Geometry. Unit 2- Reasoning and Proof. Name:

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1 Geometry Unit 2- Reasoning and Proof Name: 1

2 Geometry Chapter 2 Reasoning and Proof ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (2-1) Inductive Reasoning and Conjecture Pg 63 # 2, 4-9, 11-18, 26, 27, 30, 43, (2-2) Logic Day 1 Page 72 # 4-9, odd, (2-2) Logic Day 2 Page 72 # 30-31, (2-3) Conditional Statements Day 1 Page # (2-3) Conditional Statements Day 2 Page 79 # 40, 41, 43, 45, (2-4) Deductive Reasoning Page 85 # (2-5) Postulates and Paragraph Proofs Day 1 Page 92 # (2-5) Postulates and Paragraph Proofs Day 2 Section 2-5 Practice WS and Reading to Learn WS 9. (2-6) Algebraic Proof Day 1 Page 97 # 2 10, (2-6) Algebraic Proof Day 2 Page 97 # 14 19, 30, 31, 37, (2-7) Proving Segment Relationships Day 1 Page 104 # (2-7) Proving Segment Relationships Day 2 Section 2-7 Practice WS and Reading to Learn WS 13. (2-8) Proving Angle Relationships Day 1 Page 112 # 16 24, 38, (2-8) Proving Angle Relationships Day 2 Section 2-8 Practice WS and Reading to Learn WS 15. Chapter 2 Review WS 2

3 Section 2 1: Inductive Reasoning and Conjecture Notes Part 1 Conjecture: an guess based on known information Date: Inductive Reasoning: reasoning that uses a number of to arrive at a plausible generalization or prediction Counterexample: an example used to show that a given statement is not true Examples: Make a conjecture about the next item in each sequence. a.) 10, 20, 30,... b.) c.) 5, 10, 20,... d.) 1, 1, 2, 3, 5, 8, 13,... e.) M, T, W, T,... f.) A, M, J, J, A,... Examples: Make a conjecture based on the given information. Draw a figure to illustrate your conjecture if possible. a.) The sky becomes dark and the wind picks up. b.) Lines l and m are perpendicular. c.) 3 and 4 are a linear pair. 3

4 d.) uuur BD is an angle bisector of ABC. e.) Point S is between R and T. Examples: Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. a.) Given: I have $1 worth of change in my pocket. Conjecture: I have four quarters in my pocket. b.) Given: 1 and 2 are complementary angles. Conjecture: 1 and 2 form a right angle. c.) Given: DE = EF Conjecture: E is the midpoint of DF d.) Given: points W, X, Y, and Z Conjecture: W, X, Y, and Z are noncollinear e.) Given: noncollinear points R, S, and T Conjecture: RS, ST, and RT form a triangle 4

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6 Section 2 1: Inductive Reasoning and Conjecture Notes Part B Date: Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. 1.) A and B are supplementary. 2.) X, Y, and Z are collinear and XY = YZ. 3.) AB bisects CD at K. 4.) Point P is the midpoint of NQ. 5.) HIJK is a square. Determine whether each conjecture is true or false. conjecture. Give a counterexample for any false 6.) Given: 1 and 2 are supplementary. Conjecture: 1 and 2 form a linear pair. 6

7 7.) Given: AB + BC = AC Conjecture: AB = BC 8.) Given: 1 and 2 are adjacent angles. Conjecture: 1 and 2 form a linear pair. 9.) Given: S, T, and U are collinear and ST = TU. Conjecture: T is the midpoint of SU. 10.) Given: GH and JK form a right angle and intersect at P. Conjecture: GH JK 11.) Given: AB, BC, and AC are congruent. Conjecture: A, B, and C are collinear. 7

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9 Date: Section 2 2: Logic Notes Part A Statement: any that is either true or false, but not Truth Value: the or of a statement Negation: the negation of a statement has the meaning as well as an opposite value If a statement is represented by p, then not p is the of the statement. Symbols: Compound Statement: a statement formed by joining or more statements Conjunction: a statement formed by joining two or more statements with the word Symbols: Ex: p: Harrisburg is a city in Pennsylvania. q: Harrisburg is the capital of Pennsylvania. p and q: Disjunction: a compound statement formed by joining or more statements with the word Symbols: Ex: p: Miquel studies Geometry. q: Miquel studies English. p and q: 9

10 Example #1: Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: January 1 is the first day of the year. q: = -6 r: A triangle had three sides. a.) p and q b.) q r c.) r ~ p Example #2: Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: A right angle measures 90. q: A pentagon has 6 sides. R: A linear pair is complementary. a.) ~ q p b.) (p q) r c.) q (p or r) 10

