Week 5 Geometry Notes

Unit 2 Lesson 12 Proof Formats: Statement of the Theorem

When asked to prove a geometric statement, you will be given some facts about the situation and a statement to be proven. Many find it useful to write this information in as a conditional statement.

Remember that a conditional is a statement that takes on the form of if-then. For example, if you were given two facts about a situation and asked to prove some statement using those two facts you could say "If fact 1 and fact 2, then whatever you want to prove."

You are informed that angles 1 & 2 are congruent. You also know that segment

EF and HF are congruent. You are asked to prove that triangle EFJ and triangle HFG are congruent (Pause).

Notice that you have a couple of statements that are labeled as “Given” and another statement labeled as “Prove”. Before beginning the process of proving

the two triangles are congruent, use this information to create a conditional statement.

The conditional statement consists of the hypothesis statement “If” which are your givens and the conclusion “Then” which is your “prove” statement.

For this proof, the conditional is If angles 1 and 2 are congruent and segments EF and HF are congruent, then triangles EFJ and HFG are congruent.

You are informed that segment MP is perpendicular to segment RO and that

segments MO and PO are congruent. You are then asked to prove that the triangles MRO and PRO are congruent.

As in the last example, the “Givens” will be your hypothesis statement and the “Prove” will be your conclusion.

What is your conditional statement for this proof? For this proof, the conditional statement is If segment RO is perpendicular to

segment MP and segments MO and PO are congruent, then triangle MRO is congruent to triangle PRO.

To prove a statement in geometry means to demonstrate that the statement follows logically from other accepted statements.

Thus only definitions, postulates, or previously proven theorems can be used as general statements to support conclusions made in proving statements by deductive reasoning.

The two-column proof is used extensively in geometry. This proof enables a student to show each conclusion arrived at and the reason that supports it.

The format of a two-column proof consists of six essential parts arranged in order.

They are:

Statement of the Theorem

Much of the challenge of geometry is found in “proving things". You can use deductive reasoning to show that a conjecture is true. The set of steps you take is called “a proof". To prove a statement in geometry means to demonstrate that the statement

follows logically from other accepted statements. Thus only definitions, postulates and other previously proven theorems can be

used as general statements to support conclusions made in proving statements by deductive reasoning.

The two column proof is used extensively in geometry, and this will be the major type of proof you will use in this course of study.

This type of proof enables a student to show each conclusion arrived at and the

reason that supports it. The format of a two column proof consists of six essential parts arranged in order.

The six essential components of a proof are the “Statement of the Theorem", a diagram that illustrates what is to be proven, the information that is presented as true, also called the “givens,” a statement of what we want to prove, the plan for completing the proof, and the steps of the proof itself.

As we will be completing a number of proofs for this course, let’s look at an example of how the 6 essential parts will look. Consider the statement “Two angles that are complements of the same angle are congruent.”

o Our first step is to change the original statement into an if-then conditional: “If two angles are complements of the same angle, then the two angles are congruent". This is our “Statement of the Theorem".

o

o Next, we will create a diagram to illustrate this theorem. Recall that the hypothesis of the conditional contains the accepted true statements called

the “givens.” o So the givens for this proof are: angles 1 and 2 are complementary, and

angles 2 and 3 are complementary. From the conclusion of the conditional and the diagram, we can determine what the “prove statement” should be. We want to prove that angles 1 and 3 are congruent as they are the angles that are complementary to angle 2.

o The plan consists of a direction in which to prove that angles 1 and 3 are congruent. Using the definition of complementary, we know that the sum of the measures of angles 1 and 2 is 90 degrees.

o Also the sum of the measures of angles 2 and 3 is 90 degrees. Using substitution, we can show that angles 1 and 3 are congruent. Example proof. Given: angles 1 and angle 2 are complementary. Angle 2 and 3 are complementary. Proven: angles 1 and 3 are congruent. Plan: angles 1 and 2 = 90 degrees. Angles 2 and 3 = 90 degrees, Use substitution to show that angles one and three are congruent.

oo All we have left to do is the two column proof. For this exercise, the proof

is given in this graphic; we can see that each step is accompanied by a

reason for that step. The last step in the proof is the same as the “Prove” step earlier in the process.

Now that this theorem has been proven, we can use it as the reason for steps in future proofs.

