Formal Geometry Chapter 2 Logic and Proofs Day 1 Inductive Reasoning and Conjectures Objectives: SWBAT form a conjecture, and check it SWBAT use counterexamples to disprove a conjecture Logic the use of valid reasoning to justify a claim Inductive Finding patterns, experiments in information to justify a claim. Deductive Using laws, facts, or previously proven truths to justify a claim. Conjecture Something you think is true; an observation can be true or false Example: Make a conjecture for the following. 1. 3, 5, 9, 15, 23.. add the next even number in the sequence 2. 3+5=8, 5+1=6, 11+3=14, 7+13=20 _add two odds #s to get an even #_ 3. Add a side For a conjecture to be truth, it has to be truth in ALL Cases! Although this is impossible, we do our best and use deductive reasoning to help.
Counterexample One instance when a conjecture is false Find a counter example to the following conjectures. 1. When you add a positive number and a negative number, the answer is negative. 8 12 4 2. All bears are brown or black. 10,000 100 9,900 Polar Bears or Gummi Bears 3. All jokes are hilariously funny. What happens when you hurt your toe? You call a tow truck =D Write a conjecture that describes the pattern in each sequence. Then make a conjecture for the next term. Add $ 2. 25 $ 11. 25 Sha de t o the Left Add 45min 12 : 30 Add another circle
Day 2 Conditional Objectives: SWBAT Recognize and analyze a conditional statement Conditional Statement An if then statement If-then form must be put into this form, even if it isn t to start Hypothesis the first clause (without the if ) Conclusion the second clause (without the then ) Rewrite the conditional statements in if-then form. 1. Two points are collinear if they lie on the same line. If two points are collinear, then they lie on the same line. 2. An obtuse angle is an angle that measures more than 90 and less than 180 If an angle is obtuse, then it measures more than 90 and less than 180 3. Cheese contains Calcium. If it is cheese, then it contains calcium
Identify the hypothesis and conclusion of each conditional statement. 1. If today is Friday, then tomorrow is Saturday. Hypo Conclusion 2. If two angles are supplementary, then they add up to 180 degrees. Hypo Conclusion Inverse The negation of the Conditional statement. Think adding words like not, or isn t. Converse Switching the order of the Conditional Statement. The If and Then stay put but the Hypo and Conclusion switch. Contrapositive The negation of the Converse Bi-conditional If and only if must be true backwards and forwards
Write the following statements in if-then form. Then write the converse, contrapositive, inverse, and biconditional statements. Inverse: If there is fresh snow on the mountains, then it is a great day for skiing. If there is not fresh snow on the mountains, then it is not a great day for skiing. Converse: If it is a great day for skiing, then there is fresh snow on the mountains. Contrapositive: If it is not a great day for skiing, then there is not fresh snow on the mountains. Bi-conditional: There is fresh now on the mountains if and only if it is a great day for skiing. Write the converse, contrapositive, inverse, and biconditional statements. If today is Saturday, then we do not have school today. Converse: If we do not have school today, then today is Saturday. Inverse: If today is not Saturday, then we do have school today. Contrapositive: If we do have school today, then today is not Saturday. Bi-conditional: Today is Saturday if and only if we do not have school today.
Fun Facts Conditional and Contrapositives If the Conditional statement is true, then the Contrapositive is also true. Converses and Inverses If the converse is true, then the inverse is also true.
Day 3 Deductive Reasoning and Laws of Logic Logic Objectives: SWBAT form a conjecture, and check it SWBAT use counterexamples to disprove a conjecture Inductive Finding patterns, experiments in information to justify a claim. Deductive Using laws, facts, or previously proven truths to justify a claim. Laws of Logic The Law of Detachment If you have a true compound statement p q. If you then prove p to be true, then you can conclude that q is also true. If... p q is true p is true then... q is true The Law of Contrapositive If you have a true compound statement p q. If you then prove ~q to be true, then you can conclude that ~p is also true. The Law of Syllogism Train Property If... p q and... q r then... p r If... p q is true ~ q is true then... ~ p is true
