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LESSON 16.4 Reasoning and Proof Name Class Date 16.4 Reasoning and Proof HARDBOUND SE PAGE 647 BEGINS HERE Essential Question: How do you go about proving a statement? Common Core Math Standards The student is expected to: Resource Locker G-CO.9 Explore Prove theorems about lines and angles. Also A-REI.A.1 Exploring Inductive and Deductive Reasoning A conjecture is a statement that is believed to be true. You can use reasoning to investigate whether a conjecture is true. Inductive reasoning is the process of reasoning that a rule or statement may be true by looking at specific cases. Deductive reasoning is the process of using logic to prove whether all cases are true. Mathematical Practices MP.8 Patterns Language Objective Complete the steps to make a conjecture about the sum of three consecutive counting numbers. Have students fill in a chart explaining the meaning of conditionals, counterexample and conditional. A Write a sum to represent the first three consecutive counting numbers, starting with 1. B Is the sum divisible by 3? Essential Question: How do you go about proving a statement? C Write the sum of the next three consecutive counting Possible answer: You can make a conjecture, or statement, that you believe is true. Then through inductive or deductive reasoning, you can prove the statement is true by showing specific cases are true or by using logical steps. PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo. Discuss why the figure seems to be impossible. Then preview the Lesson Performance Task. © Houghton Mifflin Harcourt Publishing Company ENGAGE 1+2+3 Yes. 1 + 2 + 3 = 6 and 6 ÷ 3 = 2. numbers, starting with 2. 2+ 3+4 D Is the sum divisible by 3? Yes. 2 + 3 + 4 = 9 and 9 ÷ 3 = 3. E Complete the conjecture: The sum of three consecutive counting numbers is divisible by 3 . Recall that postulates are statements you accept are true. A theorem is a statement that you can prove is true using a series of logical steps. The steps of deductive reasoning involve using appropriate undefined words, defined words, mathematical relationships, postulates, or other previously-proven theorems to prove that the theorem is true. Use deductive reasoning to prove that the sum of three consecutive counting numbers is divisible by 3. F Let the three consecutive counting numbers be represented by n, n + 1, and G The sum of the three consecutive counting numbers can be written as 3n + Module 16 be ges must EDIT--Chan DO NOT Key=NL-D;CA-D Correction 3 . Lesson 4 815 gh "File info" made throu n+2 . Date Class nin 16.4 Reaso ion: How Quest Essential G-CO.9 Prove do you go theorems Explore IN1_MNLESE389762_U7M16L4.indd 815 of g and Pro Name HARDCOVER PAGES 647656 ent? g a statem about provin about lines Resource Locker .1 . 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Recall that ers using a series , defined words ing numb that the theore words prove is true cutive count undefined ms to prove conse riate theore approp of three usly-proven that the sum or other previo ning to prove +2 . tive reaso and n Use deduc n, n + 1, by 3. ented by is divisible be repres numbers . 3 counting consecutive n as 3n + Let the three can be writte numbers counting consecutive of the three The sum © Houghto n Mifflin Harcour t Publishin y g Compan Lesson 4 815 Module 16 6L4.indd 62_U7M1 ESE3897 IN1_MNL 815 Lesson 16.4 815 2/26/16 3:31 AM 2/26/16 3:30 AM ( n+1 ). H The expression 3n + 3 can be factored as 3 I The expression 3(n + 1) is divisible by J Recall the conjecture in Step E: The sum of three consecutive counting numbers is divisible by 3. EXPLORE for all values of n. 3 Look at the steps in your deductive reasoning. Is the conjecture true or false? HARDBOUND SE PAGE 648 BEGINS HERE True QUESTIONING STRATEGIES Reflect 1. When a detective solves a case, is the detective more likely to use inductive or deductive reasoning? Explain. deductive reasoning, because the solution is likely based on logical conclusions drawn from the evidence Discussion A counterexample is an example that shows a conjecture to be false. Do you think that counterexamples are used mainly in inductive reasoning or in deductive reasoning? Possible answer: A counterexample would be used in inductive reasoning to show that at least one specific case makes the conjecture false. 