES--NOTATION


POINTS, LINES AND PLANES--NOTATION





POINTS, LINES AND PLANES--COLLINEAR, COPLANAR
SPACE--the set of all points.
COLLINEAR POINTS are points all in one line.
COPLANAR POINTS are points all in one plane.



POINTS, LINES AND PLANES--INTERSECTIONS.
INTERSECT--where the figures "meet".
--the set of points shared by the figures.


SEGMENTS

RAYS




POSTULATE 1: RULER POSTULATE


1) You can pair each point on a line with a real number. The number that
corresponds to the point is called the point's "coordinate."
2) To find the distance between two points: subtract their coordinates, and
take the absolute value (make the answer a positive number).


POSTULATE 2: SEGMENT ADDITION POSTULATE

If Z is between A and B,

Then AZ + ZB = AB






EQUAL AND CONGRUENT
CONGRUENT OBJECTS have the same size and shape.
CONGRUENT SEGMENTS, for example, have equal lengths.






MIDPOINT, BISECTOR--DEFINITIONS
* The "midpoint of a segment" divides it into two congruent segments.
* The "bisector of a segment" is a line, segment, ray, or plane that
intersects the segment at its midpoint.

ANGLES:DEFINITION AND NOTATION

An angle is the figure formed by two rays that have the same endpoint.



ANGLES:CLASSIFICATION

Classify angles according to their measures.



POSTULATE 3 THE PROTRACTOR POSTULATE
For every angle, there corresponds a number between 0 and 180. This number is called the "measure" of the angle.



POSTULATE : ANGLE ADDITION POSTULATE


ANGLES: CONGRUENT, ADJACENT, BISECTOR--DEFINTIONS


ANGLES: CONCLUSIONS FROM A DRAWING



ANGLES: CONCLUSIONS FROM A DRAWING




POSTULATES ABOUT POINTS, LINES, AND PLANES.
POSTULATE: A line contains at least 2 points; a plane contains at least 3 noncollinear points; space contains at least 4 noncoplanar points.





POSTULATES ABOUT POINTS, LINES, AND PLANES.
Postulate : 2 points determine exactly one line.
Postulate : Three noncollinear points determine exactly one plane.
or
Three noncollinear points determine one and only one plane.




POSTULATES ABOUT POINTS, LINES, AND PLANES.

POSTULATE If two points are in a plane, then the line that contains those
points is also in the plane.
POSTULATE : Two planes intersect in a lines

 

THEOREMS ABOUT POINTS, LINES, AND PLANES

THEOREM : Two lines intersect in a point.

THEOREM : Exactly one plane can pass completely through a line and a
point not on that line.

THEOREM : If two lines intersect, then exactly one plane contains them.

CONCURRENT LINES

"Concurrent" lines two or more lines that intersect in one point.

THEOREM : The bisectors of the angle of a triangle intersect in a point
that is equidistant from the three sides of a triangle. This point of concurrency is called the "incenter."

THEOREM : The perpendicular bisectors of a triangle's sides intersect in a point that is equidistant from the sides of the triangle. This point of concurrency is called the "circumcenter."

THEOREM : The altitudes of a triangle intersect in a point of concurrency called the "orthocenter."

THEOREM : The medians of a triangle intersect in a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.
This point of concurrency is called the "centroid."


MEDIANS, ALTITUDES, AND ANGLE BISECTORS




INTRODUCTION: EQUIDISTANT
EQUIDISTANT means "the same distance."






INTRODUCTION: EQUIDISTANT
EQUIDISTANT means "the same distance."




MEDIANS, ALTITUDES, AND ANGLE BISECTORS
DEF: Perpendicular Bisector of a segment is a line (ray, segment) perpendicular to a segment at its midpoint.

THEOREM : If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of a segment.





MEDIANS, ALTITUDES, AND ANGLE BISECTORS
THEOREM : If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.



MEDIANS, ALTITUDES, AND ANGLE BISECTORS
THEOREM : If a point lies on the bisector of an angle, the the point is equidistant from the sides of the angle.







MAPPINGS AND FUNCTIONS
A map sets up a correspondence between places and point on a piece of paper.
All maps work this way.





MAPPINGS AND FUNCTIONS
In geometry we usually map points in the plane onto other points on the plane.


A TRANSFORMATION is a one to one mapping from the whole plane to the whole plane.

example: G : ( x, y) ---> ( 2x , y - 1)

We usually study special transformations which map segments to congruent
segments. This type of transformation is called an ISOMETRY or a CONGRUENCE MAPPING.

THEOREM : Isometries map triangles to congruent triangle
COROLLARY : Isometries map angles to congruent angles.
COROLLARY : Isometries map polygons to polygons with the same area.

ISOMETRIES ARE JUST MATHEMATICAL WAYS TO MOVE SHAPES AROUND THE PLANE WHILE PRESERVING THEIR CHARACTERISTICS.



MAPPINGS AND FUNCTIONS
In geometry, a "mapping" is a correspondence between sets of points.
In algebra, a "function" is a correspondence between sets of numbers.




Each point or number from the first set corresponds with exactly one point or number in the second set.




We can use "mapping" and "function" interchangeably.

MAPPINGS AND FUNCTIONS
A tree and its shadow is a good example of a mapping. The tree is called the
"preimage", and the shadow it casts is called the "image.".








MAPPINGS AND FUNCTIONS
The function that maps each real number to its square is expressed like
this:




ONE TO ONE MAPPINGS from set A to B. Every member of B has exactly one
preimage in A.




REFLECTIONS
THEOREM : A reflection in a line is an isometry




TRANSLATIONS
A "TRANSLATION" or "GLIDE" takes all points in a shape and moves them the same distance in the same direction.



TRANSLATIONS
A "TRANSLATION" or "GLIDE" takes all points in a shape and moves them the same distance in the same direction.





TRANSLATIONS



DEFINITION: TRANSLATION or GLIDE
A transformation T which maps points (x,y) to points (x + a, y + b) where
a and b are constants.




THEOREM : a translation is an isometry. This means that distance, angle
measure and area are all "invariant" under a translation.





GLIDE REFLECTIONS
Glide reflections are isometries.





ROTATIONS
POSITIVE ROTATION --COUNTERCLOCKWISE



THEOREM : A rotation is an
isometry.