- Biconditionals and Definitions -. Plan What You ll Learn To write biconditionals To recognize good definitions... And Why To evaluate definitions of letters used in the American Manual Alphabet, as in Exercises 33 37 5. If we go on a picnic, then the sun shines. Writing Biconditionals Vocabulary Tip Connect the conditional and its converse with and. Then compare with the if and only if form.. If three points lie on the same line, then they are collinear. The converse is also true. Three points are collinear if and only if they lie on the same line. Check Skills You ll Need GO for Help When a conditional and its converse are true, you can combine them as a true biconditional. This is the statement you get by connecting the conditional and its converse with the word and. You can write a biconditional more concisely, however, by joining the two parts of each conditional with the phrase if and only if. Writing a Biconditional Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional If two angles have the same measure, then the angles are congruent. Converse If two angles are congruent, then the angles have the same measure. The converse is also true. Since both the conditional and its converse are true, you can combine them in a true biconditional by using the phrase if and only if. Biconditional Two angles have the same measure if and only if the angles are congruent. Lesson - Identify the hypothesis and the conclusion of each conditional statement.. If x. 0, then x. 5. Hypothesis: x S 0 Conclusion: x S 5. If you live in Milwaukee, then you live in Wisconsin. Hypothesis: You live in Milwaukee. Conclusion: You live in Wisconsin. Write each statement as a conditional. 3. Squares have four sides.. All butterflies have wings. If a figure is a square, then it has four sides. If something is a butterfly, Write the converse of each statement. then it has wings. 5. If the sun shines, then we go on a picnic. See left. 6. If two lines are skew, then they do not intersect. If two lines do not 7. If x =-3, then x 3 =-7. intersect, then they are skew. If x 3 57, then x 53. New Vocabulary biconditional Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. See left. Conditional If three points are collinear, then they lie on the same line. Lesson - Biconditionals and Definitions 87 Objectives To write biconditionals To recognize good definitions Examples Writing a Biconditional Orthographic Drawing 3 Writing a Definition as a Biconditional Real-World Connection Math Background Whenever a theorem is investigated or proved in geometry, the converse also should be examined. When both a theorem and its converse are true, they can be written as a biconditional. The strict requirement in mathematics that definitions be biconditional is not always followed outside mathematics. More Math Background: p. 78C Lesson Planning and Resources See p. 78E for a list of the resources that support this lesson. Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Conditional Statements Lesson -: Examples, Extra Skills, Word Problems, Proof Practice, Ch. Converses Lesson -: Example 5 Extra Skills, Word Problems, Proof Practice, Ch. Special Needs L Have students create a summary chart listing conditional, converse, and biconditional statements. Tables should include their own examples as well as the respective symbolic forms. learning style: visual Below Level L Have students write the two conditionals that make up the definition in Example, and then discuss the truth value of each. learning style: verbal 87
. Teach Guided Instruction Teaching Tip Ask: What word with the prefix bi- have you used in this class? the word bisect Connection to Language Arts Discuss the phrase if and only if. Ask students to use each part of the phrase separately in a sentence not related to mathematics. Point out that a statement such as I play soccer only if it is Saturday is equivalent to If I play soccer, then it is Saturday but is not equivalent to If it is Saturday, then I play soccer. Additional Examples. If a number is prime, then it has only two distinct factors, and itself. If a number has only two distinct factors, and itself, then it is prime. You can write a biconditional as two conditionals that are converses of each other. Separating a Biconditional Into Parts Algebra Write two statements that form this biconditional about whole numbers: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Here are the two statements. They are converses of each other. If a number is divisible by 3, then the sum of its digits is divisible by 3. If the sum of a number s digits is divisible by 3, then the number is divisible by 3. Write two statements that form this biconditional about integers greater than : A number is prime if and only if it has only two distinct factors, and itself. See left. Key Concepts Summary Biconditional Statements Recognizing Good Definitions A biconditional combines p S q and q S p as p q. Statement Example Symbolic Form You Read It Biconditional An angle is a straight angle p q pif and only if q. if and only if its measure is 80. Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. If x = 5, then x + 5 = 0. If x ± 5 0, then x 5; x 5 if and only if x ± 5 0. Write the two statements that form this biconditional. Lines are skew if and only if they are noncoplanar. If lines are skew, then they are noncoplanar. If lines are noncoplanar, then they are skew. Polyglobs A Activity: Writing a Definition C B Not polyglobs. Use the examples at the left to identify the figures above that are polyglobs. Figure B is a polyglob.. Write a definition of a polyglob by describing what a polyglob is. Answers may vary. Sample: A polyglob has three fingers and three solid dots. 88 Chapter Reasoning and Proof Advanced Learners L Have students look up three words in a dictionary and explain why the definitions do or do not obey the rules at the top of page 89. English Language Learners ELL Discuss the meaning of the prefix bi-. In a biconditional, two if-then statements (conditionals) must be true: the conditional and its converse. 88 learning style: verbal learning style: verbal
Real-World Connection The definitions in a dictionary have to be good definitions. 3. Conditional: If an angle is a right angle, then its measure is 90. Converse: If an angle has measure 90, then it is a right angle. The two statements are true. An angle is a right angle if and only if its measure is 90. For: Biconditionals Activity Use: Interactive Textbook, - 3 3 In geometry you start with undefined terms such as point, line, and plane whose meanings you understand intuitively. Then you use those terms to define other terms such as collinear points. A good definition is a statement that can help you identify or classify an object. A good definition has several important components. A good definition uses clearly understood terms. The terms should be commonly understood or already defined. A good definition is precise. Good definitions avoid words such as large, sort of, and almost. A good definition is reversible. That means that you can write a good definition as a true biconditional. Writing a Definition as a Biconditional Show that this definition of perpendicular lines is reversible. Then write it as a true biconditional. Definition Perpendicular lines are two lines that intersect to form right angles. Conditional If two lines are perpendicular, then they intersect to form right angles. Converse If two lines intersect to form right angles, then they are perpendicular. The two conditionals converses of each other are true, so the definition can be written as a true biconditional. Biconditional Two lines are perpendicular if and only if they intersect to form right angles. Show that this definition of right angle is reversible. Then write it as a true biconditional. Definition A right angle is an angle whose measure is 90. See left. One way to show that a statement is not a good definition is to find a counterexample. Real-World Connection Language Arts Is the given statement a good definition? Explain. a. An airplane is a vehicle that flies. The statement is not a good definition because it is not reversible. A helicopter is a counterexample. A helicopter is a vehicle that flies, but a helicopter is not an airplane. b. A triangle has sharp corners. The statement is not a good definition because it uses the imprecise word sharp, and it is not reversible. Is the following statement a good definition? Explain. A square is a figure with four right angles. It is not a good definition because a rectangle has four right angles and is not necessarily a square. Lesson - Biconditionals and Definitions 89 Guided Activity We understand intuitively the meanings of some ideas, such as point, line, and plane. Discuss as a class what intuition is and how different experiences can produce different types of intuition. Connection to Law Lawyers and judges use precise definitions every day. For example, when a trial begins, the judge defines legal terms for jury members so that they can understand the law and apply it correctly. Additional Examples 3 Show that this definition of triangle is reversible. Then write it as a true biconditional. A triangle is a polygon with exactly three sides. A polygon is a triangle if and only if it has exactly three sides. Is the following statement a good definition? Explain. An apple is a fruit that contains seeds. No; while the statement is true as a description of an apple, it is not reversible because other fruits contain seeds. Resources Daily Notetaking Guide - L3 Daily Notetaking Guide - Adapted Instruction L Closure Explain the meaning of the word biconditional. Give an example of a biconditional and the two conditional statements that form it. Sample: The prefix bi- indicates that there are two conditionals. For example, the biconditional I eat if and only if I am hungry is formed by the conditional If I am hungry, then I eat and its converse If I eat, then I am hungry. 89
