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Chapter 8
Introductory Geometry
Section 8.1
Beginning Geometry
Geometry
Geometry is the study of shapes and is a part of the early childhood, middle
childhood and adolescent young adult mathematics curriculum. Geometry
analyzes patterns in shapes using both numbers and logical reasoning.
Two-Dimensional vs Three-Dimensional Figures
A plane is a flat sheet. Think of it like a very thin piece of paper that does not
bend. A plane figure is some shape that is part of or inside a plane. Plane figures
are considered to be two-dimensional shapes or flat. Three-dimensional
shapes are shapes that have length, height and depth to them.
Two-Dimensional Shapes
Three-Dimensional Shapes
Points, Lines, Segments, Rays and Angles
The geometric objects points, lines, segments, rays and angle are frequently
occurring components of other more complicated geometric objects. The
description of a these objects has both a formal definition which is used in the later
grades and an intuitive everyday situation model.
A point represents a location in a plane,
space or on an object. Cities on a map are an
everyday situation of how points are
represented. Points are thought of as dots
and usually labeled with capital letters. Points
A, B and C are shown below the map.
A
D
A line is straight and extends infinitely in both
directions. Two points on a line can name it.
A line segment is part of a line between two
points. The two points are called the endpoints
and name the segment.
A ray is the part of a line that extends infinitely
in one direction. Two points on a ray can name
it with one point being the end of the ray.
DE
E
F
H
C
B
G
I
FG
HI
Angles
Angles are made up of the union of two rays that have a common endpoint. The
two rays are called the sides of the angle and the common endpoint is called the
vertex of the angle. Using the set operations of intersection () and union () to
describe geometric shapes is very common. Angles are named with three points
with the point in the middle being the vertex and the other two being a point on
each side and we use the symbol  to mark an angle.
BAC  AB  AC
A
F
C
B
E
D
BAC  DAE  AD  AE
AC  CA  AC
Angles are measured in a unit called
degrees with a device named a protractor.
The vertex is placed at the center and one
side at the zero degree mark. The place
where the other side crosses is the
measure. The measure is denoted with m.
120
90
60
B
30
150
0
180
A
mBAC  60
C
Compass directions are often given using degrees.
The picture to the right show how to locate a plane
whose position is described a 30 south of east.
east
30

south
Types of Angles
Adjectives are used to describe certain types of angles. It can refer to its size, or
the relationship it has with another angle. Many of these have both a numerical
and geometric way of characterizing them.
D
Right Angles
These are angles that measure 90. They
break a line into two angles that are exactly
the same. DAB is exactly the same as
DAC
C
A
B
mDAB  90  mDAC
Straight Angles
These are angles that measure 180. They
form a line. To the right BAC is a straight
angle.
C
A
B
mBAC  180

Acute Angles
Obtuse Angles
These are angles
whose measure is
less than 90.
C
E
38
A
B
D
F
Reflex Angles
H
326
G
These are the angles whose measure exceeds
180. They make up the bigger part of the angle.
I
Complementary Angles
Supplementary Angles
These are pairs of angles that make
up each part of a right angle. The
measures will add up to 90.
These are pairs of angles that make
up each part of a straight angle. The
measures will add up to 180.
B
BAC and CAD
are complementary
angles.
C
68
22
A
These are angles
whose measure is
greater than 90 and
less than 180.
147
D
C
158
22
B
A
D
BAC and CAD
are supplementary
angles.
Congruent Figures
Congruent geometric figures are figures that can be slid one on top of another
so that they will exactly match up. Congruent angles are angles that can be
placed one on top of the other so that they coincide.
B
E
BAC is congruent to EDF
A
D
C
F
Intersecting Lines
Lines that intersect have exactly one point in
common. Lines (in the same plane) that do not
intersect are called parallel and are denoted
with ||. Lines that intersect at a right angle are
called perpendicular and are denoted with .
In the figure to the right we have:
AB || CD
AB  AD
B
A
C
D