**Chapter 3 **

**Theorems in
Geometry**

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Draw 2 different triangles on a piece of paper, and measure each side length with a ruler, and each angle measurement with a proctactor as shown in the example below:

Do you think there is a relationship between the lengths of the sides and the sizes of the angles? If so, this would be an important property of triangles! In the diagram above, notice the measures of the sides and angles. What appears to be true about the relationship between the side and angle measures? (If you have geometry software, you can try this on a computer. When you drag a vertex of the triangle, the lengths of the sides change.You will notice that the measures of the angles change also.)

Did you notice that the sides of the first triangle are all equal and the angles are all equal also, but in the second triangle the sides are unequal and the angles are unequal? This is an important property of triangles! Also, do you notice anything else about the angles and sides? In the triangle on the right, which side is the longest, and which angle is the largest? In that same triangle, which side is the shortest, and which angle is the smallest? If you draw more triangles and measure them, you will find that the largest angle in a triangle is always opposite the longest side, the "medium sized" angle in the triangle is opposite the "medium sized" side, and the smallest angle is opposite the shortest side. This is an important, and interesting property of triangles!

Mathematicians have, for centuries, explored geometric figures in
order to discover their properties. Sometimes we have a feeling that
we have discovered a property, something that seems to always be
true, and we call this feeling a conjecture or hypothesis or theory.
A **conjecture** is a belief that something is true. But even
though we may feel very strongly that it is true, we are not
absolutely positive that it's true: maybe there is one exception, one
type of triangle in which it isn't true!

Where do they come from? Perhaps you have taken a science class,
and done lab experiments. If you are trying to find the properties of
a piece of rock, you might experiment with the rock, trying to
determine its properties. You might see if it floats by putting it in
water, or check to see if it is flammable by lighting a match to it.
Whatever you found in your experiments you might then consider a
property of the rock. If no one had ever discovered these properties
for this particular rock, then these would be*your *
discoveries, and you could call them by your own name. If it turned
out to be a very significant scientific discovery, you might even
become famous! Much the same is true in mathematics. Mathematicians
study geometric figures, make conjectures, experiment and test the
relationships, and then try to prove that their conjectures are
true.

As a mathematician works on a math problem, he or she may notice
something interesting about the problem, something that seems to
relate to another problem or seem to be true in more than just this
one situation. The mathematician might then explore this idea in a
number of ways. One way would be to do some very accurate diagrams,
measure them, and try to check the validity of their conjecture.
Software such as the Geometer's Sketchpad can make this process much
easier, and more visual. If the conjecture seems to be true, then the
mathematician could either accept that it is true without proof and
hope that he or she is correct, or prove that it is true. A
mathematical **proof** is a written verification that a conjecture
is true. Once proved, the conjecture is called a **theorem**.
(Occasionally, a proof is discovered to be incorrect, and the theorem
is then in question, and the mathematician may be a bit
embarrassed!)

"Euclid was a Greek mathematician who lived around 300 B.C. He
established a mathematical school in Alexandria And created much of
the geometry we study today.The name of Euclid is often considered
synonymous with geometry. His book __The Elements__ is one of the
most important and influential works in the history of mathematics,
having served as the basis, if not the actual text, for most
geometrical teaching in the West for the past 2000 years. It
contributed greatly to the 'geometrization' of mathematics and set
the standard for rigor and logical structure for mathematical
works.

In the thirteen books of __The Elements__, Euclid presents, in
a very logical way, all of the elementary Greek geometrical knowledge
of his day. This includes the theorems and constructions of plane
geometry and solid geometry, along with the theory of proportions,
number theory, and a type of geometrical algebra. __The Elements__
is a textbook which gathers into one place the concepts and theorems
which constitute the foundation of Greek mathematics. Euclid was not
the first to write such a work. It is known that Hippocrates of Chios
(440 B.C.) and others had composed books of elements before him.
However, Euclid's treatise was quickly recognized as being superior
to all previous __Elements__ and none of the earlier works have
survived.

Euclid's book __The Elements__ contained Definitions,
Postulates (sometimes called Axioms), and Propositions.