Determine whether each conclusion is based on inductive or deductive reasoning. 1. Students at Reno High School must have a B average in order to participate in sports. Hank has a B average, so he concludes that he can play sports at Reno High. Deductive Reasoning 2. Holly notices that every Saturday, her neighbor mows his lawn. Today is Saturday. Holly concludes that her neighbor will mow his lawn. Inductive Reasoning Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. 3. Given: If a number is divisible by 4, when the number is also divisible by 2. Claim: 12 is divisible by four. Conclusion: 12 is divisible by two. True Statement Law of Detachment 4. If Edward stays up late, he will be tired the next day Claim: Edward is tired. Conclusion: Edward stayed up late. False Statement he could be tired for lots of reasons. 5. If you study, then you are prepared for the Celebration of Knowledge. If you are prepared for the Celebration of Knowledge, you won t panic. If you won t panic, then you will get a good grade. Claim: Stanley studied for the Celebration Conclusion: Stanley will get a good grade. True Statement Law of Syllogism
Two Points Make a Line Postulate Through any two points is exactly one line. Three Non-Collinear Points Postulate Any 3 noncollinear points will make exactly one plane. Lines on Planes Postulate If two points lie on a plane, then the entire line containing those points lies in that plane. Intersecting Lines Postulate If two lines intersect, then their intersection is exactly one point. Intersecting Planes Postulate If two planes intersect, then their intersection is a line. Inter sec ting Planes Postulate 3Non Collinear Point s Postulate Sometimes Could be a po int or line 2 Always Point s Make A line Postulate
Always 2 Point s Make A line Postulate
Day 4 Algebraic Proofs Addition Property of Equality Objectives: SWBAT form a proof using Algebra If a = b, then a + c = b + c Subtraction Property of Equality if a = b, then a c = b c Multiplication Property of Equality if a = b, then a x c = b x c Division Property of Equality if a = b, then a c = b c Distributive Property ~ a(b + c) = ab + ac Math Fact / Simplifying Like Terms Combining like terms on the same side of an equals sign Proof Basics: Given: Information given that does not need to be proved true Prove: Information that is needed to be proven true the final goal of a proof
Examples 1. Given: 10y + 5 = 25 Prove: y = 2 10y 5 25 Given 10y 20 y 2 Subtraction Property of Equality Division Property of Equality Given: 6x + 3 = 9(x 1) Prove: x = 4 6x 3 9 x 1 6x 3 9x 9 3 3x 9 12 3x Given Distributive Property Subtraction Property of Equality Addition Property of Equality x 4 Division Property of Equality Given: 6x + 7 = 8x 5 Prove: x = 6 6x 7 8x 5 7 2x 5 12 2x Given Subtraction Property of Equality Addition Property of Equality x 6 Division Property of Equality
Given: 1 m + 3 = 2m 24 5 Prove: m = 15 1 m 3 2m 24 5 m 15 10m 120 15 9m 120 145 9m 15 m m 15 Given Multiplication Property of Equality Subtraction Property of Equality Addition Property of Equality Symmetric Property Division Property of Equality 5. Given: 2m = n + 5 m = n 1 Prove: n = 7 2m = n + 5 m = n 1 2(n 1) = n + 5 2n 2 = n + 5 2n = n + 7 n = 7 Given Given Substitution Distributive Property Addition Property of Equality Subtraction Property of Equality
Day 5 - Introduction to Proofs What s a proof? Objectives: SWBAT form a proofs What makes up a proof? What can be used for the reasons? Can Be Assumed Coplanar Points Collinear Points Betweenness of Points Intersection Points Interior / Exterior of Angles Straight Lines (Linear pairs) Vertical Angles Cannot Be Assumed Perpendicular Lines Complementary Angles Congruent Angles Congruent Segments Examples: 1. Given: B is the midpoint of AC. A B C Prove: AB BC.
2. Given: BD bisects ABC Prove: ABD DBC A D B C 3. Given: m 1= 55. Prove: m 3 55 1 2 4 3 Reflexive Property Symmetric Property Substitution Property Transitive Property
