Chapter 2. Reasoning and Proof

Transcription

1 Chapter 2 Reasoning and Proof 2.1 Inductive Reasoning 2.2 Analyze Conditional Statements 2.3 Apply Deductive Reasoning 2.4 Use Postulates and Diagrams 2.5 Algebraic Proofs 2.6 Segments and Angles Proofs 2.7 Prove Angle Pair Relationships SOL G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) Identifying the converse, inverse and contrapositive of a conditional statement b)translating a short verbal argument into symbolic form c) using deductive reasoning Name Block

2 Logic Puzzles How many can you figure out?! 1. Here are four triangles, each made up of three matchsticks and labeled with three letter codes. What code corresponds with the 5th triangle? 2. Look at the following series: 3, 4, 8, 9, 10, 18, 19, 38 Which number does not belong? 3. Where does the zero go in this sequence? There was a robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that : (1) Nobody else could have been involved other than A, B and C. (2) C never commits a crime without A's participation. (3) B does not know how to drive. So, is A innocent or guilty?

3 2.1 Inductive Reasoning A is an unproven statement that is based on observations. It is likely to be true, but has yet to be proven! You use when you find a pattern in specific cases and then write a conjecture for the general case. A disproves the conjecture. Practice 1: Use reasoning to sketch the next figure in the pattern Practice 2: Describe a pattern in the numbers then write the next number in the pattern. 3. ¼, -½, 1, , 74, 68, 62 Practice 3: Show that the conjecture is false by finding a counterexample. 5. Conjecture: All birds can fly. Counterexample: 6. Conjecture: The opposite of a number is always smaller than the original number. Counterexample: 7. Conjecture: Point M is between J and K. M is in the middle of J and K. Counterexample: Challenge!! Complete the following with a partner: Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.

4 Venn Diagrams Venn Diagrams are a visual way of displaying the relationships between sets. q p q p q p All elements of p are elements of q. Some elements of p are elements of q. There is no relationship between p and q. Draw a Venn Diagram to represent each statement. Some students who take chorus also take band. Every natural number is a whole number. Irrational numbers are never rational numbers. Numbers divisible by 6 are always divisible by 3. Shade the indicated region on the Venn Diagram. q ^ p ^ q p q p q p V q p ^ ~q p q p q

5 Example: When 24 students were asked about their favorite sports, 14 said they liked softball, 18 liked basketball, and 8 liked both. How many students liked ONLY softball? How many students liked ONLY basketball? With the person next to you, complete the following Venn Diagram. In a survey of 80 students, the number of pets was surveyed. How many students do not own a pet? 52 own a dog 41 own a cat 39 own fish 25 own a dog and cat 23 own a cat and fish 30 own a dog and fish 18 own all three

6 2.2 Analyze Conditional Statements The hypothesis of a conditional statement is the phrase immediately following the word. The conclusion of a conditional statement is the phrase immediately following the word. The part is the hypothesis and the part if the conclusion A conditional statement is a statement that can be written in the form. The converse is formed by exchanging the hypothesis and conclusion of the conditional. The inverse if formed by negating both the hypothesis and conclusion of the conditional. The contrapositive is formed by negating both the hypothesis and conclusion of the converse of the conditional. Biconditional statements are true when both conditional have the same truth value. A conjunction is a compound statement formed by joining two statements with the connector. A disjunction is a compound statement formed by joining two statements with the connector. Therefore And Or

7 Directions: Write the appropriate statement to match the symbolic notation. Then, classify it as the conditional, inverse, converse or contrapositive. p: you have a library card q: you can check out books 1. p q Classify: 2. ~ q ~ p Classify: 3. q p Classify: 4. ~ p ~ q Classify: 5. p q Classify: Directions: Write the appropriate symbolic notation to match the statement. p: you see lightning q: you hear thunder a. If you see lightning, then you hear thunder b. If you hear thunder, then you see lightning c. If you don t see lightning, then you don t hear thunder d. If you don t hear thunder, then you don t see lightning

8 2.3 Apply Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. Inductive reasoning uses specific examples and patterns to form a conjecture without facts. If the hypothesis of a true conditional statement is true, then the conclusion is also true. Uses the idea of the transitive property. If hypothesis p, then conclusion, q. If hypothesis q, then conclusion r. If hypothesis p, then conclusion r. Directions: Use the Law of Detachment to make a valid conclusion in the situation. 1. If you get a hit, then your baseball team will win. You get a hit. 2. If Sophia gets promoted, then Aubree will also be promoted. Sophia is promoted. 3. If two integers are added together, then the result is an integer. You add an integer x to another integer y. Directions: Use the Law of Syllogism to write the statement that follows from the pair of statements that are given. 4. If Rylee is hungry when he goes to the pizza shop, then he ll finish a whole pizza. If Rylee eats a whole pizza, then he goes through a pitcher of soda. 5. If you mail the payment by noon, then it will arrive by tomorrow. If your payment arrives by tomorrow, then you won t be charged a late fee. 6. If a triangle has two angles of 60, then the triangle is equiangular. If a triangle is equiangular, then it is also equilateral.

9 2.4 Use Postulates and Diagrams 2.5 Algebraic Proofs Do you remember from Algebra? Properties of Real Numbers Directions: With a partner, complete the following proof by justifying each step. Note, not all properties are listed in the table above. Given: y = 2 (3x + 4) and y = 20 Prove: x = 2 Statement Reason y = 2( 3x + 4) 20 = 2( 3x + 4) 20 = 6x = 6x 2 = x x = 2

10 Points, Lines and Planes Postulates These postulates can be used in proofs! Postulate #5: Through any two points there exist exactly. Postulate #6: A line contains at least. Postulate #7: If two lines intersect, then their intersection is exactly. Postulate #8: Through any noncollinear points there exists exactly plane. Postulate #9: A plane contains at least noncollinear points. Postulate #10: If points lie in a plane, then the line containing them lies in the plane. Postulate #11: If two intersect, then their intersection is a. If M is the midpoint of AB, then AM ~ = MB. A M B Is the midpoint theorem a bi-conditional? State the postulate illustrated by the diagram. 1 2 Use the diagram to write examples of Postulates 9 and 10.

