Lesson 1.6 Paragraph Proofs

1. Lesson 1.6 Paragraph Proofs Objective: Write paragraph proofs

2. Why are we doing this? 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.

3. Example #1 Given: <x = 37 ½° <y = 37° 30’ Prove: x y Proof: Since 30’ = ½° we know that 37° 30’ = 37 ½°. Therefore ( ) . w5 or Q.E.D W5 = which was what was wanted Q.E.D. = Quod Erat Demonstrandum which means “Which was to be Demonstrated”

4. Example #2 Given: Diagram Shown Prove: D (2x)° x° 60° E C A B 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

5. One last thing to keep in mind… 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!

6. Example #3 Given: <1 is acute <2 is acute Prove: 1 2 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

7. Homework Lesson 1.6 Worksheet

8. Lesson 1.7/1.8 Deductive 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.

9. Definitions • 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: • Undefined terms • Assumptions known as postulates • Definitions • Theorems and other conclusions

10. 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:

11. Conditional Statements 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: .

12. Conditional Statement Example: Write the following statement in its conditional form: “Two straight angles are congruent”

13. The Converse 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!

14. Converse Example: Conditional Statement: “If it is raining, then worms come out.” Converse: If worms come out, then it is raining

15. Negation 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 = _____________

16. Converse, Inverse, and Contrapositive Every Conditional statement , has 3 other statements: • Converse: • Inverse: • Contrapositive:

17. The AZ Example: Write each form of the conditional and decide whether the statement is true or false. Conditional Statement: “If you live in Phoenix, then you live in AZ.” Converse: If you live in AZ, then you live in Phoenix. Inverse: If you do not live in Phoenix, then you do not live in AZ. Contrapositive: If you do not live in AZ, then you do not live in Phoenix

18. Theorem 3 If a conditional statement is true, then the contrapositive of the statement is also true. 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.

19. Chains of Reasoning Many proofs we do involve a series of steps that follow a logical form. Often times it looks something like this: 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. 

20. Example 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. Hint: Rewrite these statements using symbols, then rearrange the statements and use contrapositives to match the symbols!

21. Homework Lesson 1.7/1.8 Worksheet