Euclidean geometry/Proofs

One of the weird things about the field of mathematics in general is that all mathematically true statements are either postulates or provable from postulates. All of those equations you've heard before--like the familiar Pythagorean theorem, a² + b² = c²--have been proved at least once (and in that case many, many times over) in the history of mathematics.

Proofs are a surprising area where mathematics serves a highly useful realistic purpose, also, but not in ways you might expect! For example, the section below on conditionals is vital to understanding the most basic of computer programming--so much so, in fact, that we can't even say that a machine is entirely programmable without it. However, the act of creating mathematical proofs is also important in building a fundamental analytical mind. Most of the most basic runs of logic you go through in your daily head can be based--and improved--with an understanding of how proofs work.

In Euclidean geometry, the geometry that tends to make the most sense to people first studying the field, we deal with an axiomatic system, a system in which all theorems are derived from a small set of axioms and postulates. Most of these are relatively straightforward, e.g., it is possible to draw a straight line between any two points. Some are deceptively simple, however, and have confused mathematicians for centuries; but now isn't the time to worry about that. Let's get down to some provin'!

Conditionals

Preliminary Examples

See if you can see how all of these relate to one another. Think very simply...

If it is sunny, then I can play outside.

If there are clouds, then it will rain soon.

If you add sodium to water, then you will create an explosion.

If a statement contains if and then, then it is called a conditional.

The Conditional: Beginnings of Logic

A conditional is simply an if-then statement. If you accept the part after the "if", also called the hypothesis, then you must accept the statement after the "then", also known as the conclusion. So, all conditionals are in the form

If X [is true], then Y [is true]

There are certain relationships between conditional statements involving the same parts based on these two concepts:

Negation

Negation makes a statement false. For example, If "x=true", then the negation of x is "x=false". The negation of x is represented as ~x.

Converse

The converse of a statement basically flips the two parts of the statement. Using the example "if x is true, then y is true", the converse of this statement would be "if y is true, then x is true". These two concepts create the following five types of conditional statements:

Type of Conditional Statement	Example
1. Conditional	1. If x, then y
2. Converse	2. If y, then x
3. Inverse	3. If not x, then not y
4. Contrapositive	4. If not y, then not x

Pretty complex, right? Conditionals go even deeper than this. But before we can continue with this, you should at least have a basic grasp of what mathematical logic is.

Mathematical Logic: True or False?

In this field of mathematics, at least, logic is very simple: Either something is true, or something is false. These are also usually represented by the values 1 and 0, respectively; that is known as binary logic.

Wikibooks has an excellent book on the topic of formal logic; it is highly recommended that you read through that to get a better grasp on the topic.

Conditionals which have traveled from algebra to geometry

The following conditionals are always true. It will make sense if, while you read them, you shout, "These are an insult to my intelligence!" for in fact, that's what they'll seem like.

The first one is called the reflexive property.

If a is a number, then a=a.

The second one is called the symmetric property.

If a=b, then b=a.

The third one is called the transitive property.

If a=b and b=c, then a=c.

Now, explanations of the conditionals. The first one says any number equals itself. Strangely enough, this incredibly obvious statement will be a common sight when you write proofs.

The second one says that equations can be reversed, AKA the statement x=4 is equivalent to the statement 4=x.

The third statement says that if two numbers are equal to the same number, then they are equal, AKA if x=4 and y=4, then x=y.

The next four conditionals are very similar to each other.

The first one is called the addition property of equality.

If a=b and c=d, then a+c=b+d.

The second one is called the subtraction property of equality.

If a=b and c=d, then a-c=b-d.

The third one is called the multiplication property of equality. (Are you seeing the pattern?)

If a=b and c=d, then ac=bd.

The fourth one is called the division property of equality.

If a=b and c=d and neither c nor d equal 0, then a/c=b/d.

Now, all of these properties are very similar, as you probably can tell; for example, if x=2 and y=3, then

• x+y=2+3 by the addition property of equality.
• x-y=2-3 by the subtraction property of equality.
• xy=2(3) by the multiplication property of equality.
• x/y=2/3 by the division property of equality

You may notice that the division property of equality has an extra part in its hypothesis that isn't in any other property of equality. (Remember the name hypothesis? If hypothesis, then conclusion. If the hypothesis is accepted as true, then the conclusion must be accepted as true as well.) That is because division is undefined when the denominator is 0. 1/0 doesn't exist! Neither does 4/0, -200/0, or 0/0. The extra part of the hypothesis is to prevent division by 0.

The final conditional we will look at today is known as the substitution property, and it is incredibly useful in proofs.

If two values are equal, then they may substitute for each other.

For example, suppose we know x=y, and that x+2=4. Then we also know by the substitution property that y+2=4, since x and y can substitute for each other.

It is very important that you remember the names of these conditionals. The four "properties of equality" should be easy to remember, since they are named for the arithmetic operation that they have in their property. The substitution property also should be easy to remember, since you substitute values. That just leaves the transitive, symmetric, and reflexive property. You can remember the Transitive property if you remember that it uses Three values. If you remember lines of symmetry, then you can say that the symmetric property says that equations have lines of symetry right down their equals sign. As for reflexive? Well, you'll just have to remember it.

Also note that you can use other properties from algebra, for example

a(b+c)=ab+ac by the distributive property.

Proof

The way proofs work is that they try to prove things. Simple enough! It may be a good idea to think of proofs as a staircase. What you know is at the bottom of the staircase; what you're trying to show is at the top. The way you get there is the proof. Each step is a modification of the known information using other known information. For example, suppose at the bottom of the staircase we have a=b, and we want to show that a+b=b+a. You can do this like so.

• a=b; we're at the bottom, nothing new
• b=a; we just used the last step and the symmetric property
• a+b=b+a; this comes from the first two steps and the addition property of equality

There are three different types of proofs. They are the two column proof, the paragraph proof, and the flowchart proof

Two column proof

Given: AC=BD

Prove: AB=CD

Statements	Reasons
1. AC=BD	1. Definition of congruent segments
2. AB+BC=AC; BC+CD=BD	2. Segment addition postulate
3. AB+BC=BC+CD	3. Subsitution property
4. BC=BC	4. Reflexive property
5. AB=CD	5. Subtraction property of equality
Q.E.D.

That was a two column proof. It has five parts:

1) The diagram. This diagram gives a visual for the problem, and is immensely useful

2) The given/prove information. This shows us what is the bottom and top of our staircase

3) The statements. These tell us what the steps are.

4) The reasons. These tell us how the statements in part three are verified

5) The conclusion. It is abbreviated "Q.E.D.", which stands for "quod erat demonstrandum". This is Latin for "which was to be proven". It concludes the proof.

To make a two column proof, follow these steps

1) Draw your figure. The figure may be the most important part of a proof; it gives a visualization of what you're proving.

2) Put down what you know, and what you're trying to prove.

3) Draw your two columns; head the left one "Statements" and the right one "Reasons"

4) Now, try to visualize the path you will take to get to the end.

5) Write the steps you followed under "Statements". Verify your statements under "Reasons."