2. Exploring Inductive and Deductive Reasoning Suppose you use deductive reasoning to show that an angle is not acute. Can you conclude that the angle is obtuse? Explain. No; if the angle is not acute, I can conclude that it is right, obtuse, or straight. Explain 1 When might you want to make a conjecture about a set of numbers? If the numbers seem to form a pattern, you might want to make a conjecture based on the number pattern. Is one counterexample enough to prove that a conjecture is false? Explain. Yes, the conjecture must be true for every case. So, if even one counterexample exists, the conjecture is false. Introducing Proofs A conditional statement is a statement that can be written in the form “If p, then q” where p is the hypothesis and q is the conclusion. For example, in the conditional statement “If 3x - 5 = 13, then x = 6,” the hypothesis is “3x - 5 = 13” and the conclusion is “x = 6 .” Properties of Equality Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality If a = b, then ac = bc. Division Property of Equality If a = b and c ≠ 0, then __ac = _bc . Reflexive Property of Equality a=a Symmetric Property of Equality If a = b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Substitution Property of Equality If a = b, then b can be substituted for a in any expression. Module 16 816 © Houghton Mifflin Harcourt Publishing Company Most of the Properties of Equality can be written as conditional statements. You can use these properties to solve an equation like “3x - 5 = 13” to prove that “x = 6 .” INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss why a conjecture is like a hypothesis in the scientific method. Elicit that a conjecture, like a hypothesis, is often based on inductive reasoning. EXPLAIN 1 Introducing Proofs Lesson 4 AVOID COMMON ERRORS PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M16L4 816 Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.3, which calls for students to “construct viable arguments.” Students use deductive reasoning, and explain steps logically from definite premises to a definite general conclusion. They use inductive reasoning to make a conjecture about what is true in general by examining several cases, and they justify the falsehood of a conclusion by citing a counterexample. 4/19/14 12:06 PM Students may have difficulty identifying the correct property of equality to justify a step when solving an algebraic equation. Students may need to include steps where they show the property of equality to help them recognize how it is applied. For example, they may need to show the step where the value is added to both sides of the equation to apply the Addition Property of Equality. Reasoning and Proof 816 Will changing the order of the hypothesis and the conclusion in a true conditional statement change whether or not the statement is true? Explain. Yes, the statement may still be true but you would have to prove that it is. Do you always use deductive reasoning when you solve an equation algebraically? Explain. Yes, you should always be able to support each step in the solution using a property. Use deductive reasoning to solve the equation. Use the Properties of Equality to justify each step. Example 1 QUESTIONING STRATEGIES 14 = 3x - 4 14 = 3x - 4 HARDBOUND SE PAGE 649 18 = 3x Addition Property of Equality 6=x Division Property of Equality x= 6 Symmetric Property of Equality 9 = 17 - 4x BEGINS HERE 9 = 17 - 4x Subtraction Property of Equality 9 - 17 = -4x -8 = -4x Division =x 2 x= 2 Property of Equality Symmetric Property of Equality Your Turn Write each statement as a conditional. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital Vision/Getty Images 3. 4. All zebras belong to the genus Equus. If an animal is a zebra, then it belongs to the genus Equus. The bill will pass if it gets two-thirds of the vote in the Senate. If the bill gets two-thirds of the vote in the Senate, then it will pass. 5. Use deductive reasoning to solve the equation 3 - 4x = -5. 