Unit 2 Lesson 13 Proof Formats: The Figure

The Figure

o Each given proof should include a lettered figure drawn to illustrate the given conditions contained in the hypothesis of the statement.

o Although a diagram is not necessary to the logical reasoning of a proof, it helps to see the deductions being made. We may be able to make some inductive conclusions and then justify them deductively.

o Be sure the figure accurately depicts the given conditions. Do not add any special features that are not given, and do not draw a right triangle or an isosceles triangle when a triangle is specified. Make the figure as general as possible.

o A common mistake that students make when doing a proof is drawing an assumption from the figure when it is not actually given in the original information. Unless your information gives you specific information, you must always draw the most general case.

o Let’s take a look at an example of where creating a figure leads to much more information than was originally given.

o The corresponding angles postulate states “If a transversal intersects two parallel lines, then corresponding angles are congruent.” Notice that this is a postulate and therefore does not have to be proven in order to accept this as a true statement. The figure has two parallel lines that are cut by a transversal as this

information is contained in the hypothesis of the conditional.

The conclusion states that corresponding angles are congruent.

o From this diagram, we can also see that there are many pairs of vertical angles that are also congruent to the original corresponding angles which will lead to proofs about alternate interior angles, alternate exterior angles and many more theorems.

o By creating a diagram, connections can be made that extend beyond the problem currently at hand.

Unit 2 Lesson 14 Proof Formats: The Given Statements

Proof Formats: The Given Statement

In the previous few lessons, you have learned that it is useful to write the “Proof Statement” in if, then form and it is very important to create a diagram if one does not accompany the problem.

It’s important to consider what is given to you to begin the proof. As you begin a proof, read the problem over carefully. Write down the information

that is given to you because it will help you begin the problem. Also, make note of the conclusion to be proved because that is the final step of your proof. This step helps reinforce what the problem is asking you to do and gives you the first

and last steps of your proof.

As you begin to write the proof, you need to state the given. The given is the hypothesis and contains all the facts that are provided. The given is the what. What info have you been provided with to solve the proof? If you were given three truths as part of the problem, you would write them on the

left side of the proof format and document on the right side of the proof that these were given. 

This is just the beginning of the process of proof. The given statement is an important part of the proof writing process. Remember from this lesson to begin each proof with a list of your givens.

The Given Statement

In this diagram, we can see three important components of a two column proof: the given statement, the prove statement and the proof itself.

It is essential that we recognize the givens as they provide the “what you know and accept as truth” portion of the proof.

The “givens” are expressed in terms of the letters or numerals that are used in the figure or given in the statement to be proved. Remember that the given information is contained in the hypothesis of the statement, which is the part that follows the "if." Whenever any information is "given," we know that information is true.

A very important thing to remember when doing two column proofs in geometry is to make sure you write down ALL the given information. Do not assume that a piece of information is not important or that it will not be needed. By writing down all the given information, you have a much better chance of recognizing how to plan and proceed with the proof.

Let’s look at a few examples and identify the “givens” to the proof. Example 1 states “If a triangle has two equal angles, then the sides opposite those angles are equal.” Remember that the “givens” to a proof are found in the hypothesis of the conditional. If the statement to be proved is not in conditional form, rewrite it so that the hypothesis is easy to determine. In this example, the “given” is a triangle has two equal angles.

Example 2 begins with the statement “Glass objects are fragile.” The first step is to rewrite this statement as a conditional. The new statement becomes “If an object is glass, then it is fragile.” The “given” found in the hypothesis is “the object is glass.”

Unit 2 Lesson 15 Proof Format: To Prove StatementProof Format: To Prove Statement

In logic, a conjunction is a compound sentence formed by using the word "and" to join two simple sentences or facts. The symbol for this is an upward pointing arrow and is read as the word "and". When two sentences or facts are combined into a conjunction statement, the conjunction statement is expressed mathematically as p and q.Here's an example of a conjunction made from two simple sentences. Sentence one states "It's raining." Sentence two states "The ground is wet." The conjunction is "It's raining and the ground is wet."

Mathematicians often use symbols and tables to represent concepts in logic. The use of symbols and tables creates a shorthand method for visualizing and discussing logical sentences.

A truth table is a pictorial representation of all the possible outcomes of the truth value of a compound sentence. Letters such as p and q are used to represent sentences or facts within a compound sentence.

In order to construct a truth table for the compound statement p and q, you need to explore all the possible combinations of truth values for both p and q.When p is true, q could be either true of false. Likewise, when p is false, q could be either true or false. So, the truth table will have four rows, one for each combination of truth values for q and q. Notice that for a conjunction to be true, both sentences must be true.

Consider this quick example. Let p be the statement 'Ice is cold' and let q be the statement '5 is greater than three + four'. Is p and q true? Because q is false, p and q is also false. It doesn't matter if p is false. It only takes one of the statements to be false in order for the conjunction to be false.

