Geometry 2.pdf

2 Reasoning and Proofs 2.1 Conditional Statements 2.2 Inductive and Deductive Reasoning 2.3 Postulates and Diagrams 2.4 Algebraic Reasoning 2.5 Proving Statemen...

2 Reasoning and Proofs 2.1 Conditional Statements 2.2 Inductive and Deductive Reasoning 2.3 Postulates and Diagrams 2.4 Algebraic Reasoning 2.5 Proving Statements about Segments and Angles 2.6 Proving Geometric Relationships Airport Runway (p. 108) Sculpture S l t ((p. 104) SEE the Big Idea Cit Streett ((p. 95) City St Tiger (p (p. 81) G i Guitar ((p. 6 67)) hs_geo_pe_02co.indd 62 1/19/15 8:58 AM  Maintaining Mathematical Proficiency Finding the nth Term of an Arithmetic Sequence Example 1 Write an equation for the nth term of the arithmetic sequence 2, 5, 8, 11,.... Then find a20. The first term is 2, and the common difference is 3. an = a1 + (n − 1)d Equation for an arithmetic sequence an = 2 + (n − 1)3 Substitute 2 for a1 and 3 for d. an = 3n − 1 Simplify. Use the equation to find the 20th term. an = 3n − 1 Write the equation. a20 = 3(20) − 1 Substitute 20 for n. = 59 Simplify. The 20th term of the arithmetic sequence is 59. Write an equation for the nth term of the arithmetic sequence. Then find a50. 1. 3, 9, 15, 21,... 2. −29, −12, 5, 22,... 3. 2.8, 3.4, 4.0, 4.6,... 1 1 2 5 4. —, —, —, —,... 5. 26, 22, 18, 14,... 6. 8, 2, −4, −10,... 3 2 3 6 Rewriting Literal Equations Example 2 Solve the literal equation 3x + 6y = 24 for y. 3x + 6y = 24 Write the equation. 3x − 3x + 6y = 24 − 3x Subtract 3x from each side. 6y = 24 − 3x Simplify. 6y 24 − 3x —=— Divide each side by 6. 6 6 1 y = 4 − —x Simplify. 2 The rewritten literal equation is y = 4 − —12 x. Solve the literal equation for x. 7. 2y − 2x = 10 8. 20y + 5x = 15 9. 4y − 5 = 4x + 7 10. y = 8x − x 11. y = 4x + zx + 6 12. z = 2x + 6xy 13. ABSTRACT REASONING Can you use the equation for an arithmetic sequence to write an equation for the sequence 3, 9, 27, 81,... ? Explain your reasoning. Dynamic Solutions available at BigIdeasMath.com 63 hs_geo_pe_02co.indd 63 1/19/15 8:58 AM  Mathematical Mathematically proficient students distinguish correct reasoning from Practices flawed reasoning. Using Correct Reasoning Core Concept Deductive Reasoning When you use deductive reasoning, you start with two or more true statements and deduce or infer the truth of another statement. Here is an example. 1. Premise: If a polygon is a triangle, then the sum of its angle measures is 180°. 2. Premise: Polygon ABC is a triangle. 3. Conclusion: The sum of the angle measures of polygon ABC is 180°. This pattern for deductive reasoning is called a syllogism. Recognizing Flawed Reasoning The syllogisms below represent common types of flawed reasoning. Explain why each conclusion is not valid. a. When it rains, the ground gets wet. b. If △ABC is equilateral, then it is isosceles. The ground is wet. △ABC is not equilateral. Therefore, it must have rained. Therefore, it must not be isosceles. c. All squares are polygons. d. No triangles are quadrilaterals. All trapezoids are quadrilaterals. Some quadrilaterals are not squares. Therefore, all squares are quadrilaterals. Therefore, some squares are not triangles. SOLUTION a. The ground may be wet for another reason. b. A triangle can be isosceles but not equilateral. c. All squares are quadrilaterals, but not because all trapezoids are quadrilaterals. d. No squares are triangles. Monitoring Progress Decide whether the syllogism represents correct or flawed reasoning. If flawed, explain why the conclusion is not valid. 1. All triangles are polygons. 2. No trapezoids are rectangles. Figure ABC is a triangle. Some rectangles are not squares. Therefore, figure ABC is a polygon. Therefore, some squares are not trapezoids. 3. If polygon ABCD is a square, then it is a rectangle. 4. If polygon ABCD is a square, then it is a rectangle. Polygon ABCD is a rectangle. Polygon ABCD is not a square. Therefore, polygon ABCD is a square. Therefore, polygon ABCD is not a rectangle. 64 Chapter 2 Reasoning and Proofs hs_geo_pe_02co.indd 64 1/19/15 8:58 AM  2.1 Conditional Statements Essential Question When is a conditional statement true or false? A conditional statement, symbolized by p → q, can be written as an “if-then statement” in which p is the hypothesis and q is the conclusion. Here is an example. If a polygon is a triangle, then the sum of its angle measures is 180 °. hypothesis, p conclusion, q Determining Whether a Statement Is True or False Work with a partner. A hypothesis can either be true or false. The same is true of a conclusion. For a conditional statement to be true, the hypothesis and conclusion do not necessarily both have to be true. Determine whether each conditional statement is true or false. Justify your answer. a. If yesterday was Wednesday, then today is Thursday. b. If an angle is acute, then it has a measure of 30°. c. If a month has 30 days, then it is June. d. If an even number is not divisible by 2, then 9 is a perfect cube. Determining Whether a Statement Is True or False y 6 Work with a partner. Use the points in the A D coordinate plane to determine whether each 4 statement is true or false. Justify your answer. 