3 - 4x = -5 -4x = -8 x=2 6. Subtraction Property of Equality Division Property of Equality Identify the Property of Equality that is used in each statement. If x = 2, then 2x = 4. Multiplication Property of Equality 5 = 3a; therefore, 3a = 5 . Symmetric Property of Equality If T = 4, then 5T + 7 equals 27. Substitution Property of Equality If 9 = 4x and 4x = m, then 9 = m. Transitive Property of Equality Module 16 817 Lesson 4 COLLABORATIVE LEARNING IN1_MNLESE389762_U7M16L4 817 Small Group Activity Have students work in small groups. The first student writes a number or draws a figure. The next student writes or draws a second item, beginning a pattern. Have them continue until each student has contributed to the pattern. Then ask the first student to describe a rule for the pattern. Have the groups repeat this activity until each student has gone first. 817 Lesson 16.4 4/19/14 12:06 PM Explain 2 Using Postulates about Segments and Angles EXPLAIN 2 Recall that two angles whose measures add up to 180° are called supplementary angles. The following theorem shows one type of supplementary angle pair, called a linear pair. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. You will prove this theorem in an exercise in this lesson. Using Postulates about Segments and Angles The Linear Pair Theorem If two angles form a linear pair, then they are supplementary. 4 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Have students identify the property of 3 m∠3 + m∠4 = 180° You can use the Linear Pair Theorem, as well as the Segment Addition Postulate and Angle Addition Postulate, to find missing values in expressions for segment lengths and angle measures. Example 2 HARDBOUND SE PAGE 650 BEGINS HERE equality used as they complete each step to solve for the variable using the postulates about segments and angles. Use a postulate or theorem to find the value of x in each figure. Given: RT = 5x - 12 x+ 2 R QUESTIONING STRATEGIES 3x - 8 S 5x - 12 How are the segment and angle addition postulates applied to solve for a variable? Set the sum of the non-overlapping segments or angles equal to the measure of the whole segment or angle. T Use the Segment Addition Postulate. RS + ST = RT (x + 2) + (3x - 8) = 5x - 12 6=x x=6 Module 16 AVOID COMMON ERRORS © Houghton Mifflin Harcourt Publishing Company 4x - 6 = 5x - 12 818 When solving equations using the Segment or Angle Addition Postulates, students may forget to combine like terms or use inverse operations to solve. Review how to combine like terms and use inverse operations as needed. Lesson 4 DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M16L4 818 Kinesthetic Experience 4/19/14 12:06 PM Have students act out the Reflexive, Symmetric, and Transitive Properties. For the Reflexive Property, have students look in a mirror. For the Symmetric Property, have two students stand next to each other and then change places. For the Transitive Property, have one student give a second student a sheet of paper, and have the second student give the paper to a third. The result is the same as if the first student had given the paper directly to the third. Reasoning and Proof 818 Given: m∠RST = (15x - 10)° B R P (x + 25)° (5x + 10)° S T Use the Angle Addition Postulate. m∠RST = m∠ RSP + m∠ PST (15x − 10)° = (x + 25) ° + (5x + 10)° 15x − 10 = (6x + 35) 9 x= 45 x= 5 Reflect 7. Discussion The Linear Pair Theorem uses the terms opposite rays as well as adjacent angles. Write a definition for each of these terms. Compare your definitions with your classmates. Possible answers: Opposite rays are rays that share a common endpoint and form a line. HARDBOUND SE PAGE 651 BEGINS HERE © Houghton Mifflin Harcourt Publishing Company Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. Your Turn 8. Two angles LMN and NMP form a linear pair. The measure of ∠LMN is twice the measure of ∠NMP. Find m∠LMN. Use the Linear Pair Theorem. Substitute for m∠LMN. m∠LMN + m∠NMP = 180° (2 ⋅ m∠NMP) + m∠NMP = 180° 3 ⋅ m∠NMP = 180° m∠NMP = 60° m∠LMN = 2 ⋅ m∠NMP = 2 ⋅ 60° = 120° Module 16 819 Lesson 4 LANGUAGE SUPPORT