Ready to Go On? Skills Intervention 2-1 Using Inductive Reasoning to Make Conjectures Find these vocabulary words in Lesson 2-1 and the Multilingual Glossary. Vocabulary inductive reasoning conjecture counterexample Identifying a Pattern Find the next term in each pattern. A. 3, 6, 12, 24, B. Describe the pattern in the list. What number comes next in the pattern? Describe the pattern of the figures. Sketch the figure that will come next in the pattern. Making a Conjecture Complete the conjecture. The sum of two odd numbers is?. What is a conjecture? List some examples and look for a pattern. 1 3 3 5 5 7 A statement believed to be true based on inductive reasoning What kind of number is each sum, odd or even? The sum of two odd numbers is. Finding a Counterexample Show that the conjecture is false by finding a counterexample. If AB BC AC, then B is the midpoint of AC. What is a counterexample? What must be true for a point to be a midpoint? Sketch a figure that is a counterexample to the conjecture. 15 Holt Geometry
Ready to Go On? Problem Solving Intervention 2-1 Using Inductive Reasoning to Make Conjectures When you are given a table of data, look for a pattern to see if you can make a conjecture about the data. To treat a dog for epilepsy, a veterinarian gives the dog a dose of medication and monitors the level of medication in the dog s bloodstream every three hours. The monitoring results are given in the table. Make a conjecture about the rate at which the amount of medication in the dog s bloodstream is changing. Number of hours 0 3 6 9 Amount of medication in bloodstream (mg) 62 31 15.5 7.75 Understand the Problem 1. What data is being recorded by the veterinarian? 2. How many milligrams of medication was the dog given initially? 3. How often is the veterinarian monitoring the dog s bloodstream? Make a Plan 4. Is the amount of medication in the dog s bloodstream increasing or decreasing? 5. Describe the pattern you see in the data. 62 31 ; 31 15.5 ; 15.5 7.75 Solve 6. Complete the conjecture based on the patterns you observed in the data. The amount of medication in the dog s blood is decreasing at a rate of every. Lood Back 7. Prove your conjecture or find a counterexample to show that your conjecture is false. 62 1 ; 31 1 ; 15.5 1 2 2 2 16 Holt Geometry
Ready to Go On? Skills Intervention 2-2 Conditional Statements Find these vocabulary words in Lesson 2-2 and the Multilingual Glossary. Vocabulary conditional statement hypothesis conclusion truth value negation inverse converse contrapositive logically equivalent statements Writing a Conditional Statement Write a conditional statement: Two lines intersect in exactly one point. Identify the hypothesis. Identify the conclusion. Write the conditional. Analyzing the Truth Value of a Conditional Statement A. Determine if the conditional statement If a b, then 1 1 is true. If false, a b give a counterexample. Choose values for a and b where a b; for example a 3 and b 2. Substitute these values into the conclusion 1 1 a b. Is the conclusion true? Is the conditional statement true? B. Write the converse and inverse of the conditional statement, If a number is divisible by 3, then it is divisible by 9. Find the truth value of each. Identify the hypothesis. Identify the conclusion. What is the truth value of the statement? The converse of a conditional is formed by exchanging the hypothesis and. Write the converse: If a number is divisible by 9, then it is. Truth value? The inverse of a conditional is formed by negating the hypothesis and. Write the inverse: Truth value? 17 Holt Geometry
Ready to Go On? Skills Intervention 2-3 Using Deductive Reasoning to Verify Conjectures Find this vocabulary word in Lesson 2 3 and the Multilingual Glossary. Vocabulary deductive reasoning Verifying Conjectures by Using the Law of Detachment Determine if the conjecture is valid by the Law of Detachment. Given: If the side lengths of a rectangle are 10 m and 14 m, then the area of the rectangle is 140 m 2. Conjecture: The area of a rectangle is 140 m 2. The Law of Detachment states that if p q is a statement and, then. Identify the hypothesis of the conditional statement. Identify the conclusion of the conditional statement. Is the given conditional statement true? The statement The area of a rectangle is 140 m 2, matches the of the statement. Is the conjecture valid? Verifying Conjectures by Using the Law of Syllogism Determine if the conjecture is valid by the Law of Syllogism. Given: If C in ABC is a 90 angle, then it is a right triangle. If a triangle is a right triangle, then a 2 b 2 c 2. Conjecture: If C in ABC is a 90 angle, then a 2 b 2 c 2. The Law of Syllogism states that if p q and q r are, then is a true statement. Identify the following statements. p: q: r : What conjecture can you conclude using the Law of Syllogism? Is the conjecture valid? 18 Holt Geometry