3. Practice Assignment Guide A B -, 3, 38-6 A B 3-3, 33-37 C Challenge 7-9 Test Prep 50-5 Mixed Review 55-69 Homework To check students understanding of key skills and concepts, go over Exercises 0,, 7, 3, 36. Exercises 6 Remind students that only one counterexample is needed to prove a statement false. Error Prevention! Exercise If students think this is a good definition, ask: What other property do parallel lines have? They are coplanar. Remind students of the importance of looking for counterexamples. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Name Class Date Practice - Each conditional statement is true. Consider each converse. If the converse is true, combine the statements and write them as a biconditional.. If two angles have the same measure, then they are congruent.. If x - 5 =, then x = 8. 3. If n = 7, then ΔnΔ =7.. If a figure has eight sides, then it is an octagon. Write the two conditional statements that make up each biconditional. 5. A whole number is a multiple of 5 if and only if its last digit is either a 0 or a 5. 6. Two lines are perpendicular if and only if they intersect to form four right angles. 7. You live in Texas if and only if you live in the largest state in the contiguous United States. Explain why each of the following is not an acceptable definition. 8. An automobile is a motorized vehicle with four wheels. 9. A circle is a shape that is round. 0. The median of a set of numbers is larger than the smallest number in the set and smaller than the largest number in the set.. Cricket is a game played on a large field with a ball and a bat.. A rectangle is a very pleasing shape with smooth sides and very rigid corners. Some figures that are piggles are shown below, as are some nonpiggles. piggles nonpiggles L L Biconditionals and Definitions L3 L L3 Pearson Education, Inc. All rights reserved. EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A GO B Practice by Example for Help Example (page 87) Example (page 88) 9. See margin p. 9. Example 3 (page 89) 3. A line, segment, or ray is a perpendicular bisector of a segment if and only if it is perpendicular to the segment at its midpoint. Example (page 89) Apply Your Skills. No; a straight angle has a measure that is greater than 90, but it is not an obtuse angle. x x x 90 Chapter Reasoning and Proof Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional.. See back of book.. If two segments have the same length, then they are congruent.. Algebra If x =, then x - 5 = 9. 3. If a number is divisible by 0, then it is even.. Algebra If x = 3, then Δx«= 3. 5. In the United States, if it is July th, then it is Independence Day. 6. Algebra If x =-0, then x = 00. 5 6. See margin. Write the two statements that form each biconditional. 7 8. See margin. 7. A line bisects a segment 8. An integer is divisible by 00 if and only if the line intersects if and only if its last two digits the segment only at its midpoint. are zeros. 9. You live in Washington, D. C., 0. Two lines are parallel if and if and only if you live in the only if they are coplanar capital of the United States. and do not intersect.. Two angles are congruent if and. Algebra x = if and only if only if they have the same measure. x = or x =-. Test each statement below to see if it is reversible. If so, write it as a true biconditional. If not, write not reversible. 3. A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint. See left. x. Parallel planes are planes that do not intersect. Planes are parallel if and only if they do not intersect. 5. A Tarheel is a person who was born in North Carolina. not reversible 6. A rectangle is a four-sided figure with at least one right angle. not reversible 7. A midpoint of a segment is a point that divides a segment into two congruent segments. A point is a midpoint of a segment if and only if it divides the segment into two congruent segments. Is each statement below a good definition? If not, explain. 8 3. See margin 8. A cat is an animal with whiskers. 9. A dog is a good pet. p. 9. 0. A segment is part of a line.. Parallel lines do not intersect.. A square is a figure with two 3. An angle bisector is a ray that pairs of parallel sides. divides an angle into two congruent angles.. Language Arts Is the following a good definition? Explain. An obtuse angle is an angle whose measure is greater than 90. See left. 5. Open-Ended Choose a definition from a dictionary or from a glossary. Explain what makes the statement a good definition. See margin p. 9. 6. Writing Write a definition of a line parallel to a plane. A line is parallel to a plane if and only if it does not intersect the plane. Tell whether each of the following is a piggle. 3.. 5. 5. In the United States, if it is Independence Day, then it is July th. It is true. In the United States, 90 it is Independence Day if and only if it is July th. 6. If x 5 00, then x 50. It is false since x can also equal 0. 7. If a line bisects a segment, then the line intersects the segment only at its midpoint. If a line intersects a segment only at its midpoint, then it bisects the segment. 8. If an integer is divisible by 00, then its last two digits are zeros. If an integer s last two digits are zeros, then it is divisible by 00.
