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Unit Proof, Parallel and 1 Perpendicular Lines Unit Overview In this unit you will begin the study of an axiomatic system, Geometry. You will investigate the concept of proof and discover the importance of proof in mathematics. You will extend your knowledge of the characteristics of angles and parallel and perpendicular lines and explore practical applications involving angles and lines. Essential Questions ? Why are properties, postulates, and theorems important in mathematics? ? How are angles and parallel and perpendicular lines used in real-world settings? © 2010 College Board. All rights reserved. Academic Vocabulary Add these words and others you encounter in this unit to your Math Notebook. angle bisector midpoint of a segment complementary angles parallel conditional statement perpendicular congruent postulate conjecture proof counterexample supplementary angles deductive reasoning theorem inductive reasoning EMBEDDED ASSESSMENTS These assessments, following Activities 1.4, 1.7, and 1.9, will give you an opportunity to demonstrate what you have learned about reasoning, proof, and some basic geometric figures. Embedded Assessment 1 Conditional Statements and Logic p. 37 Embedded Assessment 2 Angles and Parallel Lines p. 65 Embedded Assessment 3 Slope, Distance, and Midpoint p. 81 1 UNIT 1 Getting Ready 1. Solve each equation. a. 5x - 2 = 8 b. 4x - 3 = 2x + 9 c. 6x + 3 = 2x + 8 6. Give the characteristics of a rectangular prism. 7. Draw a right triangle and label the hypotenuse and legs. 8. Use a protractor to find the measure of each angle. a. 2. Graph y = 3x - 3 and label the x- and y-intercepts. 3. Tell the slope of a line that contains the points (5, -3) and (7, 3). 1 4. Write the equation of a line that has slope __ 3 and y-intercept 4. b. 5. Write the equation of the line graphed below. 10 y 8 c. 6 4 2 –10 –8 –6 –4 –2 –2 2 4 6 8 10 x –4 –6 –8 © 2010 College Board. All rights reserved. –10 2 SpringBoard® Mathematics with Meaning™ Geometry Geometric Figures What’s My Name? ACTIVITY 1.1 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Group Presentation Below are some types of figures you have seen in earlier mathematics courses. Describe each figure. Using geometric terms and symbols, list as many names as possible for each figure. 1. Q 2. F 3. X Y 5. G 4. D Z m 6. E J K © 2010 College Board. All rights reserved. 7. H χ P T N A D B C TRY THESE A Identify each geometric figure. Then use symbols to write two different names for each. a. Q P b. N C Unit 1 • Proof, Parallel and Perpendicular Lines 3 ACTIVITY 1.1 Geometric Figures continued What’s My Name? SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Discussion Group, Self/Peer Revision My Notes 8. Draw four angles with different characteristics. Describe each angle. Name the angles using numbers and letters. 9. Compare and contrast each pair of angles. When you compare and contrast two figures, you describe how they are alike or different. 4 a. b. 1 SpringBoard® Mathematics with Meaning™ Geometry A 2 D B C © 2010 College Board. All rights reserved. MATH TERMS Geometric Figures ACTIVITY 1.1 continued What’s My Name? SUGGESTED LEARNING STRATEGIES: Discussion Group, Peer/Self Revision My Notes c. R D 60° 30° E d. F D T 150° X 30° E S F Y Z 10. a. The figure below shows two intersecting lines. Name two angles that are supplementary to ∠4. b. Explain why the angles you named in part a must have the same measure. 3 4 5 © 2010 College Board. All rights reserved. 6 Unit 1 • Proof, Parallel and Perpendicular Lines 5 ACTIVITY 1.1 Geometric Figures continued What’s My Name? SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge My Notes TRY THESE B Complete the chart by naming all the listed angle types in each figure. C 8 7 10 9 A D B E F Acute angles Obtuse angles Angles with the same measure Supplementary angles 11. In the circle below, draw and label each geometric term. a. Radius OA b. Chord BA c. Diameter CA 12. Refer to your drawings in circle above. What is the geometric term for point O? 6 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. Complementary angles Geometric Figures ACTIVITY 1.1 continued What’s My Name? SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge My Notes 13. In the space below, draw a circle with center P and radius 1 in. Locate a point B PQ = 1 in. Locate a point A so that PA = 1__ 2 3 in. so that PB = __ 4 14. Use your diagram to complete these statements. a. A lies _________________ the circle because ___________________________________. b. B lies _________________ the circle because ___________________________________. © 2010 College Board. All rights reserved. 15. In your diagram above, draw circles with radii PA and PB. These three circles are called _______________________. Unit 1 • Proof, Parallel and Perpendicular Lines 7 ACTIVITY 1.1 Geometric Figures continued What’s My Name? My Notes TRY THESE C Classify each segment in circle O. Use all terms that apply. ___ AF : ________________________ ___ A BO : ________________________ ___ B C CO : ________________________ ___ DO : ________________________ O ___ EO : ________________________ ___ CE : ________________________ F E D CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show Write paper. Show youryour work. 6. Compare and contrast the terms acute angle, work. obtuse angle, right angle, and straight angle. 