Unit 2 Definitions and Proofs

Transcription

1 Vocabulary Unit 2 efinitions and Proofs Inductive reasoning- reasoning based on examples, experience, or patterns to show that that a rule or statement is true Conjecture a statement you believe to be true based on inductive reasoning. Inductive reasoning does not prove your conjecture to be true examples: (a)make a conjecture about what the next three numbers in the sequence are. 1,4,9,16,,,. 1,3,6,10,15,,, (b) Every day for the last three years we have had pizza for lunch. What is your conjecture? Counterexample- One example that shows that the conjecture or conclusion is false. examples:(a) If Joey lives in Columbia, then he lives in South Carolina. counterexample? (b) If x 2 = 36, x = 6 counterexample (c) If x 2 > 0, then x > 0 counterexample Conditional Statement: A statement written in if-then form. p q ex. If an animal says moo, then it is a cow. Sometimes you have to rewrite a sentence to get it into if-then form. ex. All insects have six legs. If then. hypothesis- the phrase following if in a conditional sentence. This is represented in math as the letter p conclusion- the phrase following then in a conditional sentence. This is represented in math as the letter q The negation of a statement p is not p the symbol is ~p ex-m is the midpoint of segment AB. negation - M is not the midpoint of AB Converse (of a conditional): q p Inverse (of a conditional): ~p ~q Contrapositive (of a conditional) : ~q ~p

2 PRACTICE: Write the converse, inverse and contrapositive of the statement ALL SQUARES ARE RECTANGLES. rectangles etermine whether each statement is true or false. If a statement is false, give a counterexample. First write the conditional in if-then form. p q Conditional: If a shape is a square, then it is a rectangle q p ~p ~q ~q ~p Converse: Inverse: Contrapositive: Try this one: If two angles are complementary, then the sum of their measures is 90. p q Conditional: q p Converse: ~p ~q Inverse: ~q ~p Contrapositive Now, this one: If you play ultimate frisbee, then you are in good shape. p q q p Conditional: Converse: ~p ~q Inverse: ~q ~p Contrapositive **Note: a conditional and its contrapositive are logically equivalent;** 2

3 To determine if a conclusion is valid (not true but valid) or not, there are two rules: The Law of etachment- this law essentially restates the hypothesis. Then the conclusion is restated. Example: Conditional: If a figure is a triangle, then the sum of the measures of its angles is 180. The figure in my notebook is ABC. Conclusion: The figure in my notebook, ABC, has the sum of the measures of its angles 180. If John studies, he will do well on his test. John studies. Conclusion valid Now examples where the hypothesis is not restated If John studies, he will do well on his test. John did not study. Conclusion Not valid If John studies, he will do well on his test. John did not do well on his test. (Note: this is the beginning of the contrapositive) Conclusion valid A conditional and its contrapositive are logically equivalent More examples: If an animal is a cow it has four legs. Elsie is a cow. If an animal is a cow it has four legs. Mazie has four legs If an animal is a cow it has four legs. Wonder Woman is not a cow If an animal is a cow it has four legs. onald does not have four legs. 3

4 LAW OF SYLLOGISM- This law is a chain, like the transitive property. If a=b and b=c, then a=c. p q If I make good grades then next year I can get my driver s license. q r If I have my driver's license, then I can save for a car. p r Conclusion Biconditional: How to write a biconditional: First the conditional and its converse must be true. Next omit the words if and then. Instead write where the then was, the phrase if and only if Some books use iff instead of if and only if. p q and q p are true so the biconditional is p q If two angles have a sum of their measures equal to 180 then they are supplementary. converse biconditional Identify the hypothesis and conclusion: A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides parallel. Given: RT SU Prove: RS: = TU THINK! Use the definition of congruent segments to write the given information in terms of lengths. Next use the Segment Addition Postulate to write RT in terms of RS + ST and US as ST + TU. Substitute those into the given information and use the Subtraction Property of Equality to eliminate ST and leave RS = TU. statements reasons 1. RT SU RS + ST= RT ST + TU=US