2 a. △ABC is a right triangle. B C b. △BDC is an equilateral triangle. −6 −4 −2 2 4 6x c. △BDC is an isosceles triangle. −2 CONSTRUCTING d. Quadrilateral ABCD is a trapezoid. −4 VIABLE ARGUMENTS e. Quadrilateral ABCD is a parallelogram. To be proficient in −6 math, you need to distinguish correct logic Determining Whether a Statement Is or reasoning from that True or False which is flawed. Work with a partner. Determine whether each conditional statement is true or false. Justify your answer. a. If △ADC is a right triangle, then the Pythagorean Theorem is valid for △ADC. b. If ∠ A and ∠ B are complementary, then the sum of their measures is 180°. c. If figure ABCD is a quadrilateral, then the sum of its angle measures is 180°. d. If points A, B, and C are collinear, then they lie on the same line. e. If ⃖⃗ AB and ⃖⃗ BD intersect at a point, then they form two pairs of vertical angles. Communicate Your Answer 4. When is a conditional statement true or false? 5. Write one true conditional statement and one false conditional statement that are different from those given in Exploration 3. Justify your answer. Section 2.1 Conditional Statements 65 hs_geo_pe_0201.indd 65 1/19/15 9:00 AM  2.1 Lesson What You Will Learn Write conditional statements. Use definitions written as conditional statements. Core Vocabul Vocabulary larry Write biconditional statements. conditional statement, p. 66 Make truth tables. if-then form, p. 66 hypothesis, p. 66 conclusion, p. 66 Writing Conditional Statements negation, p. 66 converse, p. 67 Core Concept inverse, p. 67 Conditional Statement contrapositive, p. 67 equivalent statements, p. 67 A conditional statement is a logical statement that has two parts, a hypothesis p and a conclusion q. When a conditional statement is written in if-then form, the perpendicular lines, p. 68 “if” part contains the hypothesis and the “then” part contains the conclusion. biconditional statement, p. 69 truth value, p. 70 Words If p, then q. Symbols p → q (read as “p implies q”) truth table, p. 70 Rewriting a Statement in If-Then Form Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. a. All birds have feathers. b. You are in Texas if you are in Houston. SOLUTION a. All birds have feathers. b. You are in Texas if you are in Houston. If an animal is a bird, If you are in Houston, then it has feathers. then you are in Texas. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. 1. All 30° angles are acute angles. 2. 2x + 7 = 1, because x = −3. Core Concept Negation The negation of a statement is the opposite of the original statement. To write the negation of a statement p, you write the symbol for negation (∼) before the letter. So, “not p” is written ∼p. Words not p Symbols ∼p Writing a Negation Write the negation of each statement. a. The ball is red. b. The cat is not black. SOLUTION a. The ball is not red. b. The cat is black. 66 Chapter 2 Reasoning and Proofs hs_geo_pe_0201.indd 66 1/19/15 9:00 AM  Core Concept Related Conditionals Consider the conditional statement below. Words If p, then q. Symbols p→q Converse To write the converse of a conditional statement, exchange the hypothesis and the conclusion. Words If q, then p. Symbols q→p Inverse To write the inverse of a conditional statement, negate both the COMMON ERROR hypothesis and the conclusion. Just because a conditional Words If not p, then not q. Symbols ∼p → ∼q statement and its contrapositive are both Contrapositive To write the contrapositive of a conditional statement, first true does not mean that write the converse. Then negate both the hypothesis and its converse and inverse the conclusion. are both false. The converse and inverse could Words If not q, then not p. Symbols ∼q → ∼p also both be true. A conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. In general, when two statements are both true or both false, they are called equivalent statements. Writing Related Conditional Statements L p be “you are a guitar player” and let q be “you are a musician.” Write each Let sstatement in words. Then decide whether it is true or false. aa. the conditional statement p → q b. the converse q → p b cc. the inverse ∼p → ∼q d. the contrapositive ∼q → ∼p d SOLUTION S aa. Conditional: If you are a guitar player, then you are a musician. true; Guitar players are musicians. b. Converse: If you are a musician, then you are a guitar player. false; Not all musicians play the guitar. c. Inverse: If you are not a guitar player, then you are not a musician. false; Even if you do not play a guitar, you can still be a musician. d. Contrapositive: If you are not a musician, then you are not a guitar player. true; A person who is not a musician cannot be a guitar player. Monitoring Progress Help in English and Spanish at BigIdeasMath.com In Exercises 3 and 4, write the negation of the statement. 3. The shirt is green. 