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12 Date: Section 2 2: Logic Notes Part B Truth Tables Example #1: p Negation of p and q ~ p q ~ q Example #2: Conjunction Example #3: Disjunction Example #4: p ~ q 12

13 Example #5: ~ p ~ q Example #6: (p q) r 13

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15 Section 2 3: Conditional Statements Notes Part A Conditional Statement: a can be written in if-then form Sign up for a six-month fitness plan and get six months free! Get $1500 cash back when you buy a new car. Free phone with every one-year service enrollment. If Then Statement: An if-then statement is written in the form if p, then q. The phrase immediately following the word is called the hypothesis, and the phrase immediately following the word is called the conclusion. Symbol: Example# 1: Identify the hypothesis and conclusion of each statement. a.) If a polygon has 6 sides, then it is a hexagon. Hypothesis: Conclusion: b.) The Wolverines will advance to the play-offs if they win the next 2 games. Hypothesis: Conclusion: 15

16 Example #2: Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a.) An angle with a measure greater than 90 is an obtuse angle. Hypothesis: Conclusion: If-Then Form: b.) A five-sided polygon is a pentagon. Hypothesis: Conclusion: If-Then Form: Example #3: Determine the truth value of the following statement for each conditional statement. If Daryl rests for 10 days, his shoulder will heal. a.) Daryl rests for 10 days, and he still has a hurt shoulder. b.) Daryl rests for 3 days, and he still has a hurt shoulder. c.) Daryl rests for 10 days, and he does not have a hurt shoulder anymore. d.) Daryl rests for 7 days, and he does not have a hurt shoulder anymore. 16

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18 Section 2 3: Conditional Statements Notes Part B Date: Key Concept (Related Conditionals): Conditional Statement: o If p, then q. Symbol: p q 1.) If a polygon has 7 sides, then it is a heptagon. 2.) If you live in Pittsburgh, then you live in Pennsylvania. 3.) All squares are rectangles. Converse Statement: o If q, then p. Symbol: q p 1.) 2.) 3.) Inverse Statement: o If not p, then not q. Symbol: ~ p ~ q 1.) 2.) 3.) Contrapositive Statement: o If not q, then not p. Symbol: ~ q ~ p 1.) 2.) 3.) 18

19 Related Conditionals: Example# 1: Write the converse, inverse, and contrapositive of the following statement, and then determine whether each statement is true or false. If a statement is false, give a counterexample. Linear pairs of angles are supplementary. Conditional: If two angles form a linear pair, then they are supplementary. (True) Converse: Inverse: Contrapositive: Example# 2: Write the converse, inverse, and contrapositive of the following statement, and then determine whether each statement is true or false. If a statement is false, give a counterexample. An angle formed by perpendicular lines is a right angle. Conditional: If an angle is formed by perpendicular lines, then it is a right anle. (True) Converse: Inverse: Contrapositive: 19

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21 Section 2 4: Deductive Reasoning Notes Part A Law of Detachment Date: Deductive Reasoning: uses,, definitions, or properties to reach a conclusion Key Concept (Law of Detachment): If is true and is true, then is also true. Symbol: Example #1: Determine whether each conclusion is valid based on the true conditional given. If not, write invalid. Explain your reasoning. If a ray is an angle bisector, then it divides the angle into two congruent angles. a.) Given: uuur BD bisects ABC Conclusion: ABD CBD b.) Given: PQT RQS uuur uuur Conclusion: QS and QT are angle bisectors. 21

22 Example #2: Determine whether each conclusion is valid based on the true conditional given. If not, write invalid. Explain your reasoning. If two angles are complementary to the same angle, then the angles are congruent. a.) Given: A and C are complements of B Conclusion: A is congruent to C b.) Given: A C Conclusion: A and C are complements of B c.) Given: E and F are complementary to G Conclusion: E and F are vertical angles Example #3: Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment. If it does not, write invalid. (1) If the measure of an angle is greater than 90, then it is obtuse. (2) m ABC > 90 (3) ABC is obtuse (1) Vertical angles are congruent. (2) 3 4 (3) 3 and 4 are vertical angles 22

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24 Section 2 4: Deductive Reasoning Notes Part B Law of Syllogism Date: Key Concept (Law of Syllogism): If and are true, then is also true. Example #1: Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. If so, write a valid conclusion. a.) (1) If the symbol of a substance is Pb, then it is lead. (2) The atomic number of lead is 82. b.) (1) Water can be represented by H 2 O. (2) Hydrogen (H) and oxygen (O) are in the atmosphere. c.) (1) If you like pizza with everything, then you ll like Vincent s Pizza. (2) If you like Vincent s Pizza, then you are a pizza connoisseur. Example #2: Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. If a valid conclusion is possible, write it. If not, write no conclusion. a.) If two lines intersect to form a right angle, then they are perpendicular. Lines l and m are perpendicular. 24