IN1_MNLESE389762_U7M16L4 819 Connect Vocabulary For the properties of equality based on the operations, have students highlight the operation to connect to the corresponding property. For the Reflexive, Symmetric, and Transitive Properties, discuss what these words bring to mind. For example, reflexive might remind students of a reflection in a mirror. You see the same thing on both sides of a mirror, so, a = a. 819 Lesson 16.4 15/09/14 11:54 PM Explain 3 Using Postulates about Lines and Planes EXPLAIN 3 Postulates about points, lines, and planes help describe geometric figures. Postulates about Points, Lines, and Planes Using Postulates about Lines and Planes Through any two points, there is exactly one line. COOPERATIVE LEARNING Through any three noncollinear points, there is exactly one plane containing them. To help students understand the rationale behind the postulates about points, lines, and planes, ask them to draw additional examples with the points and lines in different locations to demonstrate each postulate. Share the drawings with the class. If two points lie in a plane, then the line containing those points lies in the plane. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line. © Houghton Mifflin Harcourt Publishing Company Module 16 IN1_MNLESE389762_U7M16L4 820 820 Lesson 4 4/19/14 12:06 PM Reasoning and Proof 820 QUESTIONING STRATEGIES HARDBOUND SE Example 3 PAGE 652 BEGINS HERE Use each figure to name the results described. Must any two planes intersect? Why or why not? Name planes in the classroom that support your answer. No, if the planes are parallel they will never intersect. Possible example: opposite walls in the classroom A B E C D If a line lies in a plane, how many points of intersection do the line and the plane have? an infinite number: every point that lies on the line Description VISUAL CUES Students may have difficulty interpreting the diagrams showing intersecting planes. Make a slit in the side of one piece of paper and hold the paper in a horizontal plane. Slide a second sheet in a vertical plane perpendicular to the horizontal plane to provide students with a visual demonstration of two intersecting planes. Locate points and lines as needed. Example from the figure the line of intersection of two planes Possible answer: The two planes intersect in line BD. the point of intersection of two lines The line through point A and the line through point B intersect at point C. three coplanar points Possible answer: The points B, D, and E are coplanar. three collinear points The points B, C, and D are collinear. m F H © Houghton Mifflin Harcourt Publishing Company J Description Module 16 IN1_MNLESE389762_U7M16L4 821 821 Lesson 16.4 ℓ G Example from the figure the line of intersection of two planes Possible answer: The two planes intersect in line JF. the point of intersection of two lines The line through point F and the line through point H intersect at point J. three coplanar points Possible answer: The points F, J, and H are coplanar. three collinear points The points F, J, and G are collinear. 821 Lesson 4 4/19/14 12:06 PM HARDBOUND SE Reflect 9. PAGE 653 BEGINS HERE Find examples in your classroom that illustrate the postulates of lines, planes, and points. Possible answers: walls, floors, corners, desktops, blackboard INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Point out that students are starting to build a 10. Draw a diagram of a plane with three collinear points and three points that are noncollinear. Possible answer: in the figure, B, C, and A are collinear; D, C, and A are noncollinear. B D C ELABORATE A catalog of definitions, postulates, and theorems about geometric relationships that they will use throughout the course. Discuss why it is important to become familiar with using them as tools in deductive reasoning. Elaborate 11. What is the difference between a postulate and a definition? Give an example of each. Possible answers: A postulate is a statement that is self-evident or is generally accepted to be a true statement. A definition is a statement that explains the meaning of a word in terms of previously accepted words or statements. QUESTIONING STRATEGIES Possible examples: How can you use a Property of Equality to write the equation 4x = 8 as a conditional statement? Using the Division Property of Equality, x = 2. The conditional statement is “If 4x = 8, then x = 2.” Postulate: x = x is called the Reflexive Property of Equality. Definition: An even number is a number that is divisible by 2. 