4. The shoes are not red. 5. Repeat Example 3. Let p be “the stars are visible” and let q be “it is night.” Section 2.1 Conditional Statements 67 hs_geo_pe_0201.indd 67 1/19/15 9:00 AM  Using Definitions You can write a definition as a conditional statement in if-then form or as its converse. Both the conditional statement and its converse are true for definitions. For example, consider the definition of perpendicular lines. If two lines intersect to form a right angle, then they are perpendicular lines. You can also write the definition using the converse: If two lines are perpendicular lines, then they intersect to m form a right angle. You can write “lineℓ is perpendicular to line m” asℓ⊥ m. ⊥m Using Definitions Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. B a. ⃖⃗ AC ⊥ ⃖⃗ BD b. ∠AEB and ∠CEB are a linear pair. A E C c. ⃗ EA and ⃗ EB are opposite rays. D SOLUTION a. This statement is true. The right angle symbol in the diagram indicates that the lines intersect to form a right angle. So, you can say the lines are perpendicular. b. This statement is true. By definition, if the noncommon sides of adjacent angles are opposite rays, then the angles are a linear pair. Because ⃗ EA and ⃗ EC are opposite rays, ∠AEB and ∠CEB are a linear pair. c. This statement is false. Point E does not lie on the same line as A and B, so the rays are not opposite rays. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use the diagram. Decide whether the statement is true. Explain your answer using the definitions you have learned. F G M H J 6. ∠JMF and ∠FMG are supplementary. — 7. Point M is the midpoint of FH. 8. ∠JMF and ∠HMG are vertical angles. ⃖⃗ ⊥ ⃖⃗ 9. FH JG 68 Chapter 2 Reasoning and Proofs hs_geo_pe_0201.indd 68 1/19/15 9:00 AM  Writing Biconditional Statements Core Concept Biconditional Statement When a conditional statement and its converse are both true, you can write them as a single biconditional statement. A biconditional statement is a statement that contains the phrase “if and only if.” Words p if and only if q Symbols p↔q Any definition can be written as a biconditional statement. Writing a Biconditional Statement Rewrite the definition of perpendicular lines as a single biconditional statement. Definition If two lines intersect to form a right angle, then they are perpendicular lines. SOLUTION Let p be “two lines intersect to form a right angle” s and let q be “they are perpendicular lines.” Use red to identify p and blue to identify q. Write the definition p → q. t Definition If two lines intersect to form a right angle, then they are perpendicular lines. Write the converse q → p. s⊥t Converse If two lines are perpendicular lines, then they intersect to form a right angle. Use the definition and its converse to write the biconditional statement p ↔ q. Biconditional Two lines intersect to form a right angle if and only if they are perpendicular lines. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 10. Rewrite the definition of a right angle as a single biconditional statement. Definition If an angle is a right angle, then its measure is 90°. 11. Rewrite the definition of congruent segments as a single biconditional statement. Definition If two line segments have the same length, then they are congruent segments. 12. Rewrite the statements as a single biconditional statement. If Mary is in theater class, then she will be in the fall play. If Mary is in the fall play, then she must be taking theater class. 13. Rewrite the statements as a single biconditional statement. If you can run for President, then you are at least 35 years old. If you are at least 35 years old, then you can run for President. Section 2.1 Conditional Statements 69 hs_geo_pe_0201.indd 69 1/19/15 9:00 AM  Making Truth Tables Dynamic Solutions available at BigIdeasMath.com The truth value of a statement is either true (T) or false (F). You can determine the conditions under which a conditional statement is true by using a truth table. The truth table below shows the truth values for hypothesis p and conclusion q. Conditional p q p→q T T T T F F F T T F F T The conditional statement p → q is only false when a true hypothesis produces a false conclusion. Two statements are logically equivalent when they have the same truth table. Making a Truth Table Use the truth table above to make truth tables for the converse, inverse, and contrapositive of a conditional statement p → q. SOLUTION The truth tables for the converse and the inverse are shown below. Notice that the converse and the inverse are logically equivalent because they have the same truth table. Converse Inverse p q q→p p q ∼p ∼q ∼p → ∼q T T T T T F F T T F T T F F T T F T F F T T F F F F T F F T T T The truth table for the contrapositive is shown below. Notice that a conditional statement and its contrapositive are logically equivalent because they have the same truth table. Contrapositive p q ∼q ∼p ∼q → ∼p T T F F T T F T F F F T F T T F F T T T Monitoring Progress Help in English and Spanish at BigIdeasMath.com 14. Make a truth table for the conditional statement p → ∼q. 15. Make a truth table for the conditional statement ∼(p → q). 70 Chapter 2 Reasoning and Proofs hs_geo_pe_0201.indd 70 1/19/15 9:00 AM  2.1 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY What type of statements are either both true or both false? 2. WHICH ONE DOESN’T BELONG? Which statement does not belong with the other three? Explain your reasoning. If today is Tuesday, then tomorrow is Wednesday. If it is Independence Day, then it is July. If an angle is acute, then its measure is less than 90°. If you are an athlete, then you play soccer. Monitoring Progress and Modeling with Mathematics In Exercises 3–6, copy the conditional statement. 