25 b.) If the measure of an angle is less than 90, then it is acute. If an angle is acute, then it is not obtuse. Example #3: Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not follow, write invalid and explain! a.) (1) If a student attends WHHS, then the student has an ID number. (2) Jonathan attends Woodland Hills High School. (3) Jonathan has a student ID number. b.) (1) If a rectangle has four congruent sides, then it is a square. (2) A square has diagonals that are perpendicular. (3) A rectangle has diagonals that are perpendicular. c.) (1) If Caitlin arrives at school at 6:30 am, she will get help in math. (2) If Caitlin gets help in math, then she will pass her math test. (3) If Caitlin arrives at school at 6:30 am, then she will pass her math test. 25

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27 Date: Section 2 5: Postulates and Paragraph Proofs Notes Part A Postulate: a statement that describes a fundamental between the basic terms of Postulates are accepted as. Postulates:

28 Example #1: Determine whether the following statements are always, sometimes, or never true. Explain. 1.) Three points determine a plane. 2.) Points G and H are in plane X. Any point collinear with G and H is in plane X. 3.) The intersection of two planes can be a point. 4.) Points S, T, and U determine three lines. 5.) Points A and B lie in at least one plane. 6.) If line l lies in plane P and line m lies in plane Q, then lines l and m lie in plane R. Example #2: In the figure, AC and BD lie in plane J, and BY and CX line in plane K. State the postulate that can be used to show each statement is true. 1.) C and D are collinear. 2.) XB lies in plane K. 3.) Points A, C, and X are coplanar. 4.) AD lies in plane J. 5.) X and Y are collinear. 6.) Points Y, D, and C are coplanar. 28

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30 Date: Section 2 6: Algebraic Proof Notes Part A Properties of Equality For Real Numbers Reflexive Property For every number a, a = a. Symmetric Property For all numbers a and b, if a = b, then b = a. Transitive Property For all numbers a, b, and c, if a = b and b = c, then a = c. Addition and Subtraction Property For all numbers a, b, and c, if a = b, then a + b = b + c and a b = b c. Multiplication and Division Property For all numbers a, b, and c, if a = b, then a c = b c and if c 0, a b =. c c Substitution Property For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression. Distributive Property For all numbers a, b, and c, a(b + c) = ab + ac. Example #1: State the property that justifies each statement. a.) If 2x = 5, then x = 2 5. b.) If 2 x = 7, then x = 14. c.) If x = 5 and b = 5, then x = b. d.) If XY AB = WZ AB, then XY = WZ. e.) If EF = GH and GH = JK, then EF = JK. 30

31 Example #2: Solve 3(x 2) = 42 showing every step! Be sure to number each step! Example #3: Write a two-column proof for the following. 2 Given: 5 x = 1 3 Prove: x = 6 Example #4: Write a two-column proof for the following. Given: 2(5 3a) 4(a + 7) = 92 Prove: a =

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33 NAME PERIOD Solve each problem show ALL work following the example to tell what you did in each step. EXAMPLE: Given: 3(2x+1) = 4x + 5 Prove: x = 4 Statements Reasons 1. 3(2x+1) = 4x given 2. 6x + 3 = 4x distributed 3. 6x x = 4x + 5 4x 3. subtracted 4. 2x 3 = 5 4. substitute (combine like terms) 5. 2x = addition 6. 2x = 8 6. substitute (combine like terms) 7. 2x / 2 = 8 / 2 7. division (of 2) 8. x = 4 8. substitution SOLVE each problem for the variable using the given information and the format from the example above. Given: 4x + 5 = 17 Prove: x = 3 Statements Reasons 1. 4x + 5 = Given Given: 4x + 8 = x + 2 Prove: x = -2 Statements Reasons 1. 4x + 8 = x given 33

34 4x + 6 Given: = 9 2 Prove: x = 3 Statements 4x = given Reasons Given: 3x 5 = 2x Prove: x = 5 Statements 1. 3x 5 = 2x 1. given Reasons 5x 1 Given: = 3 8 Prove: x = 5 Statements 5x 1 1. = given Reasons Given: 3x = 5 + 2(x-3) Prove: x = -1 Statements Reasons 1. 3x = 5 + 2(x-3) 1. given 34

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