11 2.6 Segments and Angles Proofs A two-column proof is one common way to organize a proof in geometry. Two-column proofs always have two columns- statements and reasons. Writing a Two-Column Proof: In a proof, you make one statement at a time until you reach the conclusion. Because you make statements based on facts, you are using. Usually the first statement-and-reason pair you write is given information. When writing your own two-column proof, keep these things in mind Never assume anything unless it is given in the picture or information!! Before starting a proof, look at the pictures and given information and fill in any details that you can based on the information given. Next, formulate your idea in your head or on paper before you begin. The left of a two-column proof are the statements (the steps to get from point A to point B) The right column is the reasons for each statement. Both the statements and the reasons should be numbered the same.

12 Intro to Proofs 1. If MN and MP are opposite rays, then I know this because of 1 2 N M P MN and MP are opposite rays. 2. If PC and PD are opposite rays ad PA and PB are opposite rays, D A then 1 2 P B C I know this because of PC and PD are opposite rays; PA and PB are opposite rays. 3. S If angles 1 and 2 are right angles, then I know this because of 1 2 P Q R 1 and 2 are right angles. 4. If ABC DBC, then I know this because of A C B D ABC DBC 5. If MN NP, then I know this because of M N P MN NP

13 Segments Proofs Reference Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Properties of Equality Substitution Property Reflexive Property Symmetric Property Transitive Property The properties above may only be used with EQUAL signs. The following properties of congruence can be applied to statements with congruence symbols. Properties of Congruence Reflexive Property of Congruence For any segment AB, Symmetric Property of Congruence If, then. Transitive Property of Congruence If and, then Definitions Definition of Congruence Segments are congruent if and only if they have the same measure. If, then. If, then. Definition of Midpoint If M is the midpoint of AB, then. Postulates Segment Addition Postulate If A, B and C are collinear points and B is between A and C, then.

14 Practice Justify each of the following statements using a property of equality, property of congruence, definition, or postulate. Property Bank: Properties of Equality: Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property Properties of Congruence: Reflexive Property Symmetric Property Transitive Property Definitions: Definition of Congruence Definition of Midpoint Postulates: Segment Addition Postulate

15 Segments Proofs Complete the proofs below by giving the missing statements and reasons. 1. Given: Prove: D E F K 2. Given: Prove: L M N 3. Given: Prove: J N K M L

16 4. Given: Prove: X Y Z T U V 5. Given: Prove: R S T 6. Given: Prove: A B C D

17 ANGLES Proofs Reference Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property Definitions Properties of Congruence Reflexive Property Symmetric Property Transitive Property Definition of Congruence m A = m B A B Definition of Angle Bisector An angle bisector divides and angle into equal parts. Definition of Complementary Angles Complementary Sum is. Definition of Supplementary Angles Supplementary Sum is. Definition of Perpendicular Perpendicular lines form. Definition of a Right Angle A right angle =. Postulates Angle Addition Postulate m ABD + m DBC = m ABC Theorems Vertical Angles Theorem Complement Theorem Supplement Theorem If two angles are vertical, then they are. If two angles form a right angle, then they are. Right angle Complementary If two angles form a linear pair, then they are. Linear pair Supplementary Congruent Complements Theorem If A is complementary to B, and C is complementary to B, then A C. Congruent Supplements Theorem If A is supplementary to B, and C is supplementary to B, then A C.

18 Practice Justify each of the following statements using a property of equality, property of congruence, definition, or postulate.

19 Angles Proofs Complete the proofs below by giving the missing statements and reasons. 1. Given: Prove: P S R Q 2. Given: Prove: Given: Prove: 1

20 B 4. Given: Prove: 1 3 A D 2 C 5. Given: Prove: Given: Prove: 1 2 3

21 2.7 Prove Angle Pair Relationships Theorem 2.3 Right Angles Congruence Theorem All right angles are congruent. Theorem 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. 1 2 If 1 and 2 are supplementary and 3 and 2 are supplementary, then Theorem 2.5 Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If 4 and 5 are complementary and 6 and 5 are complementary, then Theorem 2.6 Vertical Angles Congruence Theorem Vertical angles are congruent. Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are supplementary and 2 form a linear pair, so 1 and 2 are supplementary and m 1 + m 2 = Identify the pair(s) of congruent angles in the figure. Use a Theorem or Postulate to justify your answers. 1. B C 2. R Q S T A D P V U 3. 1 and 3 are complementary 1 and 2 are supplementary 4. 3 and 2 are supplementary 1 and 2 are complementary 2 and 3 are complementary 2 and 4 are supplementary

22 Use the diagram at the right. 5. If m 1 = 115, find m 2, m 3, and m 4 6. If m 2 = 64, find m 1, m 3, and m 4 7. If m 3 = 112, find m 1, m 2, and m If m 4 = 67, find m 1, m 2, and m 3 Find the value of the variable and the measure of each angle in the diagram x + 9 2(3y 25) 4y x y 13y 16y 27 5x + 18 In the diagram, 1 is a right angle and m 6 = 36. Complete the statement with <, > or =. 11. m 6 + m 7 m 4 + m m 6 + m 8 m 2 + m m 9 3(m 6) 14. m 2 + m 3 m