18. Let p be “you are in math class” and let q be “you are Underline the hypothesis and circle the conclusion. in Geometry.” 3. If a polygon is a pentagon, then it has five sides. 19. Let p be “you do your math homework” and let q be “you will do well on the test.” 4. If two lines form vertical angles, then they intersect. 5. If you run, then you are fast. 20. Let p be “you are not an only child” and let q be “you have a sibling.” 6. If you like math, then you like science. 21. Let p be “it does not snow” and let q be “I will run In Exercises 7–12, rewrite the conditional statement in outside.” if-then form. (See Example 1.) 22. Let p be “the Sun is out” and let q be “it is daytime.” 7. 9x + 5 = 23, because x = 2. 23. Let p be “3x − 7 = 20” and let q be “x = 9.” 8. Today is Friday, and tomorrow is the weekend. 24. Let p be “it is Valentine’s Day” and let q be “it is 9. You are in a band, and you play the drums. February.” 10. Two right angles are supplementary angles. In Exercises 25–28, decide whether the statement about the diagram is true. Explain your answer using the 11. Only people who are registered are allowed to vote. definitions you have learned. (See Example 4.) 12. The measures of complementary angles sum to 90°. 25. m∠ABC = 90° ⃖⃗ ⊥ ⃖⃗ 26. PQ ST In Exercises 13–16, write the negation of the statement. A P (See Example 2.) 13. The sky is blue. 14. The lake is cold. S T B C Q 15. The ball is not pink. 16. The dog is not a Lab. 27. m∠2 + m∠3 = 180° — 28. M is the midpoint of AB. In Exercises 17–24, write the conditional statement p → q, the converse q → p, the inverse ∼p → ∼q, and Q A M B the contrapositive ∼q → ∼p in words. Then decide whether each statement is true or false. (See Example 3.) 2 3 17. Let p be “two angles are supplementary” and let q be M N P “the measures of the angles sum to 180°.” Section 2.1 Conditional Statements 71 hs_geo_pe_0201.indd 71 1/19/15 9:00 AM  In Exercises 29–32, rewrite the definition of the term as In Exercises 39–44, create a truth table for the logical a biconditional statement. (See Example 5.) Dynamic Solutions(See statement. available Exampleat 6.) BigIdeasMath.com 29. The midpoint of a segment is the point that divides the 39. ∼p → q segment into two congruent segments. 40. ∼q → p 30. Two angles are vertical angles when their sides form two pairs of opposite rays. 41. ∼(∼p → ∼q) 31. Adjacent angles are two angles that share a common 42. ∼( p → ∼q) vertex and side but have no common interior points. 43. q → ∼p 32. Two angles are supplementary angles when the sum of their measures is 180°. 44. ∼(q → p) In Exercises 33–36, rewrite the statements as a single 45. USING STRUCTURE The statements below describe biconditional statement. (See Example 5.) three ways that rocks are formed. 33. If a polygon has three sides, then it is a triangle. Igneous rock is formed If a polygon is a triangle, then it has three sides. from the cooling of molten rock. 34. If a polygon has four sides, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. Sedimentary rock is 35. If an angle is a right angle, then it measures 90°. formed from pieces of If an angle measures 90°, then it is a right angle. other rocks. 36. If an angle is obtuse, then it has a measure between 90° and 180°. If an angle has a measure between 90° and 180°, then it is obtuse. Metamorphic rock is formed by changing 37. ERROR ANALYSIS Describe and correct the error in temperature, pressure, rewriting the conditional statement in if-then form. or chemistry. ✗ Conditional statement All high school students take four English courses. a. Write each statement in if-then form. b. Write the converse of each of the statements in If-then form part (a). Is the converse of each statement true? If a high school student takes Explain your reasoning. four courses, then all four are c. Write a true if-then statement about rocks that is English courses. different from the ones in parts (a) and (b). Is the converse of your statement true or false? Explain your reasoning. 38. ERROR ANALYSIS Describe and correct the error in writing the converse of the conditional statement. 46. MAKING AN ARGUMENT Your friend claims the ✗ statement “If I bought a shirt, then I went to the mall” can be written as a true biconditional statement. Your Conditional statement sister says you cannot write it as a biconditional. Who If it is raining, then I will bring is correct? Explain your reasoning. an umbrella. Converse 47. REASONING You are told that the contrapositive If it is not raining, then I will not of a statement is true. Will that help you determine bring an umbrella. whether the statement can be written as a true biconditional statement? Explain your reasoning. 72 Chapter 2 Reasoning and Proofs hs_geo_pe_0201.indd 72 1/19/15 9:00 AM  48. PROBLEM SOLVING Use the conditional statement to 53. MATHEMATICAL CONNECTIONS Can the statement identify the if-then statement as the converse, inverse, “If x2 − 10 = x + 2, then x = 4” be combined with or contrapositive of the conditional statement. Then its converse to form a true biconditional statement? use the symbols to represent both statements. 