Undefined Terms, Definitions, Postulates, Segments, and Angles Section 2.1
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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.
-Statements of meaning.
-Our simplest and most fundamental statements. Given without proof.
-These are the statements that we will prove.
Definitions
Postulates
Theorems
A circular dot that is shrunk until it has no size.
A “wire of points” stretched as tightly as possible.
A sheet of paper of points with no thickness and extending infinitely in all directions.
DEFINED as the set of all points.
Since we haven’t established any defined terms yet, the first terms we introduce must be undefined. But they can be described.
Point
Line
Plane
Space
Postulate 2.1
Every line contains at least two distinct points.
A
B
Postulate 2.2
Two Points are contained in one and only one line.
A
B Not Possible
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.
A
B Not Possible
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.
A B
C
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.
A B
C
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.
A B
C
D
A
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.
-2 -1 0 1 2 3 -3
We measure distances between points using the coordinates of the points. We define the distance from A to B as the non-negative difference between their coordinates. In the figure, the distance from A to B is 3 - (-2) = 5
B
If A and B are two points on a line, then the line segment
determined by the endpoints A and B is written or .
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.
AB BA
A
B C
D
CDAB
CDAB
A portion of a line that has one endpoint and extends indefinitely in one direction is called a ray.
A
B
ABRay
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.
A
B
C
BAC
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
A
B
C
BAC
E
D
F
DEF
030030
DEFmBACm
DEFBAC
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
28
152
Adjacent Complementary angles
Complementary angles need not be adjacent.
Adjacent Supplementary angles (also called a linear pair)
Supplementary angles also need not be adjacent.
- an enclosure composed of line segments
Number of Sides
Type of Polygon
3
4
5
6
7
8
9
10
n
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
N-gon
Polygons
Types of Triangles
_ _
|
_ _ _ _
-triangles can be classified in terms of their sides or angles
Sides
Angles
Equilateral (or regular)
Isosceles
Scalene
Acute – three acute angles
Right – one right angle
Obtuse – one obtuse angle
Equiangular – three congruent angles
Types of Quadrilaterals
_ _
|
|
|
| _ _ _ _
_ _
|
|
>
>
>
>
_ _ >
>
|
|
|
|
|
|
Square (or regular)
Rectangle
Rhombus
Parallelogram – opposite sides parallel
Trapezoid – one pair of 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).
Rectangular Bases Triangular Bases
base
lateral face
he
igh
t
base edge
lateral edge
Polyhedra (plural) -An enclosed portion of space (without holes), composed of polygonal regions.
Right Triangular Prism Oblique Triangular Prism
he
igh
t
he
igh
t
A prisms height is measured by the perpendicular distance between bases.
Right Hexagonal Regular Prism Oblique Hexagonal Regular Prism
Polyhedra (plural) -An enclosed portion of space (without holes), composed of polygonal regions.
Another type of polyhedron is the pyramid.
height Slant height
Equilateral (Regular) Triangular Right Pyramid
Square (Regular) Right Pyramid
Apex
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?
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