12. Give an example of a diagram illustrating the Segment Addition Postulate. Write the Segment Addition Postulate as a conditional statement. R S T If S is between R and T, then RS + ST = RT. CONNECT VOCABULARY © Houghton Mifflin Harcourt Publishing Company 13. Explain why photographers often use a tripod when taking pictures. Through any three noncollinear points, there is only one plane, so the feet of the tripod are all always flat against the plane of the ground, which steadies the camera. 14. Essential Question Check-In What are some of the reasons you can give in proving a statement using deductive reasoning? You can use given facts, definitions, postulates or properties, and Connect conditionals in geometry to conditional statements in everyday life. For example: “If I walk in the rain without an umbrella, then I will get wet.” Have students express their own conditional statements tied to reality so that they can see this connection. previously-proven theorems. Module 16 IN1_MNLESE389762_U7M16L4 822 SUMMARIZE THE LESSON 822 Lesson 4 4/19/14 12:06 PM How can you use deductive reasoning to establish a conclusion? Provide a logical sequence of statements supported by postulates and established theorems in which each statement logically follows from the preceding statement up to the conclusion. Reasoning and Proof 822 Evaluate: Homework and Practice EVALUATE • Online Homework • Hints and Help • Extra Practice Explain why the given conclusion uses inductive reasoning. 1. Find the next term in the pattern: 3, 6, 9. The next term is 12 because the previous terms are multiples of 3. The conclusion is based on observing three numbers. 2. ASSIGNMENT GUIDE 3 + 5 = 8 and 13 + 5 = 18, therefore the sum of two odd numbers is an even number. The conclusion is based on two examples. Concepts and Skills Practice Explore Exploring Inductive and Deductive Reasoning Exercises 1–12 Example 1 Introducing Proofs Exercises 13–16 Example 2 Using Postulates about Segments and Angles Exercises 17–20 Example 3 Using Postulates about Lines and Planes Exercises 21–26 3. My neighbor has two cats and both cats have yellow eyes. Therefore when two cats live together, they will both have yellow eyes. The conclusion is based on two observations. 4. The conclusion is based on a limited number of observations. Give a counterexample for each conclusion. 5. If x is a prime number, then x + 1 is not a prime number. Counterexample: 2; if x = 2, then x + 1 = 3, which is a prime number. 6. The difference between two even numbers is positive. Counterexample: 6 − 10 = −4, which is negative. 7. Points A, B, and C are noncollinear, so therefore they are noncoplanar. When I draw three points that are noncollinear, I can draw a single plane through all three points, so they are coplanar after all. Have students compare counterexamples used to demonstrate conjectures that are not true. GRAPHIC ORGANIZERS HARDBOUND SE PAGE 654 BEGINS HERE © Houghton Mifflin Harcourt Publishing Company COMMUNICATING MATH Students work in pairs to complete a chart. The chart has three columns and each column is labeled with the highlighted vocabulary. Students must discuss in depth the meaning of each word, and take notes on their discussion. Then they write down their agreed-upon ideas under the word in each column. It always seems to rain the day after July 4th. 8. The square of a number is always greater than the number. 1 __ 1 The square of __ is 19 , which is less than __ . 3 3 In Exercises 9–12 use deductive reasoning to write a conclusion. 