2. Describe all acceptable ways to name a plane. 3. Compare and contrast collinear and coplanar points. 4. What are some acceptable ways to name an angle? 5. a. Why is there a problem with using ∠B to name the angle below? b. How many angles are in the figure? A C D B 8 SpringBoard® Mathematics with Meaning™ Geometry 7. The measure of ∠A is 42°. a. What is the measure of an angle that is complementary to ∠A? b. What is the measure of an angle that is supplementary to ∠A? 8. Draw a circle P. a. Draw a segment that has one endpoint on the circle but is not a chord. b. Draw a segment that intersects the circle in two points, contains the center, but is not a radius, diameter, or chord. 9. MATHEMATICAL How can you use the R E F L E C T I O N figures in this activity to describe real-world objects and situations? Give examples. © 2010 College Board. All rights reserved. 1. Describe all acceptable ways to name a line. Logical Reasoning Riddle Me This ACTIVITY 1.2 SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Look for a Pattern My Notes The ability to recognize patterns is an important aspect of mathematics. But when is an observed pattern actually a real pattern that continues beyond just the observed cases? In this activity, you will explore patterns and check to see if the patterns hold true beyond the observed cases. Inductive reasoning is the process of observing data, recognizing patterns, and making a generalization. This generalization is a conjecture. 1. Five students attended a party and ate a variety of foods. Something caused some of them to become ill. JT ate a hamburger, pasta salad, and coleslaw. She became ill. Guy ate coleslaw and pasta salad but not a hamburger. He became ill. Dean ate only a hamburger and felt fine. Judy didn’t eat anything and also felt fine. Cheryl ate a hamburger and pasta salad but no coleslaw, and she became ill. Use inductive reasoning to make a conjecture about which food probably caused the illness. ACADEMIC VOCABULARY conjecture inductive reasoning 2. Use inductive reasoning to make a conjecture about the next two terms in each sequence. Explain the pattern you used to determine the terms. © 2010 College Board. All rights reserved. a. A, 4, C, 8, E, 12, G, 16, b. 3, 9, 27, 81, 243, , , CONNECT TO HISTORY c. 3, 8, 15, 24, 35, 48, d. 1, 1, 2, 3, 5, 8, 13, , , The sequence of numbers in Item 2d is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics. Unit 1 • Proof, Parallel and Perpendicular Lines 9 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Group Presentation My Notes 3. Use the four circles on the next page. From each of the given points, draw all possible chords. These chords will form a number of non-overlapping regions in the interior of each circle. For each circle, count the number of these regions. Then enter this number in the appropriate place in the table below. Number of Points on the Circle 2 3 4 5 Number of Non-Overlapping Regions Formed 4. Look for a pattern in the table above. a. Describe, in words, any patterns you see for the numbers in the column labeled Number of Non-Overlapping Regions Formed. © 2010 College Board. All rights reserved. b. Use the pattern that you described to predict the number of non-overlapping regions that will be formed if you draw all possible line segments that connect six points on a circle. 10 SpringBoard® Mathematics with Meaning™ Geometry Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Visualization, Look for a Pattern © 2010 College Board. All rights reserved. My Notes Unit 1 • Proof, Parallel and Perpendicular Lines 11 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Discussion Group My Notes 5. Use the circle on the next page. Draw all possible chords connecting any two of the six points. a. What is the number of non-overlapping regions formed by chords connecting the points on this circle? b. Is the number you obtained above the same number you predicted from the pattern in the table in Item 4b? d. Try to find a new pattern for predicting the number of regions formed by chords joining points on a circle when two, three, four, five, and six points are placed on a circle. Describe the pattern algebraically or in words. 12 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. c. Describe what you would do to further investigate the pattern in the number of regions formed by chords joining n points on a circle, where n represents any number of points placed on a circle. Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Visualization, Look for a Pattern © 2010 College Board. All rights reserved. My Notes Unit 1 • Proof, Parallel and Perpendicular Lines 13 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Think/Pair/Share My Notes 6. Write each sum. 1+3= 1+3+5= 1+3+5+7= 1+3+5+7+9= 1 + 3 + 5 + 7 + 9 + 11 = 7. Look for a pattern in the sums in Item 6. a. Describe algebraically or in words the pattern you see for the sums in Item 6. c. Check the patterns in Items 7a and 7b to see if they predict the next few sums beyond the list of sums given in Item 6. Record or explain your findings below. 