54. CRITICAL THINKING The largest natural arch in Conditional statement the United States is Landscape Arch, located in If I rode my bike to school, then I did not Thompson, Utah. It spans 290 feet. walk to school. If-then statement If I did not ride my bike to school, then I walked to school. p q ∼ → ↔ USING STRUCTURE In Exercises 49–52, rewrite the conditional statement in if-then form. Then underline the hypothesis and circle the conclusion. 49. a. Use the information to write at least two true conditional statements. b. Which type of related conditional statement must also be true? Write the related conditional statements. c. What are the other two types of related conditional 50. statements? Write the related conditional statements. Then determine their truth values. Explain your reasoning. 55. REASONING Which statement has the same meaning as the given statement? Given statement You can watch a movie after you do your 51. homework. A If you do your homework, then you can watch a ○ movie afterward. B If you do not do your homework, then you can ○ watch a movie afterward. C If you cannot watch a movie afterward, then do ○ 52. your homework. D If you can watch a movie afterward, then do not ○ do your homework. 56. THOUGHT PROVOKING Write three conditional statements, where one is always true, one is always false, and one depends on the person interpreting the statement. Section 2.1 Conditional Statements 73 hs_geo_pe_0201.indd 73 1/19/15 9:00 AM  57. CRITICAL THINKING One example of a conditional 60. DRAWING CONCLUSIONS You measure the heights of statement involving dates is “If today is August 31, your classmates to get a data set. then tomorrow is September 1.” Write a conditional a. Tell whether this statement is true: If x and y are statement using dates from two different months so the least and greatest values in your data set, then that the truth value depends on when the statement the mean of the data is between x and y. is read. b. Write the converse of the statement in part (a). Is the converse true? Explain your reasoning. 58. HOW DO YOU SEE IT? The Venn diagram represents all the musicians at a high school. Write three c. Copy and complete the statement below using conditional statements in if-then form describing mean, median, or mode to make a conditional the relationships between the various groups statement that is true for any data set. Explain of musicians. your reasoning. If a data set has a mean, median, and a musicians mode, then the _______ of the data set will always be a data value. chorus band jazz band 61. WRITING Write a conditional statement that is true, but its converse is false. 62. CRITICAL THINKING Write a series of if-then statements that allow you to find the measure of each angle, given that m∠1 = 90°. Use the definition of 59. MULTIPLE REPRESENTATIONS Create a Venn diagram linear pairs. representing each conditional statement. Write the converse of each conditional statement. Then determine whether each conditional statement and its 4 1 converse are true or false. Explain your reasoning. 3 2 a. If you go to the zoo to see a lion, then you will see a cat. 63. WRITING Advertising slogans such as “Buy these b. If you play a sport, then you wear a helmet. shoes! They will make you a better athlete!” often c. If this month has 31 days, then it is not February. imply conditional statements. Find an advertisement or write your own slogan. Then write it as a conditional statement. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the pattern. Then draw the next two figures in the sequence. (Skills Review Handbook) 64. 65. Find the pattern. Then write the next two numbers. (Skills Review Handbook) 66. 1, 3, 5, 7,... 67. 12, 23, 34, 45,... 4 8 16 68. 2, —3 , —9 , — 27 ,... 69. 1, 4, 9, 16,... 74 Chapter 2 Reasoning and Proofs hs_geo_pe_0201.indd 74 1/19/15 9:00 AM  2.2 Inductive and Deductive Reasoning Essential Question How can you use reasoning to solve problems? A conjecture is an unproven statement based on observations. Writing a Conjecture Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern. 1 2 3 4 5 6 7 a. b. CONSTRUCTING c. VIABLE ARGUMENTS To be proficient in math, you need to justify your conclusions and Using a Venn Diagram communicate them to others. Work with a partner. Use the Venn diagram to determine whether the statement is true or false. Justify your answer. Assume that no region of the Venn diagram is empty. a. If an item has Property B, then it has Property A. Property A b. If an item has Property A, then it has Property B. Property C Property B c. If an item has Property A, then it has Property C. d. Some items that have Property A do not have Property B. e. If an item has Property C, then it does not have Property B. f. Some items have both Properties A and C. g. Some items have both Properties B and C. Reasoning and Venn Diagrams Work with a partner. Draw a Venn diagram that shows the relationship between different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, rhombuses, and kites. Then write several conditional statements that are shown in your diagram, such as “If a quadrilateral is a square, then it is a rectangle.” Communicate Your Answer 4. How can you use reasoning to solve problems? 