5 2.5 and 2.6 Algebraic and Geometric Proofs Each statement below has reference to one of the two diagrams given. Suppose each statement is TRUE. Name the reason for each true statement. efinitions must be written out 1. If 1 2, then XZ bisects < MXY. 2. m 1 + m 2 = m MXY 3. C = C 4. If AB = C, then AB + BC = C + BC 5. If AB = BC and BC = EF, then AB = EF 6. If is the midpoint of CE, then C E. 7. If m< A = m B, then m B = m A. 8. If B = CE and C = C, then B C = CE C. 9. AB + BE = AE 10. If XZ bisects MXY, then If m 1 = m 2 and m 2 = m 3, then m 1 = m E + EF = F 13. If AB + BC = AC and BC = EF, then AB + EF = AC If E is the midpoint of F, then line l bisects F. II. Complete the reasons in each algebraic proof. 1. Prove that if 2(x 3) = 8, then x = 7. Statements Reasons a) 2 (x 3) = 8 a) b) 2x 6 = 8 b) c) 2x = 14 c) d) x = 7 d) 5

6 1 2. Prove that if 3x 4 = x + 6, then x = 4. 2 Statements Reasons 1 a) 3x 4 = x + 6 a) 2 1 b) 2(3x 4) = 2( x + 6) b) 2 c) 6x 8 = x + 12 c) d) 5x 8 = 12 d) e) 5x = 20 e) f) x = 4 f) 3. You can use the Angle Addition Postulate Given: m AOC = 139 Prove: x=23 Statement Property 1) m AOB + m BOC = m AOC 1) 2) x + 2x + 10 = 139 2) 3) 3x + 10 = 139 3) 4) 3x = 129 4) 5) x = 23 5) 4. You can use the definition of an angle bisector. Given: ray LM bisects KLN prove: x=20 Statement Property 1) raylm bisects KLN 1) 2) KLM MLN 2) 3) m KLM = m MLN 3) 4) 2x + 40 = 4x 4) 5) 40 = 2x 5) 6) 20 = x 6) 7) x=20 7) 6

7 5. You can use the Segment Addition Postulate. 2y 3y-9 given: AC = 21 prove: y=6 A B C CStatement Property 1) AB + BC = AC 1) 2) 2y + 3y 9 = 21 2) 3) 5y 9 = 21 3) 4) 5y = 30 4) 5) y = 6 5) Practice with Proofs: Fill in the given statements or reasons A) Given: XY = YZ Prove: Y is the midpoint of Statements Reasons 1) XY = YZ 1) 2) 2) 3) Y is the midpoint of 3) B) Given: B is the midpoint of is the midpoint of Prove: AB + C = BC + E STATEMENTS REASONS 1. B is the midpoint of is the midpoint of AB = BC C = E AB + C = BC + E 4. 7

8 C) P Q R S T Given: PQ = RS STATEMENTS Prove: PQ + QR = QS REASONS 1. PQ = RS QR + RS = QS QR + PQ = QS 3. ) Given: l bisects Prove: AB + E = BC + EF l bisects E is the midpoint of 2 B is the midpoint of AB = BC 4. E = EF 5. AB + E = BC + EF 5. E) Given: Prove: RS = TU Statements R S T U Reasons RT = SU RT = RS + ST 3. SU = ST + TU 4. RS + ST = ST + TU ST = ST RS = TU 6. 8

9 F) Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1 3 G) Given: 1 and 2 are supplementary, and same figure as F 1 3 Prove: 3 and 2 are supplementary. Statements Reasons H) Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1 3 Statements Reasons 9

10 I) Given: HKJ is a straight angle. ray KI bisects HKJ. Prove: IKJ is a right angle. Statements Reasons 1. a. 1. Given 2. m HKJ = b. 3. c. 3. Given 4. IKJ IKH m IKJ = m IKH d. 6. Add. Post. 7. m IKJ +m IKJ = e 8. 2m IKJ = f 9..m IKJ = g 10. IKJ is a right angle. 10.h 10