14 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. b. Compare your description above with the descriptions that others in your group or class have written. Below, record any descriptions that were different from yours. Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Predict and Confirm, Look for a Pattern, Group Presentation, Discussion Group My Notes 8. Consider this array of squares. A B C D E F B B C D E F C C C D E F D D D D E F E E E E E F F F F F F F © 2010 College Board. All rights reserved. a. Round 1: Shade all the squares labeled A. How many A squares did you shade? How many total squares are shaded? b. Round 2: Shade all the squares labeled B. How many B squares did you shade? How many total squares are shaded? c. Complete the table for letters C, D, E, and F. Letter Round 1: A Round 2: B Round 3: C Round 4: D Round 5: E Round 6: F Squares Shaded in this Round 1 3 Total Number of Shaded Squares 1 4 Unit 1 • Proof, Parallel and Perpendicular Lines 15 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This My Notes SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Predict and Confirm, Look for a Pattern, Group Presentation, Discussion Group 9. Think about the process you used to complete the table for Item 8c. Use your findings in Item 8 to answer the following questions. a. One method of describing the pattern in Item 6 would be to say, “The new sum is always the next larger perfect square.” Analyze the process you used in Item 8 and then write a convincing argument that this pattern description will continue to hold true as the array is expanded and more squares are shaded. © 2010 College Board. All rights reserved. b. Another method of describing the pattern in Item 6 would be to say, “The new sum is always the result of adding the next odd integer to the preceding sum.” Analyze the process you used in Item 8 and write a convincing argument that this pattern description will continue to hold true as the array is expanded and more squares are shaded. 16 SpringBoard® Mathematics with Meaning™ Geometry Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Marking the Text My Notes In this activity, you have described patterns and made conjectures about how these patterns would extend beyond your observed cases, based on collected data. Some of your conjectures were probably shown to be untrue when additional data were collected. Other conjectures took on a greater sense of certainty as more confirming data were collected. In mathematics, there are certain methods and rules of argument that mathematicians use to convince someone that a conjecture is true, even for cases that extend beyond the observed data set. These rules are called rules of logical reasoning or rules of deductive reasoning. An argument that follows such rules is called a proof. A statement or conjecture that has been proven, that is, established as true without a doubt, is called a theorem. A proof transforms a conjecture into a theorem. Below are some definitions from arithmetic. Even integer: An integer that has a remainder of 0 when it is divided by 2. Odd integer: An integer that has a remainder of 1 when it is divided by 2. In the following items, you will make some conjectures about the sums of even and odd integers. ACADEMIC VOCABULARY deductive reasoning proof theorem © 2010 College Board. All rights reserved. 10. Calculate the sum of some pairs of even integers. Show the examples you use and make a conjecture about the sum of two even integers. 11. Calculate the sum of some pairs of odd integers. Show the examples you use and make a conjecture about the sum of two odd integers. 12. Calculate the sum of pairs of integers consisting of one even integer and one odd integer. Show the examples you use and make a conjecture about the sum of an even integer and an odd integer. Unit 1 • Proof, Parallel and Perpendicular Lines 17 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Look for a Pattern, Use Manipulatives My Notes The following items will help you write a convincing argument (a proof) that supports each of the conjectures you made in Items 10–12. 13. Figures A, B, C, and D are puzzle pieces. Each figure represents an integer determined by counting the square pieces in the figure. Use these figures to answer the following questions. Figure A Figure B Figure C Figure D a. Which of the figures can be used to model an even integer? c. Compare and contrast the models of even and odd integers. 18 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. b. Which of the figures can be used to model an odd integer? Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Discussion Group My Notes 14. Which pairs of puzzle pieces can fit together to form rectangles? Make sketches to show how they fit. © 2010 College Board. All rights reserved. 