To Prove

It is essential to determine what you know and what is to be proven in order to

have a correct proof. For many proof exercises, this information is given just as it

is represented in this graphic. If the statement to be proved is written as a conditional, the conclusion contains

what is to be proven. If the statement is not written as a conditional, it can always be rewritten in If,

Then format. This graphic also shows three of the main components in completing a two column proof.

Two column proof formats provide an easy to see and easy to follow guide for completing any proof. The two column format enables us to show each conclusion arrived at and the reason that supports that conclusion. Notice the first two headings in the graphic are “Given” and “Prove".

This theorem is already written as a conditional so the “givens” are found in the hypothesis and the “prove” is contained in the conclusion. The “Given” portion of the two column proof is “two parallel lines cut by a third line” and the Prove statement is “alternate interior angles are congruent”.

The hypothesis becomes the “Given”, and the conclusion of the sum of the angles equals 180 degrees is what needs to be proven.

Unit 2 Lesson 16 Proof Formats: The Plan of the ProofProof Formats: The Plan of the Proof

In this example, you are given a diagram of two supplementary angles. 

(The Angle-Addition-Postulate or AAP allows you to add the two angles and set them equal to 180 degrees.) Through algebraic properties of subtraction and division we can prove that x = 40. With a quick check, you can know that your answer of x = 40 satisfies the condition that the angles are supplementary. Always begin a proof with a plan in mind.

Plan of the Proof

As we begin a proof, it is helpful to take a moment to devise a strategy for attacking the problem.

One way to look at a problem is to reason backwards from the conclusion. In reasoning backwards, we assume the conclusion is true if some fact A is true;

Fact A in turn will be true if fact B is true. We continue with this line of thinking until we have made a connection with one or more of the givens.

We can also look back at previously proven theorems for possible methods of proof. Two-column proofs will challenge your logical thinking skills, but do not become discouraged - with time and practice, proficiency can be achieved.

Another method that will help develop a plan is to closely look at the given

information and the figure that accompanies the proof.Many times the given information will provide a hint about the direction the proof should follow.For example, look at the following given information:The Given is Lines a and b intersect at point C.

Let's start with the first piece of information: we are given two lines a and b that intersect at point C. What else can we determine from this information?Well, we know that two lines can only intersect at one point. [Given] That may not seem too helpful, but it just might give enough information to start a proof. We also know that intersecting lines create vertical angles which are congruent.

What about the given information "Triangle XYZ is an isosceles triangle"?

What other information can you deduce from this statement? What are the properties of an isosceles triangle? 

An isosceles triangle is a triangle with two equal sides. The fact that this triangle has two equal sides may lead us to discover the path our proof should take.

When coming up with a plan, don't overlook the information contained in the "given" statement. 

The actual proof will be a series of numbered statements in one column with a like numbered column next to it for the reasons. 

Each step must have a reason associated with it. You must explain the "why" behind the step.

The only things that can be used as reasons are given data, definitions, postulates, previously proven theorems, and algebraic properties.

We know each step in the Statement column is true because we are using deductive reasoning and listing the general principle in the Reason column.

In this two-column Model:

Step 1 is true because it was the given equation. Step 2 is true because we used the distributive property on the equation in Step

1. Step 3 is true since we added 6 to both sides of the equation in Step 2. Step 5 is correct because we used the division property on step 4.

Each proof will be different from the previous proof. Unfortunately, it is not a matter of memorizing certain steps to a process (similar to algebra), but rather it is a matter of developing your logical thinking ability.

Unit 2 Lesson 17 Indirect Proof Formats: The Paragraph Proof

The type of proof discussed in the previous section is known as a direct proof. Each step follows directly from one before it until the desired conclusion is reached. Another type of proof is often used to prove theorems. This type of proof is called an indirect proof.

The Paragraph Proof

Indirect proofs are sort of a weird relative of regular proofs. With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false.

For the most part, an indirect proof is very similar to a regular proof. What makes it different is the way it begins and ends. And except for the beginning and end, to solve an indirect proof, you use the same techniques and theorems that you would use on regular proofs.

The question "How do I know when to use an indirect proof?" is a very common one. The general rule for when to use an indirect proof is the presence of the word "not", a "not symbol" as in not congruent or "not equal" or an inequality symbol.

Indirect proofs can be written as a paragraph proof or as a two column proof.

Now let's look at a two column indirect proof. You are given a diagram of quadrilateral

PRVE.

Indirect proofs do take a little practice and patience. This is a powerful tool in reasoning. Read through as many of these proofs as you can to familiarize yourself with them, their formats and their goals.