Day 7 Segment Proofs Part 1 Objectives: SWBAT form a proofs involve Segments and Segment Addition Key Postulates / Theorems to Remember Definition of Congruent Segments Segment Bisector Definition of Midpoint The first Statement and Reason for a proof is. The last statement of a proof is always. When do Segments get hats?. Segment Proofs: fill in the blanks of the following segment proofs. 1. Given: Y is the midpoint of XZ. Prove: XY YZ 2) 2) XY YZ 2. Given: M is the midpoint of LP. Prove: LM = MP 2) 2) DE EF 3) 3)
3. Given: AB bisects DF Prove: DE = EF 2) 2) 3) DE = EF 3) Transitive Property Reflexive Property 4. Given: HJ KL, KL MN Prove: HJ MN 2) 2) Given 3) HJ MN 3) 4. Given: XY WZ, WZ ZY, ZY 11 Prove: XY 11 2) 2) Given 3) 3) Given 4) 4) 5) 5) 6) XY 11 6)
6. Given : AB DE B is the midpoint of AC E is the midpoint of DF Prove : BC EF 2) 2) Given 3) 3) Given 4) AB BC 4) 5) 5) Definition of Midpoint 6) BC EF 6)
Day 8 Segment Proofs Part 2 Objectives: SWBAT form a proofs involve Segments and Segment Addition Segment Addition Postulate Definition of Congruent Segments Transitive Property Reflexive Property Substitution Property Definition of Congruent Segments 1. Given: Diagram Prove: EF FG EG 2) EF FG EG 2) 2. Given: TU = 14, UV = 21 Prove: TV = 35 2) 2) Given 3) 3) 4) 4) 5) 5)
3. Given: AB = x, BC = y Prove: AC = x + y 2) 2) Given 3) 3) 4) 4) Segment Addition Proof Pattern
4. Given:, MP EG, NP EF Prove: MN FG 2) 2) Given 3) MP EG 3) 4) NP EF 4) 5) 5) Segment Addition Postulate 6) 6) Segment Addition Postulate 7) MN NP EF FG 7) 8) MN FG 8) 9) MN FG 9) 5. Given: XY BC, YZ AB Prove: XZ AC 2) 2) Given 3) XY AB 3) 4) YZ AB 4) 5) 5) Segment Addition Postulate 6) 6) Segment Addition Postulate 7) XY YZ AB BC 7) 8) XZ AC 8) 9) XZ AC 9)
6. Given: HJ KL Prove: HK JL 2) HJ KL 2) 3) 3) Segment Addition Postulate 4) HJ JK HK 4) 5) JK JK 5) 6) 6) Definition of Congruent Segments 7) JK HJ JK KL 7) 8) 8) Substitution 9) 9) 10) HK JL 10) 7. Given: AB BC, BC CD Prove: AC BD 2) 2) Given 3) AB BC 3) 4) BC CD 4) 5) AC AB BC 5) 6) BC BC CD 6) 7) 7) Reflexive Property 8) 8) Definition of Congruent Segments 9) 9) Addition Prop of = 10) 10) Substitution 11) AC BD 11)
Day 9 Angle Proofs Part 1 Objectives: SWBAT form a proofs involve Angle Addition Angle Addition Postulate Definition of Congruent Segments Transitive Property Reflexive Property Substitution Property Definition of Congruent Angles 1) Given: m RST = 50 m TSV = 40 Prove: m < RSV = 90 2) 2) Given 3) 3) Angle Addition Postulate 4) m < RSV = 40 + 50 4) 5) m < RSV = 90 5) 2) Given: m 1 = 20 m 2 = 40 m 3 = 30 Prove: m < XYZ = 90 2) m 2 = 40 2) 3) 3) Given 4) 4) Angle Addition Postulate 5) m < XYZ = 20 + 40 + 30 5) 6) m < XYZ = 90 6)
3) Given: HGJ KGL Prove: HGK JGL 2) 2) Definition of Congruent Angles 3) 3) Angle Addition Postulate 4) 4) Angle Addition Postulate 5) 5) Reflexive Property 6) 6) Definition of Congruent Angles 7) 7) Addition Property of Equality 8) 8) Substitution 9) 9) Definition of Congruent Angles 4) Given: AEC BED Prove: AEB CED 2) 2) 3) 3) Angle Addition Postulate 4) 4) 5) 5) Substitution 6) 6) Subtraction Property of Equality 7) 7)
5) Given: 1 4 2 3 Prove: ABC DEF 1) 1 4 1) 2) 2) 3) m 1 = m 4 3) 4) 4) 5) m 1 + m 2 = m ABC 5) 6) 6) 7) m 1 + m 2 = m DEF 7) 8) 8) 9) ABC DEF 9)
Day 10 Angle Proofs Part 2 Objectives: SWBAT form a proofs involve Angles and Angle Addition Angle Bisector Definition of Complementary Angles Linear Pair Definition of Supplementary Angles Vertical Angles Right Angle Theorem Angle Proofs: fill in the blanks of the following angle proofs. 1) Given: m CDE = 110 m FGH = 110 Prove: CDE FGH 2) 2) Given 3) 3) Substitution / Transitive Property 4) CDE FGH 4) 2) Given: m RST = 50 m TSV = 40 Prove: < RST and < TSV are Complementary 1) m RST = 50 1) 2) 2) Given 3) 3) Angle Addition Postulate 4) m < RSV = 40 + 50 4) 5) m < RSV = 90 5) 6) < RST and < TSV are Complementary 6)
3) Given: m CDE = 110 m FGH = 70 Prove: < CDE and < FGH are Supplementary 1) m CDE = 110 1) 2) m FGH = 70 2) 3) 110 + 70 = 180 3) Math Fact 4) m CDE + m FGH = 180 4) 5) < CDE and < FGH are Supplementary 5) 4) Given: WY bisects < XWZ Prove: m 1 = m 3 2) m 1 = m 2 2) 3) 3) Vertical Angles Theorem 4) m 1 = m 3 4) 5) 5) Substitution / Transitive Property 5. Given: 1 2, 1 4 Prove: FH bisects < EFG 1) 1) 2 2) 3) 3) 4) 4)
6. Given: A D A and B are supp. C and D are supp. Prove: B C 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 7. Given: V YRX, Y TRV Prove: V Y 1) 1) 2) 2) 3) 3) 4) 4)