Geometry/Chapter 2

Section 2.1 - Proofs
Proofs are set up to let the user understand what steps were taken in order to receive a given output. There are three types of proofs depending on which is easiest to the student.

Two-Column proofs
Two-column proofs (also known as formal proofs) are set up in a two-value table, one being "Statement" and the other being "Reason". To prove a simple problem using this method, set up a table like the following:

Be sure to leave room for values to go in both columns. In geometry, the first row is the 'given' of the problem. This is the information that is given about a certain problem without using a picture. The last row should be the conclusion of what you are trying to prove.

Example of a two-column proof
Now, suppose a problem tells you to solve $$x + 1 = 2$$ for $$x$$, showing all steps made to get to the answer. A proof shows how this is done:

We use "Given" as the first reason, as it is "given" to us in the problem.

Written Proof
Written proofs (also known as informal proofs, paragraph proofs, or 'plans for proof') are written in paragraph form. Other than this formatting difference, they are similar to two-column proofs.

Sometimes it is helpful to start with a written proof, before formalizing the proof in two-column form. If you're having trouble putting your proof into two column form, try "talking it out" in a written proof first.

Example of a Written Proof
We are given that x + 1 = 2, so if we subtract one from each side of the equation (x + 1 - 1 = 2 - 1), then we can see that x = 1 by the definition of subtraction.

Flowchart Proof
A flowchart proof or more simply a flow proof is a graphical representation of a two-column proof. Each set of statement and reasons are recorded in a box and then arrows are drawn from one step to another. This method shows how different ideas come together to formulate the proof.

Section 2.2 - Reasons
Every concept in geometry flows in a logical progression. One simply cannot go from A to B without explaining how or why. For instance, the following is not a proof:

$$5(x+1)^2-x$$ $$x=4$$

Also, we cannot make up reasons why we made the next step so. Therefore, we can only use certain information as our reasons. These include:

1. Given: This is generally either the problem (equation) we are trying to solve, or some piece of important information given in the problem.

2. Properties: These for the most part are the basic mathematical functions of adding, subtracting, multiplying, and dividing, such as the second reason in the example above (Property of Subtraction).

3. Definitions: Again, saying "Because it is" is not a reason. This sort of reasoning is not seen as often as other reasons. By using definitions, sometimes the answer or part of the working of a proof can be shortened. For example, by using the reason "definition of a bisector" (and being already able to prove through either given information or earlier parts of the proof), you can prove that two adjoining angles are congruent without having to go through a more lengthy proof.

4. Postulates: They hold the same value as theorems (explained next), except that they cannot be proven. However, these generalized rules have proven correct for a very long time and can be accepted with proof of their validity. An example is "Through any two points, there is exactly one line". While it cannot be proven through a proof (although the authors dare anyone to disprove it), it is accepted as a reason. There are few of these, so as good as it may sound, if you make it up, someone will notice.

5. Theorems: Theorems are statements that have been proven true through proofs of their own. They are especially helpful shortcuts in their own right as by stating a theorem, a great many things are proven and you do not have to do all the work of re-proving the theorem. Theorems can be simple ("If two lines intersect, they intersect in exactly one point.") or very complex ("The composite of two isometries is an isometry." [Don't panic if you don't understand; it will be explained later on]). Sometimes, you will be given the proofs for theorems; othertimes, as part of the exercises, you will be asked to prove it yourself.

6. Axioms: For most purposes, the same as postulates. The difference is that axioms are algebraic in nature, while postulates come mainly from geometry.

7. Corollaries: These are statements that stem from what becomes proven in theorems and definitions and do not require (though usually have) separate proofs themselves.

In many textbooks, the proofs are numbered for an index at the back of the book. When doing correct geometric proofs, it is NOT OK to write down "Theorem 15". Write out the statement exactly as it was given to you (yes, you have to do some memorization for geometry). You have to make sure that the information in the box is related to the earlier box.

Exercises
Answers to each exercise can be found separately in the appendix.

1.



Given: Prove: Find the measures of the other seven angles in the accompanying figure (above).
 * $$r || s$$ r is parallel to s
 * $$\angle 1=60^\circ$$ Angle 1 = 60 degrees.

2.



Given: Prove: Lines r and s are parallel.
 * $$\angle 2 \cong \angle 3$$ Angles 2 and 3 are congruent

3.



Given: Prove: Lines a and b in the figure are parallel.
 * $$\angle 1 = \angle 2 = 90^\circ$$ Angles 1 and 2 are both 90 ⁰

4.



Given:
 * $$\overline{GH} || \overrightarrow{DK}$$ Line GH is parallel to ray DK
 * $$\angle 6 = 75^\circ$$ Angle 6 = 75 degrees.
 * $$\angle 2 = 30^\circ$$ Angle 2 = 30 degrees.

Prove: Find the measure of each numbered angle in the figure above.

Section 2.4: Proof by contradiction
Proof by contradiction, also known as indirect proofs, prove that a statement is true by showing that the proposition's being false would imply a contradiction.

A classic example: Proving that the square root of 2 is irrational

 * 1) Assume that $$\sqrt{2}$$ is a rational number, meaning that there exists an integer $$a$$ and an integer $$b$$ in general such that $$a/b = \sqrt{2}$$.
 * 2) Then, $$\sqrt{2}$$ can be written as a simplified fraction $$a/b$$ such that $$a$$ and $$b$$ are coprime integers, that is, their greatest common divisor being 1.
 * 3) It follows that $$a^2 / b^2 = 2$$ and $$a^2 = 2 b^2$$. &emsp; (&thinsp;$$(a/b)^n = a^n/b^n$$ &thinsp;)
 * 4) Therefore $$a^2$$ is even, as it is equal to $$2 b^2$$. ($$2 b^2$$ is necessarily even, as it is 2 times another whole number, and even numbers are multiples of 2.)
 * 5) It follows that $$a$$ must be even (as squares of odd integers are never even).
 * 6) Because $$a$$ is even, there exists an integer $$k$$ that fulfills: $$a = 2k$$.
 * 7) Substituting $$2k$$ from step 6 for $$a$$ in the second equation of step 3: $$(2k)^2 = 2 b^2$$ is equivalent to $$4k^2 = 2b^2$$, which is equivalent to $$2k^2 = b^2$$.
 * 8) Because $$2k^2$$ is divisible by two and therefore even, and as $$2k^2 = b^2$$, it follows that $$b^2$$ is also even, which means that $$b$$ is even.
 * 9) By steps 5 and 8 $$a$$ and $$b$$ are both even, which contradicts that $$a/b$$ is irreducible as stated in step 2.

As there is a contradiction, the assumption (1) that $$\sqrt{2}$$ is a rational number must be false. By the law of excluded middle, that is, that a proposition can only be true or false, the opposite is proven: $$\sqrt{2}$$ is irrational.