Connecting Geometry©

Chapter 3

Theorems in Geometry


In our study of geometry, we will be deal with many geometric figures such as triangles and circles, and we will be concerned mostly with their properties. A property of a geometric figure is some interesting or important thing that is true about the figure. For example, a property of a triangle is that it has 3 sides; this property comes from the definition of a triangle. But once we define "triangle", we might notice that it has other properties.

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!

How do we discover geometric properties?

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 beyour 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!)

And now, a bit of history!

"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:

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html


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 Paragraph Proof. Sometimes proofs are written in a more formal style: a form called a Two-Column Proof. An example of a Two -Column Proof is shown below. Both forms of proof shown here prove the same theorem.


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.

Project

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://forum.swarthmore.edu/

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


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