Name: Geometry. Chapter 2 Reasoning and Proof

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1 Name: 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, (2-1) Inductive Reasoning and Conjecture 4. (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 # (2-6) Algebraic Proof Day 1 Page 97 # 2 10, (2-6) Algebraic Proof Day 2 Page 97 # 14 19, 30, 31, 37, Chapter 2 Review WS 1

2 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. 2

3 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 3

4 Section 2.1 Day 2 - Finding the n th term Warm Up: Complete the following tables. Find the differences between consecutive values. a. n n b. c. Difference: n n Difference: n n Difference: Example 1: Consider the sequence 20, 27, 34, 41, 48, 55 What are the next 3 terms? What is the common difference? We need to find a rule to get the n th term of the pattern. Since the common difference is, part of the rule is. To find the rest of the rule, substitute n = 1 into the rule and fix the rule so that your answer is the 1 st term in the sequence. Check your rule with the rest of the sequence. Example 2: Find the rule for the sequence 7, 2, -3, -8, -13, -18 4

5 Exit slip: Complete the following tables: 1. n n 20 f(n) n n 20 f(n) n n 20 f(n)

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: BC AB,, and AC are congruent. Conjecture: A, B, and C are collinear. 7

8 CRITICAL THINKING The expression n 2 n + 41 has a prime value for n = 1, n = 2, and n = 3. Based on this pattern, you might conjecture that this expression always generates a prime number for any positive integral value of n. Try different values of n to test the conjecture. Answer true if you think the conjecture is always true. Answer false and give a counterexample if you think the conjecture is false. Show all work! 8

9 Section 2 3: Conditional Statements Notes Part A Date: 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: 9

10 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. 10

11 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.) 11

12 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: 12

13 CRITICAL THINKING Write a false conditional statement. Is it possible to insert the word not into your conditional to make it true? If so, write the true conditional. 13

14 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. 14

15 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 15

16 Section 2 4: Deductive Reasoning 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. 16

17 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. 17

18 CRITICAL THINKING An advertisement states that If you like to ski, then you ll love Snow Mountain Resort. Stacey likes to ski, but when she went to Snow Mountain Resort, she did not like it very much. If you know that Stacey saw the ad, explain how her reasoning was flawed. 18

19 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. 19

20 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 =

21 CRITICAL THINKING Use the following information. Below is a family tree of the Gibbs family. Clara, Carol, Cynthia, and Cheryl are all daughters of Lucy. Because they are sisters, they have a transitive and symmetric relationship. That is, Clara is a sister of Carol, Carol is a sister of Cynthia an, so Clara is a sister of Cynthia. 1. What other relationships in a family have reflexive, symmetric, or transitive relationships? Explain why. Remember that the child or children of each person are listed beneath that person s name. Consider relationships such as first cousin, ancestor or descendent, aunt or uncle, sibling, or any other relationship. 2. Construct your family tree on one or both sides of your family and identify the reflexive, symmetric, or transitive relationships. 21

22 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 22

23 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 23

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