GO 8. No; / and / are not suppl. 9. Yes; / and / share a side and a vertex, and are suppl. 30. No; / and / do not share a vertex. 3. No; / and / do not share a side, and are not suppl. Real-World nline Homework Help GPS Visit: PHSchool.com Web Code: aue-00 Connection The five letters above form a word to think about. 7. Writing Use the figures below to write a good definition of linear pair. Answers may vary. Sample: Two angles are a linear pair if and only if they share a side 5 6 and a vertex 3 and are supplementary. 5 6 3 Linear pairs Not linear pairs Do angles and form a linear pair? Explain. (Hint: See Exercise 7.) 8. 9. 30. 3. 3. Multiple Choice Which conditional and its converse form a true biconditional? D If x. 0, then u x u. 0. If x 3 5 5, then x 5 5. If x 5 3, then x 5 9. If x 5 9, then x 3 5 35. The American Manual Alphabet For Exercises 33 37, use the chart below. Decide whether the description of each letter is a good definition. If not, provide a counterexample by giving another letter that could fit the definition. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 33. The letter D is formed by pointing straight up with the finger beside the thumb and folding the other fingers and the thumb so that they all touch. good definition 3. The letter K is formed by making a V with the two fingers beside the thumb. V is a counterexample. 35. You have formed the letter Y if and only if the thumb and one finger are extended up and the other fingers are folded into the palm of your hand. L is a counterexample. 36. You have formed the letter I if and only if the smallest finger is sticking up and the other fingers are folded into the palm of your hand with your thumb folded over them, and your hand is held still. good definition 37. You form the letter B by holding all four fingers tightly together and pointing them straight up while your thumb is folded into the palm of your hand. good definition Lesson - Biconditionals and Definitions 9 Teaching Tip Exercise 3 For answer choice C, students may forget that x = 9 has two possible solutions, 3 and -3. Exercises 33 37 If any students are adept at the American Manual Alphabet, have them demonstrate it for the rest of the class. Exercise 7 Point out that once a row or column contains two X s, the remaining box identifies the instrument the musician plays. Visual Learners Exercise 8 Seeing that a biconditional becomes a single circle in a Venn diagram can deepen students understanding and make abstract ideas concrete.. If x 5, then x 5 or x 5. If x 5 or x 5, then x 5. 8 3. Answers may vary. Samples are given. 8. No; it is not reversible; a mouse is a counterexample. 9. No; it is not reversible; a cat is a counterexample. 0. No; it is not precise; a ray or pt. could be part of a line.. No; it is not reversible; skew lines are not parallel.. No; it is not reversible; stop sign is a counterexample. 3. good definition 5. Answers may vary. Sample: An acute angle is an angle whose measure is between 0 and 90. The terms are clearly understood. It is precise and it is reversible. 9. If you live in Washington, D.C., then you live in the capital of the United States. If you live in the capital of the United States, then you live in Washington, D.C. 0. If two lines are parallel, then they are coplanar and do not intersect. If two lines are coplanar and do not intersect, then they are parallel.. If two angles are congruent, then they have the same measure. If two angles have the same measure, then they are congruent. 9
. Assess & Reteach Lesson Quiz. Write the converse of the statement. If it rains, then the car gets wet. If the car gets wet, then it rains.. Write the statement above and its converse as a biconditional. It rains if and only if the car gets wet. 3. Write the two conditional statements that make up the biconditional. Lines are skew if and only if they are noncoplanar. If lines are skew, then they are noncoplanar; if lines are noncoplanar, then they are skew. Is each statement a good definition? If not, find a counterexample.. The midpoint of a line segment is the point that divides the segment into two congruent segments. yes 5. A line segment is a part of a line. No; the statement is not reversible; a ray. Alternative Assessment Have students work with partners. Assign partners the task of writing good definitions for the terms hat and automobile. Then discuss the definitions as a class, encouraging students to critique and defend the definitions.. If la and lb are right angles, then la and lb are supplementary angles.. If la and lb are supplementary angles, then la and lb are right angles. 3. la and lb are right angles if and only if la and lb are supplementary angles. 9 Vocabulary Tip The expressions vice versa and conversely are synonyms. Real-World C Challenge Connection Careers Music educators are well-versed in both traditional and modern music. 9. Answers may vary. Sample: If the two hats in front of Alan were blue, he would know he was wearing red. Ben can tell from Alan s response that there are or red hats in front of Alan. Since Ben can t tell his hat color, Cal s hat must be red. 9 Chapter Reasoning and Proof. la and lb are supplementary angles if and only if la and lb are right angles. 5. la and lb are right angles if and only if la and lb are adjacent angles. Write each statement as a biconditional. Angles are congruent if and only 38. Congruent angles are angles with equal measure. if they have equal measure. 39. When the sum of the digits of an integer is divisible by 9, the integer is divisible by 9 and vice versa. The sum of the digits of an integer is divisible by 9, if and only if the integer is divisible by 9. 