15. Explain how the figures (when used as puzzle pieces) can be used to show that each of the conjectures in Items 10 through 12 is true. In Items 14 and 15, you proved the conjectures in Items 10 through 12 geometrically. In Items 16 through 18, you will prove the same conjectures algebraically. Unit 1 • Proof, Parallel and Perpendicular Lines 19 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Discussion Group My Notes This is an algebraic definition of even integer: An integer is even if and only if it can be written in the form 2p, where p is an integer. (You can use other variables, such as 2m, to represent an even integer, where m is an integer.) CONNECT TO PROPERTIES • A set has closure under an operation if the result of the operation on members of the set is also in the same set. 16. Use the expressions 2p and 2m, where p and m are integers, to confirm the conjecture that the sum of two even integers is an even integer. • In symbols, distributive property of multiplication over addition can be written as a(b + c) = a(b) + a(c). This is an algebraic definition of odd integer: An integer is odd if and only if it can be written in the form 2t + 1, where t is an integer. (Again, you do not have to use t as the variable.) 18. Use expressions for even and odd integers to confirm the conjecture that the sum of an even integer and an odd integer is an odd integer. 20 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. 17. Use expressions for odd integers to confirm the conjecture that the sum of two odd integers is an even integer. Logical Reasoning ACTIVITY 1.2 continued Riddle Me This SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Summarize/Paraphrase/Retell, Create Representations, Group Presentation My Notes In Items 14 and 15, you developed a geometric puzzle-piece argument to confirm conjectures about the sums of even and odd integers. In Items 16–18, you developed an algebraic argument to confirm these same conjectures. Even though one method is considered geometric and the other algebraic, they are often seen as the same basic argument. 19. Explain the link between the geometric and the algebraic methods of the proof. © 2010 College Board. All rights reserved. 20. State three theorems that you have proved about the sums of even and odd integers. Unit 1 • Proof, Parallel and Perpendicular Lines 21 ACTIVITY 1.2 Logical Reasoning continued Riddle Me This CHECK YOUR UNDERSTANDING Writeyour youranswers answersonon notebook paper. Show Write notebook paper. Show youryour work. 4. a. Draw the next shape in the pattern, work. b. Write a sequence of numbers that could be used to express the pattern, 1. Use inductive reasoning to determine the next two terms in the sequence. c. Verbally describe the pattern of the 1 , __ 1 , __ 1 , ___ 1 ,… sequence. a. __ 2 4 8 16 5. Use expressions for even and odd integers to b. A, B, D, G, K, … confirm the conjecture that the product of 2. Write the first five terms of two different an even integer and an odd integer is an even sequences that have 10 as the second term. integer. 3. Generate a sequence using the description: the 6. MATHEMATICAL Suppose you are given the first term in the sequence is 4 and each term is R E F L E C T I O N first five terms of a three more than twice the previous term. numerical sequence. What are some approaches you could use to determine the rule for the sequence and then write the next two terms of the sequence? You may use an example to illustrate your answer. © 2010 College Board. All rights reserved. Use this picture pattern for Item 4 at the top of the next column. 22 SpringBoard® Mathematics with Meaning™ Geometry The Axiomatic System of Geometry Back to the Beginning ACTIVITY 1.3 SUGGESTED LEARNING STRATEGIES: Close Reading, Quickwrite, Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall My Notes Geometry is an axiomatic system. That means that from a small, basic set of agreed-upon assumptions and premises, an entire structure of logic is devised. Many interactive computer games are designed with this kind of structure. A game may begin with basic set of scenarios. From these scenarios, a gamer can devise tools and strategies to win the game. In geometry, it is necessary to agree on clear-cut meanings, or definitions, for words used in a technical manner. For a definition to be helpful, it must be expressed in words whose meanings are already known and understood. Compare the following definitions. Fountain: a roundel that is barry wavy of six argent and azure. Guige: a belt that is worn over the right shoulder and used to support a shield. © 2010 College Board. All rights reserved. 1. Which of the two definitions above is easier to understand? Why? For a new vocabulary term to be helpful, it should be defined using words that have already been defined. The first definitions used in building a system, however, cannot be defined in terms of other vocabulary words because no other vocabulary words have been defined yet. In geometry, it is traditional to start with the simplest and most fundamental terms— without trying to define them—and use these terms to define other terms and develop the system of geometry. These fundamental undefined terms are point, line, and plane. MATH TERMS The term, line segment, can be defined in terms of undefined terms: A line segment is part of a line bounded by two points on the line called endpoints. 