A **Postulate** or** Axiom** is a mathematical property that
is assumed to be true without proof. (The Greeks made a distinction
between postulates and axioms, but modern mathematicians usually do
not.)

Some of the original axioms of Euclid, which he calls "common notions", are the following:

1.Things which are equal to the same thing are also equal to one another.

2.If equals are added to equals, the wholes are equal.

3.If equals are subtracted from equals, the remainders are equal.

4.The whole is greater than the part."

**Propositions**

A proposition is usually a statement about the properties of a
geometric object. We call them **theorems**, in modern geometry.
They are accepted as true after having been proved. Most of what we
will be studying in modern geometry are theorems, and the great
majority of these theorems are based on Euclid's original
Propositions. __The Elements __ contained 353 Propositions. It is
really amazing that these Propositions are still used today, after so
many centuries!

According to another author, "Euclid's __Elements__ form one of
the most beautiful and influential works of science in the history of
humankind. Its beauty lies in its logical development of geometry and
other branches of mathematics. It has influenced all branches of
science but none so much as mathematics and the exact sciences.__
The Elements__ have been studied 24 centuries in many languages
starting, of course, in the original Greek, then in Arabic, Latin,
and many modern languages."

This quote was taken from a very interesting website. If you would like to visit this site, click on the link below:

**How do Mathematicians Decide if a Conjecture is
True?**

In this chapter of Connecting Geometry, you will use The Geometer's Sketchpad to experiment with some geometric figures. You will make discoveries about these figures, and you will be asked to write a brief paragraph proving that your conjecture is true. In future chapters, you will often be asked to experiment with other geometric figures. Your conjectures will be verified using various methods. A description of one of these methods of verification is written below.

**Deductive proof** is one way to prove things in mathematics.
It depends on a kind of reasoning called **deductive reasoning**.
When we use deductive reasoning, we write a series of statements,
each of which is either some information that we were given (called
the Given), a definition, a postulate, or a previously proved
theorem. In the proof of the right angle theorem below, you can
probably see which is the Given. A definition used in this proof is
the definition of right angle, a postulate used in this proof is
Euclid's first postulate. The final result of this proof is the Right
Angles Theorem.

The sequential steps in this proof form what is called a **chain
of reasoning**. In a chain of reasoning, each thought is dependent
on, and follows from the previous thought, and all lead to the final
conclusion. At the end, we have proved that which we set out to
prove. This is what we mean by deductive proof. Notice that the
theorem is written using the words **if** . . . and **then** .
. . This is the standard way in which theorems are written. Although
the theorem may seem to be obviously true, to be called a Theorem it
nevertheless requires proof.The proof may seem short and lacking in
real substance, but we do want to start off with a nice, short, easy
one. Don't worry, they will get more interesting (and, more
complex!).

__Theorem: if two angles are right angles then they are
equal.__

GIVEN: angle A is a right angle, angle B is a right angle

PROVE: angle A = angle B

PROOF: since angles A and B are each right angles, angle A= 90° and angle B=90° by the definition of right angle. Since angle A and B are equal to the same thing (90°), they are equal to each other.

Some deductive proofs are written informally, as in the proof above: this informal, wordy style is called a

__Theorem: if two angles are right angles then they are
equal.__

Mathematicians prove theorems in a variety of ways. Sometimes the
proofs of theorems can be very brief and informal, such as the
Paragraph Proof. Some are more formal, such as the Two Column
Proof.** **Proofs can also be written using algebra and using
coordinate geometry. In this course, we will explore each of these
different types of proof as they apply to our studies in
geometry.

Your project is to do some research of your own on the internet, on the History of Geometry. I would suggest you begin with the following links, to get warmed up. Then do a Net Search using Geometry as the Key Word. Write a 2 to 3 page report on what you discover, typing in a GSP file. Please do not just copy and print pages off the internet. You should search, read what you find, think about it, and write a summary/discussion of what you find. You may include direct quotations (a few sentences or even a paragraph), but be sure to put these portions in quotation marks, and include the web addresses to give credit to your sources as I have done above.

http://www-groups.dcs.st-and.ac.uk/~history/

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