Lesson 1.6Paragraph Proofs

Objective:

Write paragraph proofs

Although most proofs we do in this class are two-column, you also need to be familiar with paragraph

proofs.

Paragraph proofs are useful to know because they help us to think logically through a problem, and put a solution in a form that everyone can understand

and follow.

We are going to see how to write a paragraph proof, as well as how to show that a conclusion cannot be

proved.

Why are we doing this?

Proof:

Since 30’ = ½° we know that 37° 30’ = 37 ½°. Therefore ( ) .

w5 or Q.E.D

Example #1

x y

Given: <x = 37 ½°

<y = 37° 30’

Prove: x y

x y

W5 = which was what was wanted

Q.E.D. = Quod Erat Demonstrandum which means

“Which was to be Demonstrated”

Proof:

According to the diagram, <ABC is a straight angle. Therefore, 2x + x = 180

3x = 180

x = 60

Since <DBC = 60° and <E = 60°, the angles are congruent. Q.E.D

Example #2

D

E

Given: Diagram Shown

Prove: DBC E

A B C60°(2x)° x°

Not all proofs can be proved. If this happens, we use what’s called a counter-example. We assume that the original statement is true, and then use a specific

example to show that it is not possible.

Remember, it only takes one false example to disprove a statement!

One last thing to keep in mind…

Proof:

Since <1 is acute, let it be 50°, and since <2 is acute, let it be 30°. Therefore, by counter-example, it

cannot be proved that . Q.E.D

Example #3

Given: <1 is acute

<2 is acute

Prove: 1 2

21

1 2

Lesson 1.6 Worksheet

Homework

Lesson 1.7/1.8Deductive Structure and Statements of

Logic

Objective:

Recognize that geometry is based on deductive structure, identify undefined terms, postulates, and definitions,

understand the characteristics of theorems, recognize conditional statements, recognize the negation of a

statement, the converse, inverse, and contrapositive, and draw conclusions using the chain rule.

Def. Deductive Structure is a system of thought in which conclusions are justified by means of previously assumed or proved statements.

Note: every deductive structure contains 4 elements:

1.Undefined terms

2.Assumptions known as postulates

3.Definitions

4.Theorems and other conclusions

Definitions

Def. A Postulate is an unproven assumption (In other words, it is so obvious, it does not need to be proved)

Def. A Definition states the meaning of a term or idea.

Note: Definitions are reversible!

Example:

Original Definition:

Reversed Definition:

Definitions

Midpoint Segments

Segments Midpoint

All definitions are stated in a specific form:

“If p, then q”

This type of sentence is called a Conditional Statement (or an Implication)

The “if” part = the hypothesis

The “then” part = the conclusion

We write this mathematically as: .

Conditional Statements

p q

Write the following statement in its conditional form:

“Two straight angles are congruent”

Conditional Statement Example:

The converse of

is:

To write the converse of a statement, you reverse parts p and q.

Important Note!

Because definitions can be reversed, the conditional statement (the original) and the converse will

always be true. This is not always the case for theorems and postulates!

The Converse

p q q p

Conditional Statement:

“If it is raining, then worms come out.”

Converse:

Converse Example:

If worms come out, then it is raining

The negation of any statement p is the statement “not p.”

The symbol for “not p” is “~p”

Ex. If p = It is raining then ~p = _____________

Negation

Every Conditional statement ,

has 3 other statements:

Converse, Inverse, and Contrapositive

If p then q

If q, then p

If ~p, then ~q

If ~q, then ~p

1. Converse:

2. Inverse:

3. Contrapositive:

Conditional Statement:

“If you live in Phoenix, then you live in AZ.”

The AZ Example:Write each form of the conditional and decide whether the

statement is true or false.

If you live in AZ, then you live in Phoenix.

If you do not live in Phoenix, then you do not live in AZ.

If you do not live in AZ, then you do not live in Phoenix

Converse:

Inverse:

Contrapositive:

If a conditional statement is true, then the contrapositive of the statement is also true.

Theorem 3

Note:

Often times mini Venn Diagrams are useful in determining whether or not a conditional statement

and its converse, inverse, or contrapositive are logically equivalent.

Try making Venn Diagrams for each example written on the last slide.

Many proofs we do involve a series of steps that follow a logical form. Often times it looks

something like this:

Chains of Reasoning

This is called the chain rule, and a series of conditional statements is known as a chain of reasoning.

Example:

If you study hard, then you will earn a good grade, and if you earn a good grade, then your family will be happy.

We can conclude: If you study hard, your family will be happy.

If p q, q r, and r s, then p s

Draw a conclusion from the following statements:

If gremlins grow grapes, then elves eat earthworms.

If trolls don’t tell tales, then wizards weave willows.

If trolls tell tales, then elves don’t eat earthworms.

Example

Hint:

Rewrite these statements using symbols, then rearrange the statements and use contrapositives

to match the symbols!

Lesson 1.7/1.8 Worksheet

Homework