2. Define each term using the undefined terms. a. Ray b. Collinear points c. Coplanar points Unit 1 • Proof, Parallel and Perpendicular Lines 23 ACTIVITY 1.3 The Axiomatic System of Geometry continued Back to the Beginning SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Activating Prior Knowledge My Notes After a term has been defined, it can be used to define other terms. For example, an angle is defined as a figure formed by two rays with a common endpoint. 3. Define each term using the already defined terms. ACADEMIC VOCABULARY a. Complementary angles complementary angles supplementary angles postulate b. Supplementary angles Addition Property of Equality If a = b and c = d, then a + c = b + d. Subtraction Property of Equality If a = b and c = d, then a - c = b - d. Multiplication Property of Equality The process of deductive reasoning, or deduction, must have a starting point. A conclusion based on deduction cannot be made unless there is an established assertion to work from. To provide a starting point for the process of deduction, a number of assertions are accepted as true without proof. These assertions are called axioms, or postulates. When you solve algebraic equations, you are using deduction. The properties you use are like postulates in geometry. 4. Using one operation or property per step, show how to solve the equation 4x + 9 = 18 - _12 x. Name each operation or property used to justify each step. If a = b, then ca = cb. Division Property of Equality a = __ b If a = b and c ≠ 0, then __ c c. Distributive Property a(b + c) = ab + ac Reflexive Property a=a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then a can be substituted for b in any equation or inequality. 24 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. CONNECT TO ALGEBRA The Axiomatic System of Geometry ACTIVITY 1.3 continued Back to the Beginning SUGGESTED LEARNING STRATEGIES: Think/Pair/Share My Notes You can organize the steps and the reasons used to justify the steps in two columns with statements (steps) on the left and reasons (properties) on the right. This format is called a two-column proof. EXAMPLE 1 Given: 3(x + 2) - 1 = 5x + 11 Prove: x = -3 Statements 1. 3(x + 2) - 1 = 5x + 11 2. 3(x + 2) = 5x + 12 3. 3x + 6 = 5x + 12 4. 6 = 2x + 12 5. -6 = 2x 6. -3 = x 7. x = -3 Reasons 1. Given equation 2. Addition Property of Equality 3. Distributive Property 4. Subtraction Property of Equality 5. Subtraction Property of Equality 6. Division Property of Equality 7. Symmetric Property of Equality TRY THESE A © 2010 College Board. All rights reserved. a. Supply the reasons to justify each statement in the proof below. 6+x x - 3 = _____ Given: _____ 5 2 Statements 6+x x - 3 = _____ _____ 1. 5 2 6+x x - 3 = 10 _____ 2. 10 _____ 5 2 ( ) ( Prove: x = 9 Reasons 1. ) 2. 3. 5(x - 3) = 2(6 + x) 3. 4. 5x - 15 = 12 + 2x 4. 5. 3x - 15 = 12 5. 6. 3x = 27 6. 7. x=9 7. Unit 1 • Proof, Parallel and Perpendicular Lines 25 ACTIVITY 1.3 The Axiomatic System of Geometry continued Back to the Beginning My Notes SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Think/Pair/Share, Interactive Word Wall, Marking the Text, Group Presentation TRY THESE A (continued) b. Complete the Prove statement and write a two-column proof for the equation given in Item 4. Number each statement and corresponding reason. 1 Given: 4x + 9 = 18 - __ 2x Prove: ACADEMIC VOCABULARY conditional statement Rules of logical reasoning involve using a set of given statements along with a valid argument to reach a conclusion. Statements to be proved are often written in if-then form. An if-then statement is called a conditional statement. In such statements, the if clause is the hypothesis, and the then clause is the conclusion. WRITING MATH The letters p and q are often used to represent the hypothesis and conclusion, respectively, in a conditional statement. The basic form of an if-then statement would then be, “If p, then q.” EXAMPLE 2 Conditional statement: If 3(x + 2) - 1 = 5x + 11, then x = -3. Hypothesis Conclusion 3(x + 2) - 1 = 5x + 11 x = -3 TRY THESE B a. What is the hypothesis? b. What is the conclusion? c. State the property of equality that justifies the conclusion of the statement. 26 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. Use the conditional statement: If x + 7 = 10, then x = 3. The Axiomatic System of Geometry ACTIVITY 1.3 continued Back to the Beginning SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall My Notes Conditional statements may not always be written in if-then form. You can restate such conditional statements in if-then form. 5. Restate each conditional statement in if-then form. a. I’ll go if you go. READING MATH Forms of conditional statements include: b. There is smoke only if there is fire. • If p, then q. • p only if q • p implies q. c. x = 4 implies x2 = 16. • q if p An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. This type of example is a counterexample. ACADEMIC VOCABULARY counterexample 6. This is a false conditional statement. If two numbers are odd, then their sum is odd. © 2010 College Board. All