Ready To Go On? Skills Intervention 2-4 Biconditional Statements and Definitions Find these vocabulary words in Lesson 2-4 and the Multilingual Glossary. Vocabulary biconditional statement definition polygon triangle quadrilateral Writing a Biconditional Statement For each conditional, write the converse and a biconditional statement. A. If an animal has four legs, then it is a dog. Write the converse of the statement. A biconditional statement is a statement that can be written in the form. Write a biconditional statement using the conditional statement and its converse. Is the biconditional true? B. If 5x 2 3x 10, then x 6. Write the converse of the statement. Write a biconditional statement using the conditional statement and its converse. Is the biconditional true? Analyzing the Truth Value of a Biconditional Statement Determine if the biconditional Two angles are complementary if and only if the sum of their angle measures is 90, is true. If false, give a counterexample. Write the biconditional statement as a conditional statement. Write the converse of the conditional statement. Is the conditional statement true? Is the converse true? Is the biconditional statement true? If not, give a counterexample. 19 Holt Geometry
Ready to Go On? Quiz 2-1 Using Inductive Reasoning to Make Conjectures Find the next term in each pattern. 1. 17, 13, 9, 5 2. Mon, Wed, Fri, 3. 2 7, 4 9, 6 11, 4.,,,... 5. A biologist recorded the following data about the weight of baby giraffes in a wildlife park. Use the table to make a conjecture about the average weight of a baby giraffe. 6. Show that the conjecture If a number is a prime number, then it is an odd number is false by finding a counterexample. ID Number Weight (pounds) FR0504F 146 FR0610M 155 FR0612F 147 FR0615F 152 FR0626M 150 2-2 Conditional Statements 7. Identify the hypothesis and conclusion of the conditional statement Two angles are supplementary angles if the sum of their measures is 180. Write a conditional statement from each of the following. 8. Sixteen-year-olds are eligible to drive. Reptile 9. Snake 10. The sides of a square are congruent. Determine if each conditional is true. If false, give a counterexample. 11. If an angle is obtuse, then it has a measure of 150. 12. If 5x 3 8x 15, then x 4.. 20 Holt Geometry
13. Write the converse, inverse, and contrapositive of the statement If a ray divides an angle into two congruent angles, then it is an angle bisector. Find the truth value of each. converse inverse contrapositive Truth value? Truth value? Truth value? 2-3 Using Deductive Reasoning to Verify Conjectures 14. Determine if the following conjecture is valid by the Law of Detachment. Given: If Ron finishes washing the dishes, he can go to the batting cage. Ron finishes washing the dishes. Conjecture: Ron goes to the batting cage. 15. Determine if the following conjecture is valid by the Law of Syllogism. Given: If two angles lie in the same plane and have a common vertex and a common side, but no common interior points, then they are adjacent angles. If two adjacent angles are a linear pair then their noncommon sides are opposite rays. Conjecture: If two angles lie in the same plane and have a common vertex and a common side and no common interior points, then their noncommon sides are opposite rays. Biconditional Statements and Definitions 16. For the conditional, If a point divides a segment into two congruent segments, then the point is the midpoint of the segment, write the converse and a biconditional statement. Converse Ready to Go On? Quiz continued Biconditional statement 17. Determine if the biconditional, A number is divisible by 6 if and only if it is divisible by 3 is true. If false, give a counterexample. 21 Holt Geometry
Ready to Go On? Enrichment Exploring Patterns Prime Numbers For centuries, mathematicians around the word have been trying to find a pattern in the set of prime numbers. 1. What is a prime number? 2. List the first ten prime numbers. 3. One popular conjecture about prime numbers is The rule 2 n 1 generates a list of prime numbers. Make a table of values for the conjecture where n is a whole number. n 1 2 3 4 5 6 7 8 9 10 2 n 1 1 4. Can you find a counterexample to the conjecture? If so, what is it? 5. Use the following conjecture to generate a list of primes: Substitute prime numbers into the formula 2p 1 to find new primes. Use this method to generate a list. 6. Can you find a counterexample to the conjecture? If so, what is it? Pascal s Triangle Blaise Pascal, a 17 th century mathematician, designed a triangle made up of numbers. Many number patterns are contained in the triangle. A portion of the triangle is shown below: 1 7. What numbers will be in the next row? 1 2 1 1 3 3 1 1 4 6 4 1 8. Make a conjecture about how the numbers in the rows are generated. 9. Describe any patterns you see along the diagonals of Pascal s Triangle. 22 Holt Geometry
