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

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-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

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| _ _ _ _

_ _

|

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>

>

>

>

_ _ >

>

|

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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?