rights reserved. a. Identify the hypothesis of the statement. b. Identify the conclusion of the statement. c. Give a counterexample for the conditional statement and justify your choice for this example. Unit 1 • Proof, Parallel and Perpendicular Lines 27 ACTIVITY 1.3 The Axiomatic System of Geometry continued Back to the Beginning SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share My Notes converse inverse contrapositive CONNECT TO AP When both a statement and its converse are true, you can connect the hypothesis and conclusion with the words “if and only if.” Conditional: If p, then q. Converse: If q, then p. Inverse: If not p, then not q. Contrapositive: If not q, then not p. 7. Given the conditional statement: If a figure is a triangle, then it is a polygon. Complete the table. Form of the statement Conditional statement Converse of the conditional statement Inverse of the conditional statement Contrapositive of the conditional statement 28 SpringBoard® Mathematics with Meaning™ Geometry Write the statement True or False? If the statement is false, give a counterexample. If a figure is a triangle, then it is a polygon. © 2010 College Board. All rights reserved. MATH TERMS Every conditional statement has three related conditionals. These are the converse, the inverse, and the contrapositive of the conditional statement. The converse of a conditional is formed by interchanging the hypothesis and conclusion of the statement. The inverse is formed by negating both the hypothesis and the conclusion. Finally, the contrapositive is formed by interchanging and negating both the hypothesis and the conclusion. The Axiomatic System of Geometry ACTIVITY 1.3 continued Back to the Beginning SUGGESTED LEARNING STRATEGIES: Group Presentation My Notes If a given conditional statement is true, the converse and inverse are not necessarily true. However, the contrapositive of a true conditional is always true, and the contrapositive of a false conditional is always false. Likewise, the converse and inverse of a conditional are either both true or both false. Statements with the same truth values are logically equivalent. MATH TERMS The truth value of a statement is the truth or falsity of that statement 8. Write a true conditional statement whose inverse is false. 9. Write a true conditional statement that is logically equivalent to its converse. When a statement and its converse are both true, they can be combined into one statement using the words “if and only if ”. All definitions you have learned can be written as “if and only if ” statements. 10. Write the definition of perpendicular lines in if and only if form. TRY THESE C Use this statement: Numbers that do not end in 2 are not even. © 2010 College Board. All rights reserved. a. Rewrite the statement in if-then form and state whether it is true or false. b. Write the converse and state whether it is true or false. If false, give a counterexample. c. Write the inverse and state whether it is true or false. d. Write the contrapositive and state whether it is true or false. If false, give a counterexample. Unit 1 • Proof, Parallel and Perpendicular Lines 29 ACTIVITY 1.3 The Axiomatic System of Geometry continued Back to the Beginning CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show your work. Write paper. Show 9. Which of the following is a counterexample of your work. this statement? 2. Complete the prove statement and write a two column proof for the equation: Given: x - 2 = 3(x - 4) Prove: 3. Write the statement in if-then form. Two angles have measures that add up to 90° only if they are complements of each other. Use the following statement for Items 4–7. If a vehicle has four wheels, then it is a car. a. a 100° angle b. a 70° angle c. an 80° angle d. a 90° angle 10. Give an example of a false statement that has a true converse. 11. A certain conditional statement is true. Which of the following must also be true? a. converse b. inverse c. contrapositive d. all of the above 12. Given: (1) If X is blue, then Y is gold. (2) Y is not gold. 4. State the hypothesis. Which of the following must be true? 5. Write the converse. a. Y is blue. b. Y is not blue. 6. Write the inverse. c. X is not blue. d. X is gold. 7. Write the contrapositive. 8. Given the false conditional statement: If a vehicle has four wheels, then it is a car. Write a counterexample. 30 If an angle is acute, then it measures 80°. SpringBoard® Mathematics with Meaning™ Geometry 13. MATHEMATICAL Some test questions ask you R E F L E C T I O N to tell if a statement is sometimes true, always true, or never true. How can counterexamples help you to answer these types of questions? © 2010 College Board. All rights reserved. 1. Identify the property that justifies the statement: If 4x - 3 = 7, then 4x = 10. Truth Tables The Truth of the Matter ACTIVITY 1.4 SUGGESTED LEARNING STRATEGIES: Close Reading, Activating Prior Knowledge, Group Discussion, Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share My Notes Symbolic logic allows you to determine the validity of statements without being distracted by a lot of text. You can use symbols, such as p and q, to represent simple statements. A compound statement is formed when two or more simple statements are connected as a conditional (if-then) a biconditional (if and only if), a conjunction (and), or a disjunction (or). 