9. If a number is divisible by 2, then it is even. The number 14 is divisible by 2. The number 14 is an even number. Module 16 Exercise IN1_MNLESE389762_U7M16L4 823 Lesson 16.4 Depth of Knowledge (D.O.K.) Mathematical Practices 1–6 1 Recall of Information MP.3 Logic 7–8 2 Skills/Concepts MP.3 Logic 9–16 1 Recall of Information MP.3 Logic 17–20 1 Recall of Information MP.2 Reasoning 21–22 1 Recall of Information MP.3 Logic 23–24 2 Skills/Concepts MP.3 Logic 3 Strategic Thinking MP.3 Logic 25 823 Lesson 4 823 4/19/14 12:06 PM Use deductive reasoning to write a conclusion. AVOID COMMON ERRORS 10. If two planes intersect, then they intersect in exactly one line. Planes ℜ and ℑ intersect. Students may have difficulty identifying the hypothesis and the conclusion given a conditional statement when then is missing from the statement. Suggest that students first rewrite the statement in the “if, . . . then” form before they identify the hypothesis and conclusion. Planes ℜ and ℑ intersect in exactly one line. 11. Through any three noncollinear points, there is exactly one plane containing them. Points W, X, and Y are noncollinear. There is exactly one plane containing points W, X, and Y. 12. If the sum of the digits of an integer is divisible by 3, then the number is divisible by 3. The sum of the digits of 46,125 is 18, which is divisible by 3. The number 46,125 is divisible by 3. AVOID COMMON ERRORS Identify the hypothesis and conclusion of each statement. Remind students to combine like terms when they solve equations by applying the Segment and Angle Addition Postulates. 13. If the ball is red, then it will bounce higher. Hypothesis: the ball is red Conclusion: it will bounce higher 14. If a plane contains two lines, then they are coplanar. Hypothesis: a plane contains two lines Conclusion: the lines are coplanar INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 After students solve for the variable using the 15. If the light does not come on, then the circuit is broken. Hypothesis: the light does not come on Conclusion: the circuit is broken 16. You must wear your jacket if it is cold outside. Hypothesis: it is cold outside Conclusion: you must wear your jacket Segment and Angle Addition Postulates, have them use the value to find the actual lengths or angle measures represented by the expressions. They can also use this method to check their work. Use a definition, postulate, or theorem to find the value of x in the figure described. 17. Point E is between points D and F. If DE = x - 4, EF = 2x + 5, and DF = 4x − 8, find x. © Houghton Mifflin Harcourt Publishing Company Use the Segment Addition Postulate; DE + EF = DF; (x − 4) + (2x + 5) = 4x − 8; 3x + 1 = 4x − 8; 9 = x _ 18. Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 2x + 1, find x. ― Because Y is the midpoint of XZ, XY = YZ. Use this fact and the Segment Addition Postulate; XY + YZ = XZ; (2x + 1) + (2x + 1) = 8x − 2; 4x + 2 = 8x − 2; 4 = 4x; 1 = x → ‾ is an angle bisector of ∠RST. If m∠RSV = (3x + 5)° and m∠RST = (8x − 14)°, find x. 19. SV → ‾ is an angle bisector of ∠RST, m∠RSV = m∠VST. Use this fact and Because SV the Angle Addition Postulate; m∠RSV + m∠VST = m∠RST; (3x + 5) + (3x + 5) = 8x - 14; 6x + 10 = 8x - 14; 24 = 2 x; 12 = x 20. ∠ABC and ∠CBD are a linear pair. If m∠ABC = m∠CBD = 3x - 6, find x. Use the Linear Pair Theorem.; m∠ABC + m∠CBD = 180°; (3x - 6)°+ (3x - 6)° = 180°; 6x - 12 = 180; 6x = 192; x = 32 Module 16 Exercise IN1_MNLESE389762_U7M16L4 824 Lesson 4 824 Depth of Knowledge (D.O.K.) Mathematical Practices 26 2 Skills/Concepts MP.3 Logic 27 3 Strategic Thinking MP.3 Logic 28 2 Skills/Concepts MP.6 Precision 15/09/14 11:59 PM Reasoning and Proof 824 Use the figure for Exercises 21 and 22. PEERTOPEER DISCUSSION 21. Name three collinear points. Have students use a ruler to draw a line segment made from two non-overlapping segments. Have them label the lengths of one section and the total segment. Then have them exchange papers and explain how to use deductive reasoning to find the missing length. S Possible answer: P, R, and T P R 22. Name two linear pairs. Possible answers: ∠PRQ and ∠QRT, ∠STR and ∠UTR Q T U Explain the error in each statement. 