Download Undefined Terms, Definitions, Postulates, Segments, and Angles

Undefined Terms, Definitions, Postulates,
Segments, and Angles
Section 2.1
Geometry, as an axiomatic system,
begins with a small set of undefined
terms that can be described and builds
through the addition of definitions
and postulates to the point where
mathematical theorems can be
proven.
Definitions -Statements of meaning.
Postulates -Our simplest and most fundamental
statements. Given without proof.
Theorems -These are the statements that we will prove.
Since we haven’t established any
defined terms yet, the first terms we
introduce must be undefined. But
they can be described.
Point
A circular dot that is shrunk until it has no size.
Line
A “wire of points” stretched as tightly as possible.
Plane
A sheet of paper of points with no thickness
and extending infinitely in all directions.
Space DEFINED as the set of all points.
Postulate 2.1
Every line contains at least two distinct points.
B
A
Postulate 2.2
Two Points are contained in one and only one line.
Not Possible
B
A
Postulate 2.2 suggests that all lines are straight because
otherwise more than one line could be drawn through two
points (not possible).
Postulate 2.3
If two points are in a plane, then the line containing these points is
also in the plane, and since the line is straight, the plane is flat.
Not Possible
B
A
This postulate suggests a connection between the
“straightness” of lines and the “flatness” of planes.
Collinear
Points are said to be collinear if there is a line containing all the
points and noncollinear if there is no line that contains all the
points.
C
B
A
D
What set of points in the diagram are collinear and noncollinear?
Postulate 2.4
Three noncollinear points are contained in one and only one
plane, and every plane contains at least three noncollinear
points.
C
A
B
Postulate 2.5
In space, there exist at least four points that are not all coplanar.
This postulate assures us that space is not just a single plane;
that space is three-dimensional.
D
C
A
B
The Ruler Postulate (2.6)
Every line can be made into an exact copy of the real number
line using a 1-1 correspondence (superimposing the number
line directly onto the line).
The real number associated with a point on the line is called the
coordinate of that point.
B
A
-3 -2 -1 0
1
2
3
We measure distances between points using the coordinates of
the points. We define the distance from A to B as the nonnegative difference between their coordinates. In the figure,
the distance from A to B is 3 - (-2) = 5
If A and B are two points on a line, then the line segment
determined by the endpoints A and B is written AB or BA .
The length of the line segment, written as AB (without the bar),
is the distance from A to B.
Two segments are said to be congruent if they have the same
length.
C
B
AB  CD
A
AB  CD
D
A portion of a line that has one endpoint and extends
indefinitely in one direction is called a ray.
Ray AB
B
A
Angles and Their Measure
Two line segments or rays meeting at a common endpoint form
an angle. The common endpoint is called the vertex (plural
vertices) of the angle and the segments or rays are called the
sides.
BAC
B
A
C
The protractor postulate is the angle version of the ruler
postulate. You can superimpose a protractor on an angle and
measure it in degrees in the same way that you can
superimpose a ruler onto a line.
Angles and Their Measure
BAC
DEF
B
A
300
D
E
C
300
F
mBAC  mDEF
BAC  DEF
Types of Angles
Type of Angle
Measure in Degrees
Acute
Between 0 and 90
Right
Exactly 90
Obtuse
Between 90 and 180
Straight
Exactly 180
Reflex
Between 180 and 360
Two angles that have the same measure are said to be congruent.
Two angles whose sum is 90 degrees are said to be complementary.
Two angles whose sum is 180 degrees are said to be supplementary.
Two angles who share a common vertex and side are said to be adjacent.
Types of Angles
62
28
Adjacent
Complementary angles
Complementary angles need not be
adjacent.
28
152
Adjacent Supplementary
angles
(also called a linear pair)
Supplementary angles also need not
be adjacent.
Polygons - an enclosure composed of line segments
Number
of Sides
3
4
5
6
Type of Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
n
N-gon
Types of Triangles
-triangles can be classified in terms of their sides or angles
Angles
Sides
_ _
|
_ _
Acute – three
acute angles
Equilateral
(or regular)
Right – one
right angle
Isosceles
Obtuse –
one obtuse
angle
Scalene
Equiangular –
three congruent
angles
Types of Quadrilaterals
>
|
_
_
Square
(or regular)
>
>
__
>
>
|
|
__
Rectangle
|
_
Trapezoid – one
pair of opposite
sides parallel
_
>
|
_
|
_
Rhombus
Parallelogram –
opposite sides
parallel
|
||
|
||
Isosceles
Trapezoid
Kite – two pairs
of adjacent sides
congruent
Polyhedra (plural)
-An enclosed portion of space (without holes),
composed of polygonal regions.
Prisms are the most common type of polyhedron (singular).
base edge
base
lateral face
height
lateral
edge
Rectangular Bases
Triangular Bases
Polyhedra (plural)
-An enclosed portion of space (without holes),
composed of polygonal regions.
Right Triangular Prism
Oblique Hexagonal Regular Prism
height
height
Right Hexagonal Regular Prism
Oblique Triangular Prism
A prisms height is measured by the perpendicular distance
between bases.
Polyhedra (plural)
-An enclosed portion of space (without holes),
composed of polygonal regions.
Another type of polyhedron is the pyramid.
Apex
Slant height
height
Equilateral (Regular)
Triangular Right Pyramid
Square (Regular) Right Pyramid
Hexagonal Oblique Regular Pyramid
(the base is a regular hexagon)
Polyhedra (plural)
-if all the faces are regular polygons of the exact same
size and shape these are called regular polyhedra.
Greek mathematicians discovered that only five exist:
A soccer ball is an example of a semiregular
polyhedron, as it composed of two different
regular polygons as faces. Which are they?