1. The symbol → represents a conditional. You read p → q as, “if p, then q,” or “p implies q.” Let p represent the statement “you arrive before 7 PM,” and let q represent the statement “you will get a good seat.” Use the information in the play poster to the right to write the statement that is represented by p → q. © 2010 College Board. All rights reserved. Truth tables are used to determine the conditions under which a statement is true or false. This truth table displays the truth values for p → q, which are dependent on the truth values for p and q. Row 1 Row 2 Row 3 Row 4 p q T T F F T F T F p→q T F T T 2. Row 1 addresses the case when both p and q are true. Refer to the play announcement that you completed in Item 1. Explain why p → q would be true if both p and q are true. 3. Row 2 addresses the case when p is true and q is false. In terms of the play announcement, explain why p → q would be false if p is true and q is false. 4. Refer to Rows 3 and 4. In terms of the play announcement, explain why p → q is true if p is false. Unit 1 • Proof, Parallel and Perpendicular Lines 31 ACTIVITY 1.4 Truth Tables continued The Truth of the Matter SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Think/Pair/Share My Notes The symbol ~p is the negation of p and can be read as “not p.” This truth table shows the conditions under which ~p is true or false. When p is true, ~p is false, and when p is false, ~p is true. p ~p T F F T 5. Let p represent the simple statement “Triangles are convex,” which is a true statement. Write the statement denoted by ~p and state whether it is true or false. 6. Create a simple statement p that you know to be false. Write the statement ~p and state whether is true or false. EXAMPLE Make a truth table for ~p → q. Step 1 Write down all possible T and F combinations for p and q. Step 3 Add a column for ~p → q and evaluate ~p → q. 32 SpringBoard® Mathematics with Meaning™ Geometry Step 2 Add a column for ~p and negate p. p q T T F F T F T F p q ~p T T F F T F T F F F T T ~p → q T T T F p q ~p T T F F T F T F F F T T © 2010 College Board. All rights reserved. 7. Let p represent “you are not in the band” and let q represent “you can go on the trip.” Write the compound statement represented by ~p → q. Truth Tables ACTIVITY 1.4 continued The Truth of the Matter SUGGESTED LEARNING STRATEGIES: Close Reading, Think/Pair/Share, Create Representations, Identify a Subtask, Activating Prior Knowledge, Interactive Word Wall, Vocabulary Organizer My Notes 8. Follow the steps and complete the truth table for ~(p → q). Step 1 Find all possible T and F combinations for p and q. Step 1 Step 2 Step 3 p p→q ~(p → q) q Step 2 Evaluate p → q. Step 3 Negate p → q. 9. Write the steps and complete the truth table for ~q → ~p. Step 1 © 2010 College Board. All rights reserved. p q Step 2 ~p ~q Step 3 ~q → ~p 10. The symbol ↔ represents a biconditional. You read it as “if and only if.” Let p represent “A chord in a circle contains the center,” and let q represent “The chord is a diameter.” Write the statement that is represented by p ↔ q. Unit 1 • Proof, Parallel and Perpendicular Lines 33 ACTIVITY 1.4 Truth Tables continued The Truth of the Matter My Notes Because biconditionals are true only when both parts have the same truth value, many definitions are written in the form of a biconditional. SUGGESTED LEARNING STRATEGIES: Close Reading, Think/ Pair/Share, Create Representations, Identify a Subtask, Discussion Group, Group Presentation, Interactive Word Wall, Vocabulary Organizer The truth table for p ↔ q is shown to the right. Notice that p ↔ q is true only when p and q are both true or both false. p q T T F F T F T F p↔q T F F T 11. Construct a truth table for (p → q) ↔ ~q. A rectangle is a parallelogram with a right angle. A rhombus is a parallelogram with consecutive congruent sides. 12. The symbol ∧ represents a conjunction. You read it as “and.” Let p represent “the figure is a rectangle,” and let q represent “the figure is a rhombus.” a. Write the statement that is represented by p ∧ q. b. Write the statement that is represented by p ∧ ~q. The truth table for p ∧ q is shown to the right. Notice that p ∧ q is only true when both p and q are true. 34 SpringBoard® Mathematics with Meaning™ Geometry p q T T F F T F T F p∧q T F F F © 2010 College Board. All rights reserved. MATH TERMS Truth Tables ACTIVITY 1.4 continued The Truth of the Matter SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Identify a Subtask My Notes 13. The symbol ∨ represents a disjunction. You read it as “or.” Let p represent “the figure is not a rectangle,” and let q represent “the figure is a rhombus.” a. Write the statement that is represented by p ∨ q. b. Write the statement that is represented by ~p ∨ q. The truth table for p ∨ q is shown to the right. Notice that p ∨ q is only false when both p and q are false. p q T T F F T F T F p∨q T T T F © 2010 College Board. All rights reserved. 