23. Two planes can intersect in a single point. When two planes cross, they intersect each other at an infinite number of points, i.e., in a line. JOURNAL 24. Three points have to be collinear. Have students write examples of the following: a conjecture, a counterexample to a conjecture, inductive reasoning, and deductive reasoning. The three points could be the vertices of a triangle. 25. A line is contained in exactly one plane A line can be in more than one plane. 26. If x 2 = 25, then x = 5. The value of x could also be -5. HARDBOUND SE H.O.T. Focus on Higher Order Thinking PAGE 655 © Houghton Mifflin Harcourt Publishing Company BEGINS HERE 27. Analyze Relationships What is the greatest number of intersection points 4 coplanar lines can have? What is the greatest number of planes determined by 4 noncollinear points? Draw diagrams to illustrate your answers. Four coplanar lines can intersect in up to 6 points. Module 16 IN1_MNLESE389762_U7M16L4 825 825 Lesson 16.4 825 Up to four planes can be determined by 4 noncollinear points. Lesson 4 4/19/14 12:06 PM 28. Justify Reasoning Prove the Linear Pair Theorem. Given: ∠MJK and ∠MJL are a linear pair of angles. Prove: ∠MJK and ∠MJL are supplementary. LANGUAGE SUPPORT M K Complete the proof by writing the missing reasons. Choose from the following reasons. J Angle Addition Postulate Definition of linear pair Substitution Property of Equality Given Statements A conjecture is an opinion or proposition that is supported by evidence but has not been proven. It begins with an observation, such as: “I added twenty pairs of odd numbers, and the sums were all even.” The conjecture based on this might be, “The sum of two odd numbers is always even.” A conjecture doesn’t have to be true. For example: “I saw eleven kids with apples at lunch today, and all the apples were red. Conjecture: All apples are red.” L Reasons 1. ∠MJK and ∠MJL are a linear pair. → → ‾ are opposite rays. ‾ and JK 2. JL → → ‾ and JK ‾ form a straight line. 3. JL 1. Given 4. m∠LJK = 180° 4. Definition of straight angle 5. m∠MJK + m∠MJL = m∠LJK 5. Angle Addition Postulate 6. m∠MJK + m∠MJL = 180° 6. Substitution Property of Equality 7. ∠MJK and ∠MJL are supplementary. 7. Definition of supplementary angles 2. Definition of linear pair 3. Definition of opposite rays INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to state a possible conjecture for Lesson Performance Task HARDBOUND SE PAGE 656 BEGINS HERE If two planes intersect, then they intersect in exactly one line. Find a real-world example that illustrates the postulate above. Then formulate a conjecture by completing the following statement: . Justify your conjecture with real-world examples or a drawing. Possible example: a wall and the ceiling Students may make any of the following conjectures: If three planes intersect, then they intersect in a point. If three planes intersect, then they intersect in a line. If three planes intersect, then they intersect in either a point or a line. Obviously the third conjecture is the most complete. Possible examples: two walls and the ceiling intersect at a point; the pages of a book are planes that intersect in a line, the spine of the book. Module 16 826 “I’ve seen thousands of creatures with wings and all of them could fly.” Conjecture: If a creature has wings, then it can fly. Refute: Penguins and ostriches have wings but cannot fly. “I’ve seen hundreds of bicycles and all of them had two wheels.” Conjecture: If a vehicle is a bicycle, then it has two wheels. Support: The prefix bimeans two, and cycle refers to a wheel. So, the definition of bicycle includes the requirement that the vehicle have two wheels. © Houghton Mifflin Harcourt Publishing Company If three planes intersect, then the statement, then support or refute it: Lesson 4 EXTENSION ACTIVITY IN1_MNLESE389762_U7M16L4 826 The “impossible” triangle in the photo in the online Engage activity is an example of an optical illusion. Have students research optical illusions and either sketch or print ones that especially intrigue them. For each illusion, students should describe what’s intriguing about it, and then explain—if they can!—how the illusion is accomplished. 4/19/14 12:06 PM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Reasoning and Proof 826