0. The whole numbers are the nonnegative integers. A number is a whole number if and only if it is a nonnegative integer. Reading Math Let statements p, q, and r, be as follows. p: la and lb are right angles. q: la and lb are supplementary angles. r: la and lb are adjacent angles. Substitute for p, q, and r, and write each statement the way you would read it.. p S q. q S p 3. p q. q p 5. p r 6. r q 6. See margin. 7. Reasoning In a band, Amy, Bob, and Carla are the drummer, guitarist, and keyboard player. Use the clues to find the instrument that each one plays. Carla and the drummer wear different-colored shirts. The keyboard player is older than Bob. Amy, the youngest band member, lives next door to the guitarist. You can solve this type of logic puzzle by eliminating possibilities. Copy the grid below. Put an X in a box once you eliminate it as a possibility. Amy plays the drums. Instrument Amy Bob Carla Bob plays Drums x x the guitar. Carla plays Guitar x x the keyboard. Keyboard x x 8. You have illustrated true conditional statements with Venn diagrams. You can do the same thing with true biconditionals. Consider the following statement. An integer is divisible by 0 if and only if its last digit is 0. a f. See margin pp. 9 93. a. Write the two conditional statements that make up this biconditional. b. Illustrate the first conditional from part (a) with a Venn diagram. c. Illustrate the second conditional from part (a) with a Venn diagram. d. Combine your two Venn diagrams from parts (b) and (c) to form a Venn diagram representing the biconditional statement. e. What must be true of the Venn diagram for any true biconditional statement? f. Reasoning How does your conclusion in part (e) help to explain why a good definition can be written as a biconditional? 9. Reasoning Alan, Ben, and Cal are seated as shown with their eyes closed. Diane places a hat on each of their heads from a box they know contains 3 red and blue hats. They open their eyes and look forward. Alan says, I cannot deduce what color hat I m wearing. Hearing that, Ben says, I cannot deduce what color I m wearing, either. Cal then says, I know what color I m wearing! How does Cal know the color of his hat? See left. 6. la and lb are adjacent angles if and only if la and lb are supplementary angles. 8.a. If an integer is divisible by 0, then its last digit is 0. If an integer s last digit is 0, then it is b c. Alan Ben Cal divisible by 0. with Last Digit of 0 Divisible by 0
GO Test Prep Multiple Choice Short Response Extended Response Mixed Review for Help Lesson - Lessons -5 and -6 Lesson -3 6. AB and BC 63. AB and CG 6. AB and CD lesson quiz, PHSchool.com, Web Code: aua-00 50. Which statement is a good definition? C A. Skew lines are lines that do not intersect. B. Parallel lines are lines that do not intersect. C. A square is a rectangle with four congruent sides. D. Right angles are angles formed by two intersecting lines. 5. Which statement is NOT true? G F. If two lines are parallel, then they lie in one plane and do not intersect. G. Two lines lie in one plane if and only if the lines are parallel. H. If two coplanar lines do not intersect, then the lines are parallel. J. Two lines lie in one plane and do not intersect if and only if the two lines are parallel. 5. Which statement is NOT true? B A. If x =, then x =. B. If x =, then x =. C. If x =-, then x =. D. x = if and only if x = or x =-. 53. Write the two conditionals that form this biconditional: See margin. You can go to the movies if and only if you do your homework. 5. Here is a true conditional statement: If a person is 8 years old, that person is old enough to vote. a. Write the converse. a c. See margin. b. Determine whether the converse is true or false. c. If the converse is false, give a counterexample to show that it is false. If the converse is true, combine the original statement and its converse by writing a biconditional. Write each statement as a conditional. 55. Whole numbers that end in zero are even. If a whole number ends in 0, then it is even. 56. When x =-5, x = 5. If x 5, then x 5. 57. Sunday is a weekend day. If a day is Sunday, then it is a weekend day. 58. All prime numbers greater than are odd. If a prime number is greater than, then it is odd. 59. Draw a segment XY. Construct a bisector of XY. See margin. 60. Draw an acute angle, &. Construct an angle congruent to &. See margin. 6. Draw an obtuse angle, &CAD. Construct the bisector of &CAD. See back of book. Use the figure at the right to name each of the following. 6 69. Answers may vary. Samples are given. 6. two intersecting lines 63. two skew lines 6. two parallel lines 65. two parallel planes ABC, EFG 66. three coplanar points 67. two intersecting planes A, B, C AEF, BFG 68. a plane that contains H EFG g 69. the intersection of two planes BC D C Lesson - Biconditionals and Definitions 93 A H E B F G Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 6 Test-Taking Strategies with Transparencies circles of its Venn diagram can be the hypothesis of a conditional, and the other can be the conclusion. 53. [] If you can go to the movies, then you did your homework. If you do your homework, then you can go to the movies. [] just one of the conditionals 5. [] a. If a person is old enough to vote, then that person is 8 years old. b. false c. A 0-year-old is a counterexample. A 0-year-old is old enough to vote, but is not 8 years old. (OR equivalent conditionals) [3] predominantly correct but with one error [] at least one correct answer, and some appropriate information for one other part [] some correct information 59. X Y Divisible by 0 with Last Digit of 0 d. e. Answers may Divisible by 0 vary. Sample: The two with circles Last Digit of 0 coincide. f. Answers may vary. Sample: A good definition may be written as a biconditional because either of the coinciding 60. Line < bisects XY. l Ol 93