5. Give an example of how you used reasoning to solve a real-life problem. Section 2.2 Inductive and Deductive Reasoning 75 hs_geo_pe_0202.indd 75 1/19/15 9:01 AM  2.2 Lesson What You Will Learn Use inductive reasoning. Use deductive reasoning. Core Vocabul Vocabulary larry conjecture, p. 76 Using Inductive Reasoning inductive reasoning, p. 76 counterexample, p. 77 deductive reasoning, p. 78 Core Concept Inductive Reasoning A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. Describing a Visual Pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Figure 1 Figure 2 Figure 3 SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. Figure 4 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Sketch the fifth figure in the pattern in Example 1. Sketch the next figure in the pattern. 2. 3. 76 Chapter 2 Reasoning and Proofs hs_geo_pe_0202.indd 76 1/19/15 9:01 AM  Making and Testing a Conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. SOLUTION Step 1 Find a pattern using a few groups of small numbers. 3 + 4 + 5 = 12 = 4 3 ⋅ 7 + 8 + 9 = 24 = 8 3 ⋅ 10 + 11 + 12 = 33 = 11 3 ⋅ 16 + 17 + 18 = 51 = 17 3 ⋅ Step 2 Make a conjecture. Conjecture The sum of any three consecutive integers is three times the second number. Step 3 Test your conjecture using other numbers. For example, test that it works with the groups −1, 0, 1 and 100, 101, 102. −1 + 0 + 1 = 0 = 0 3 ⋅ ✓ 100 + 101 + 102 = 303 = 101 3 ⋅ ✓ Core Concept Counterexample To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding just one counterexample. A counterexample is a specific case for which the conjecture is false. Finding a Counterexample A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture. Conjecture The sum of two numbers is always more than the greater number. SOLUTION To find a counterexample, you need to find a sum that is less than the greater number. −2 + (−3) = −5 −5 ≯ −2 Because a counterexample exists, the conjecture is false. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 4. Make and test a conjecture about the sign of the product of any three negative integers. 5. Make and test a conjecture about the sum of any five consecutive integers. Find a counterexample to show that the conjecture is false. 6. The value of x2 is always greater than the value of x. 7. The sum of two numbers is always greater than their difference. Section 2.2 Inductive and Deductive Reasoning 77 hs_geo_pe_0202.indd 77 1/19/15 9:01 AM  Using Deductive Reasoning Core Concept Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture. Laws of Logic Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Law of Syllogism If hypothesis p, then conclusion q. If these statements are true, If hypothesis q, then conclusion r. If hypothesis p, then conclusion r. then this statement is true. Using the Law of Detachment If two segments have the same length, then they are congruent. You know that BC = XY. Using the Law of Detachment, what statement can you make? SOLUTION Because BC = XY satisfies the hypothesis of a true conditional statement, the conclusion is also true. — ≅ XY So, BC —. Using the Law of Syllogism If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. a. If x2 > 25, then x2 > 20. If x > 5, then x2 > 25. b. If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all its sides are congruent. SOLUTION a. Notice that the conclusion of the second statement is the hypothesis of the first statement. The order in which the statements are given does not affect whether you can use the Law of Syllogism. So, you can write the following new statement. If x > 5, then x2 > 20. b. Neither statement’s conclusion is the same as the other statement’s hypothesis. You cannot use the Law of Syllogism to write a new conditional statement. 78 Chapter 2 Reasoning and Proofs hs_geo_pe_0202.indd 78 1/19/15 9:01 AM  Using Inductive and Deductive Reasoning What conclusion can you make about the product of an even integer and any other integer? SOLUTION Step 1 Look for a pattern in several examples. Use inductive reasoning to make a MAKING SENSE conjecture. OF PROBLEMS (−2)(2) = −4 (−1)(2) = −2 2(2) = 4 3(2) = 6 In geometry, you will (−2)(−4) = 8 (−1)(−4) = 4 2(−4) = −8 3(−4) = −12 frequently use inductive reasoning to make Conjecture Even integer Any integer = Even integer conjectures. You will also Step 2 Let n and m each be any integer. Use deductive reasoning to show that the use deductive reasoning conjecture is true. to show that conjectures are true or false. You will 2n is an even integer because any integer multiplied by 2 is even. need to know which type 2nm represents the product of an even integer 2n and any integer m. of reasoning to use. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer. Comparing Inductive and Deductive Reasoning Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. a. Each time Monica kicks a ball up in the air, it returns to the ground. So, the next time Monica kicks a ball up in the air, it will return to the ground. b. All reptiles are cold-blooded. Parrots are not cold-blooded. Sue’s pet parrot is not a reptile. SOLUTION a. Inductive reasoning, because a pattern is used to reach the conclusion. b. Deductive reasoning, because facts about animals and the laws of logic are used to reach the conclusion. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. If 90° < m∠R < 180°, then ∠R is obtuse. The measure of ∠R is 155°. Using the Law of Detachment, what statement can you make? 9. Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. If you get an A on your math test, then you can go to the movies. If you go to the movies, then you can watch your favorite actor. 10. Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show that the conjecture is true. 11. Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. All multiples of 8 are divisible by 4. 64 is a multiple of 8. So, 64 is divisible by 4. Section 2.2 Inductive and Deductive Reasoning 79 hs_geo_pe_0202.indd 79 1/19/15 9:01 AM  2.2 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY How does the prefix “counter-” help you understand the term counterexample? 2. WRITING Explain the difference between inductive reasoning and deductive reasoning. Monitoring Progress and Modeling with Mathematics In Exercises 3–8, describe the pattern. Then write or In Exercises 17–20, use the Law of Detachment to draw the next two numbers, letters, or figures. determine what you can conclude from the given (See Example 1.) information, if possible. (See Example 4.) 3. 1, −2, 3, −4, 5,... 4. 0, 2, 6, 12, 20,... 17. If you pass the final, then you pass the class. You passed the final. 5. Z, Y, X, W, V,... 6. J, F, M, A, M,... 18. If your parents let you borrow the car, then you will 7. go to the movies with your friend. You will go to the movies with your friend. 19. If a quadrilateral is a square, then it has four right 8. angles. Quadrilateral QRST has four right angles. 20. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point P — into two congruent line segments. divides LH In Exercises 9–12, make and test a conjecture about the In Exercises 21–24, use the Law of Syllogism to write a given quantity. (See Example 2.) new conditional statement that follows from the pair of 9. the product of any two even integers true statements, if possible. (See Example 5.) 21. If x < −2, then ∣ x ∣ > 2. If x > 2, then ∣ x ∣ > 2. 10. the sum of an even integer and an odd integer 1 1 22. If a = 3, then 5a = 15. If —2 a = 1—2 , then a = 3. 11. the quotient of a number and its reciprocal 23. If a figure is a rhombus, then the figure is a 12. the quotient of two negative integers parallelogram. If a figure is a parallelogram, then the figure has two pairs of opposite sides that In Exercises 13–16, find a counterexample to show that are parallel. the conjecture is false. (See Example 3.) 13. The product of two positive numbers is always greater 24. If a figure is a square, then the figure has four than either number. congruent sides. If a figure is a square, then the figure has four right angles. n+1 14. If n is a nonzero integer, then — is always greater n In Exercises 25–28, state the law of logic that than 1. is illustrated. 15. If two angles are supplements of each other, then one 25. If you do your homework, then you can watch TV. If of the angles must be acute. you watch TV, then you can watch your favorite show. 16. — into two line segments. So, the A line s divides MN If you do your homework, then you can watch your —. line s is a segment bisector of MN favorite show. 80 Chapter 2 Reasoning and Proofs hs_geo_pe_0202.indd 80 1/19/15 9:01 AM  26. If you miss practice the day before a game, then you 37. REASONING The table shows the average weights will not be a starting player in the game. of several subspecies of tigers. What conjecture can you make about the relation between the weights of You miss practice on Tuesday. You will not start the female tigers and the weights of male tigers? Explain game Wednesday. your reasoning. 27. If x > 12, then x + 9 > 20. The value of x is 14. So, x + 9 > 20. Weight Weight of female of male 28. If ∠1 and ∠2 are vertical angles, then ∠1 ≅ ∠2. (pounds) (pounds) If ∠1 ≅ ∠2, then m∠1 = m∠2. Amur 370 660 If ∠1 and ∠2 are vertical angles, then m∠1 = m∠2. Bengal 300 480 In Exercises 29 and 30, use inductive reasoning to South China 240 330 make a conjecture about the given quantity. Then use Sumatran 200 270 deductive reasoning to show that the conjecture is true. (See Example 6.) Indo-Chinese 250 400 29. the sum of two odd integers 30. the product of two odd integers 38. HOW DO YOU SEE IT? Determine whether you can make each conjecture from the graph. Explain In Exercises 31–34, decide whether inductive reasoning your reasoning. or deductive reasoning is used to reach the conclusion. Explain your reasoning. (See Example 7.) U.S. High School Girls’ Lacrosse Number of participants 31. Each time your mom goes to the store, she buys milk. y 140 So, the next time your mom goes to the store, she will (thousands) buy milk. 100 32. Rational numbers can be written as fractions. 60 Irrational numbers cannot be written as fractions. 20 So, —12 is a rational number. 1 2 3 4 5 6 7x 33. All men are mortal. Mozart is a man, so Mozart Year is mortal. 34. Each time you clean your room, you are allowed to a. More girls will participate in high school lacrosse go out with your friends. So, the next time you clean in Year 8 than those who participated in Year 7. your room, you will be allowed to go out with b. The number of girls participating in high your friends. school lacrosse will exceed the number of boys participating in high school lacrosse in Year 9. ERROR ANALYSIS In Exercises 35 and 36, describe and correct the error in interpreting the statement. 