GEOMETRY – AN OVERVIEW OF THE SUBJECT
How old is geometry?
Geometry is one of the oldest branches of mathematics. It flourished in Ancient Greece
from about 600 B.C.E., but was used by ancient Babylonians, Mayans, and Chinese. You
will hear names like Pythagoras, Euclid, Archimedes, Euler, and Gauss. In case you
think all of geometry is ancient, you should be aware that new branches of geometry were
developed in the 1800’s (non-Euclidean geometries) and in the late 1900’s (fractal
geometry).
What is geometry?
You can think of geometry as being composed of two intertwining strands:
1) the study of the physical world we live in
2) the creation and use of a logical system
The first strand comes from the fact that the word geometry literally means “earth
measure” and was developed to help solve problems in everyday life. It includes
practical applications such as computing angle measures, areas, and volumes of objects
we create and use. The second strand involves using definitions, postulates, theorems
and deductive reasoning to prove statements. It is like learning the vocabulary and rules
associated with a game (football, chess) and then playing the game! Working within a
logical system and creating proofs are great examples of using critical thinking!
What is the difference between a postulate and a theorem?
A postulate is a statement we agree to be true without proof. It is an accepted, agreeupon statement. (These will hopefully be agreeable to everyone or else you have to go
off and invent your own geometry!)
A theorem is a statement that we prove to be true by using logical reasoning and
previously proved statements. Most of our statements will be of this type because we
don’t just want to agree to everything!
Example of a postulate: Through two distinct points there is exactly
one line. (agree to this?)
Example of a theorem: The sum of the measures of the interior angles
of a triangle is 180. (prove it to me!)
(You will need to study which statements are postulates and which are
theorems!)
Can I write my own definitions of words we use?
No, not really. Definitions must be precise so that all of us are clear about what we are
discussing and we are only using words that have been defined previously. So, our
definitions need to be memorized and not faked.
Example: Can you determine which definition below is the good one?
Circle – a perfectly round object that has no beginning and no end.
Circle – set of all points in the plane at a fixed distance from a fixed point.
Circle – an arc that does a 360 spin to connect itself.
Circle – figure that looks like this:
What will the homework be like?
Let me first say that it will NOT be like the typical homework in algebra courses. Most
algebra homework focuses on drilling a small number of skills, with examples of each
type of problem. In geometry, don’t expect a sample of each type of homework problem
in the book. You will have knowledge of basics, but you will be required to apply that
knowledge to unique situations. Besides the homework problems, part of your
homework is always to memorize the definitions, postulates, and theorems we developed
that day. Don’t let these accumulate since it is impossible to memorize everything the
night before the exam. Also, we will be using these statements throughout the course.
What sort of skills will I need in order to do well in this course?
 Knowledge of beginning algebra (solving equations, working with radicals, etc.)
 Ability to write clear sentences
 Ability to memorize definitions, postulates and theorems
(notecards are very helpful!)
 Ability to break down problems into sequential steps
 Manipulate drawing instruments and visualize situations
 Manage your time and keep up with the material on a daily basis
It is my hope that after you finish this course, you would look at the world through a
geometer’s eye. Your visual skills and critical thinking skills should mature. You will
notice shapes and forms of objects around you and be able to think clearly through a
problem and its solution. Finally, I hope that you will remember the class as being one
of the more fun math classes you had!
Student Learning Outcomes for Math 110
Upon completion of this course, the student will be able to:
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clearly state Euclidean definitions, postulates, and theorems.
find missing side lengths and angle measures in a diagram using appropriate
theorems.
use compass and straightedge to perform constructions and use a protractor to
measure angles.
write 2-column direct proofs and indirect proofs using the definitions, postulates,
and theorems of Euclidean geometry.
apply appropriate formulas when finding areas of planar figures, surface area and
volume of solids, and when analyzing diagrams in the Cartesian coordinate
system