2B Ready To Go On? Skills Intervention 2-5 Algebraic Proof Find this vocabulary word in Lesson 2-5 and the Multilingual Glossary. Solving an Equation in Algebra Solve the equation 13 5x 2. Write a justification for each step. Vocabulary proof What is the first step in solving this equation? 13 5x 2 What property allows you to do this? Simplify. The next step is to both sides by. 5x 5x What property allows you to do this? Simplify. What property allows you to reorder the solution? x x Identifying Properties of Equality and Congruence Identify the property that justifies each statement. A. AB AB Is this a statement of equality or congruence? What property justifies this statement? B. PQR XYZ, and XYZ JKL, so PQR JKL. Is this a statement of equality or congruence? What property justifies this statement? C. m WXY m CDE, so m CDE m WXY. Is this a statement of equality or congruence? What property justifies this statement? 23 Holt Geometry
2B Ready To Go On? Skills Intervention 2-6 Geometric Proof Find these vocabulary words in Lesson 2-6 and the Multilingual Glossary. Vocabulary theorem two-column proof Completing a Two-Column Proof Fill in the blanks to complete the two-column proof. Given: m 1 m 3 90, m 1 m 2 90 Prove: m 2 m 3 The reasons listed in a two column proof can be in the form of 2 1 3. It is given that m 1 m 3 90 and m 1 m 2 90, so you know that angles and angles are because of the definition of complementary angles. You can conclude that 2 and 3 are congruent because of the Theorem. Since 2 and 3 are, you know that m 2 m 3 because of the definition of. Use this information to complete the two-column proof. Statements Reasons 1. m 1 m 3 90, m 1 m 2 90 1. 2. 2. Def. of Complementary s. 3. 2 3 3. 4. m 2 m 3 4. Def. of 24 Holt Geometry
2B Ready to Go On? Skills Intervention 2-7 Flowchart and Paragraph Proofs Find these vocabulary words in Lesson 2-7 and the Multilingual Glossary. Vocabulary flowchart proof paragraph proof Writing a Flowchart Proof Use the given two-column proof to write a flowchart proof. Given: m 2 m 3 180, 3 4 Prove: m 1 m 4 4 3 1 2 Statements 1. m 2 m 3 180, 3 4 1. Given 2. 2 and 3 are supplementary 2. Def. of supp. s 3. 1 and 2 are supplementary 3. Lin. Pair Thm. 4. 1 3 4. Supps. Thm. 5. 1 4 5. Trans. Prop. Of 6. m 1 m 4 6. Def. of s Reasons A flowchart proof uses to show the structure of the proof. The steps in a flowchart move from to right or from to bottom, shown by the in each box. The justification for each step is written below the. supplementary Given Given 2 and 3 are supplementary Def. of supp. 1 3 Trans. Prop. Of 25 Holt Geometry
2B Ready to Go On? Quiz 2-5 Algebraic Proof Solve each equation. Write a justification for each step. 1. t 9 17 2. 4m 7 23 3. b 4 6 Identify the property that justifies each statement. 4. 5 6, so 6 5 5. m 1 m B and m B 53 so m 1 53 6. _ JK _ JK 7. x 5, so 5 x 2-6 Geometric Proof 8. Fill in the blanks to complete the two-column proof. Given: m 1 m 2 90, m 2 m 3 90 Prove: m 1 m 3 Proof: 1 2 3 Statements Reasons 1. 1. Given 2. 1 and 2 are complementary. 2 and 3 are complementary. 2. 3. 3. Comp. Thm. 4. 4. Def. of congruence 26 Holt Geometry
2B Ready to Go On? Quiz continued 9. Use the given plan to write a two-column proof. Given: PR bisects QPS. PS bisects RPT. Q Prove: m 1 m 3 Plan: By the definition of angle bisectors, 1 2 and 2 3. Use the Transitive Property of Congruence to show that 1 3. Use the definition of congruent angles to show that m 1 m 3. P 1 2 3 R T S Statements Reasons 1. 1. 2. 1 2 and 2 3 2. Def. of bisectors 3. 1 3 3. Trans. Prop. of Congruence 4. m 1 m 3 4. Def. s 2-7 Flowchart and Paragraph Proofs Use the given two-column proof to write the following. Given: 1 3 Prove: m 2 m 3 180 1 2 3 Statements Reasons 1. 1 3 1. Given 2. m 1 m 3 2. Def. of s 3. 1 and 2 are supplementary. 3. Lin. Pair Thm. 4. m 1 m 2 180 4. Def. of supp. s 5. m 3 m 2 180 5. Subs. Prop. Of Equality 10. a flowchart proof 11. a paragraph proof 27 Holt Geometry
2B Ready to Go On? Enrichment Proofs Write a two-column proof. 1. Given: 2 and 3 are supplementary; 3 4 Prove: 1 and 5 are supplementary. 3 1 2 4 5 6 Statements 1. 2 and 3 are supp., 3 4 1. Given 2. 2. Reasons 3. 1 3 3. 4. 1 4 4. 5. m 1 m 4 5. 6. m 4 m 5 180 6. 7. m 1 m 5 180 7. 8. 1 and 5 are supplementary. 8. 2. Given: m 1 m 2 m 3 180 Prove: m 1 m 2 m 4 1 2 3 4 Statements Reasons 1. m 1 m 2 m 3 180 1. 2. m 3 m 4 180 2. 3. m 1 m 2 m 3 3. 4. m 1 m 2 m 4 4. 28 Holt Geometry