35. If a figure is a rectangle, then the figure has four sides. 39. MATHEMATICAL CONNECTIONS Use inductive A trapezoid has four sides. reasoning to write a formula for the sum of the first n positive even integers. ✗ Using the Law of Detachment, you can conclude that a trapezoid is a rectangle. 40. FINDING A PATTERN The following are the first nine Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34,... 36. Each day, you get to school before your friend. a. Make a conjecture about each of the Fibonacci ✗ numbers after the first two. Using deductive reasoning, you can b. Write the next three numbers in the pattern. conclude that you will arrive at school before your friend tomorrow. c. Research to find a real-world example of this pattern. Section 2.2 Inductive and Deductive Reasoning 81 hs_geo_pe_0202.indd 81 1/19/15 9:01 AM  41. MAKING AN ARGUMENT Which argument is correct? 45. DRAWING CONCLUSIONS Decide whether each Explain your reasoning. conclusion is valid. Explain your reasoning. Argument 1: If two angles measure 30° and 60°, Yellowstone is a national park in Wyoming. then the angles are complementary. ∠ 1 and ∠ 2 are You and your friend went camping at complementary. So, m∠ 1 = 30° and m∠ 2 = 60°. Yellowstone National Park. Argument 2: If two angles measure 30° and 60°, then When you go camping, you go canoeing. the angles are complementary. The measure of ∠ 1 is If you go on a hike, your friend goes with you. 30° and the measure of ∠ 2 is 60°. So, ∠ 1 and ∠ 2 are complementary. You go on a hike. There is a 3-mile-long trail near your campsite. 42. THOUGHT PROVOKING The first two terms of a a. You went camping in Wyoming. sequence are —14 and —12. Describe three different possible patterns for the sequence. List the first five terms for b. Your friend went canoeing. each sequence. c. Your friend went on a hike. d. You and your friend went on a hike on a 43. MATHEMATICAL CONNECTIONS Use the table to 3-mile-long trail. make a conjecture about the relationship between x and y. Then write an equation for y in terms of x. 46. CRITICAL THINKING Geologists use the Mohs’ scale Use the equation to test your conjecture for other to determine a mineral’s hardness. Using the scale, values of x. a mineral with a higher rating will leave a scratch on a mineral with a lower rating. Testing a mineral’s x 0 1 2 3 4 hardness can help identify the mineral. y 2 5 8 11 14 Mineral Talc Gypsum Calcite Fluorite 44. REASONING Use the pattern below. Each figure is made of squares that are 1 unit by 1 unit. Mohs’ 1 2 3 4 rating a. The four minerals are randomly labeled A, B, C, and D. Mineral A is scratched by Mineral B. Mineral C is scratched by all three of the other 1 2 3 4 5 minerals. What can you conclude? Explain your reasoning. a. Find the perimeter of each figure. Describe the pattern of the perimeters. b. What additional test(s) can you use to identify all the minerals in part (a)? b. Predict the perimeter of the 20th figure. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Determine which postulate is illustrated by the statement. (Section 1.2 and Section 1.5) D E A B C 47. AB + BC = AC 48. m∠ DAC = m∠ DAE + m∠ EAB 49. AD is the absolute value of the difference of the coordinates of A and D. 50. m∠ DAC is equal to the absolute value of the difference between the ⃗ and ⃗ real numbers matched with AD AC on a protractor. 82 Chapter 2 Reasoning and Proofs hs_geo_pe_0202.indd 82 3/9/16 9:12 AM  2.3 Postulates and Diagrams Essential Question In a diagram, what can be assumed and what needs to be labeled? Looking at a Diagram Work with a partner. On a piece of paper, draw two perpendicular lines. Label them ⃖⃗ ⃖⃗. Look at the diagram from different angles. Do the lines appear AB and CD perpendicular regardless of the angle at which you look at them? Describe all the angles at which you can look at the lines and have them appear perpendicular. C A B B C D A view from m D upper rightt ATTENDING view from above TO PRECISION To be proficient in math, Interpreting a Diagram you need to state the meanings of the symbols Work with a partner. When you draw a you choose. diagram, you are communicating with others. It is important that you include sufficient A D information in the diagram. Use the diagram to determine which of the following statements C you can assume to be true. Explain B G your reasoning. I F a. All the points shown are coplanar. b. Points D, G, and I are collinear. E c. Points A, C, and H are collinear. H EG and ⃖⃗ d. ⃖⃗ AH are perpendicular. e. ∠BCA and ∠ACD are a linear pair. f. ⃖⃗ AF and ⃖⃗ BD are perpendicular. g. ⃖⃗ EG and ⃖⃗ BD are parallel. h. ⃖⃗ ⃖⃗ are coplanar. AF and BD i. ⃖⃗ EG and ⃖⃗ BD do not intersect. j. ⃖⃗ ⃖⃗ intersect. AF and BD k. ⃖⃗ EG and ⃖⃗ BD are perpendicular. l. ∠ACD and ∠BCF are vertical angles. m. ⃖⃗ AC and ⃖⃗ FH are the same line. Communicate Your Answer 3. In a diagram, what can be assumed and what needs to be labeled? 4. Use the diagram in Exploration 2 to write two statements you can assume to be true and two statements you cannot assume to be true. Your statement