14. Construct a truth table for p ∨ ( p ∧ q). Unit 1 • Proof, Parallel and Perpendicular Lines 35 ACTIVITY 1.4 Truth Tables continued The Truth of the Matter SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Identify a Subtask, Create Representations, Identify a Subtask My Notes A tautology is a statement that is true for all cases of p and q. The corresponding truth table will have only T’s in the last column. 15. Construct a truth table for (p → q) ↔ (p ∧ q). Is this a tautology? CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show Write paper. Show youryour work. 4. (p → q) ∧ p work. 5. q ∧ (p → ~q) Construct a truth table for the compound statements. 6. (p → q) ↔ (~q → ~p) Identify which statements, if any, are tautologies. 7. MATHEMATICAL How could a truth table 1. ~p ∧ q R E F L E C T I O N help you to analyze a campaign promise made by a person running 2. q → (p ∨ ~p) for office? Think of a conditional statement 3. p ∨ (q → p) starting, “If I am elected, …” 36 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. 16. Construct a truth table for ~(p ∨ q) ↔ (~p ∧ ~q). Is this a tautology? Conditional Statements and Logic Embedded Assessment 1 HEALTHY HABITS Use after Activity 1.4. Statements given in advertisements often involve conditional statements. Analyze the advertisement below. “If you want to feel your very best, take one SBV tablet every day. People who take SBV vitamins care about their health. You owe it to those close to you to take care of yourself. Begin taking SBV today!” Write your answers on notebook paper. Show your work. 1. Consider the first statement in the advertisement: “If you want to feel your very best, take one SBV tablet every day.” a. Identify the hypothesis and conclusion of the statement. b. Write the converse of the statement. Discuss the validity of the converse and why it might be used to influence people to buy the vitamin. 2. Consider the second statement: “People who take SBV vitamins care about their health.” a. Rewrite the statement in if-then form. b. Write the inverse of the statement. Discuss the validity of the inverse and why it might be used to influence people to buy the vitamin. © 2010 College Board. All rights reserved. According to current guidelines, people are advised to maintain cholesterol levels below 200 units. Four friends; Juana, Bea, Les, and Hugh, all have cholesterol levels above the recommended limit. They have all heard cholesterol-lowering strategies discussed on talk shows and in commercials. Those strategies included: taking two 1 hour red yeast rice tablets daily; eating oatmeal daily; practicing meditation for __ 2 each day; and getting at least twenty minutes of aerobic exercise daily. 3. For one month, they each used strategies daily. They had their cholesterol rechecked at the end of the month. Juana meditated and ate oatmeal. Her cholesterol level fell 5 points. Hugh exercised and meditated. His cholesterol lowered 10 points. Les took red yeast rice tablets and meditated. His cholesterol level stayed the same. Bea took red yeast rice tablets and ate oatmeal. Her cholesterol level went down 7 points. Which technique(s) appear to be most effective in lowering cholesterol levels? Explain your reasoning. Unit 1 • Proof, Parallel and Perpendicular Lines 37 Embedded Assessment 1 Conditional Statements and Logic Use after Activity 1.4. HEALTHY HABITS Exemplary Proficient Emerging The student: The student: • Correctly • Correctly identifies the identifies the hypothesis or hypothesis and conclusion but conclusion. (1a) not both. • Writes the correct • Writes a correct converse of the if-then form but statement. (1b) not the correct • Writes a correct inverse. if-then form and inverse of the statement. (2a, b) The student: • Correctly identifies neither the hypothesis nor the conclusion. • Writes an incorrect converse of the statement. • Writes neither a correct ifthen form nor a correct inverse. Problem Solving #3 The student correctly identifies the seemingly most effective technique(s). (3) The student identifies a technique that is somewhat effective. The student identifies a technique that is not effective. Communication #1b, 2b, 3 The student: • Gives a complete discussion of the validity of the converse. (1b) • Gives a complete discussion of the validity of the inverse. (2b) • Gives logical reasoning for his/ her choice(s) of technique(s). (3). The student: • Gives an incomplete discussion of the validity of the converse. • Gives an incomplete discussion of the validity of the inverse. • Gives incomplete reasoning for the choice. The student: • Gives an irrelevant discussion of the validity of the converse. • Gives an irrelevant discussion of the validity of the inverse. • Gives illogical reasoning for the choice. © 2010 College Board. All rights reserved. Math Knowledge #a, b; 2a, b 38 SpringBoard® Mathematics with Meaning™ Geometry Segment and Angle Measurement It All Adds Up ACTIVITY 1.5 SUGGESTED LEARNING STRATEGIES: Close Reading My Notes If two points are no more than one foot apart, you can find the distance between them by using an ordinary ruler. (The inch rulers below have been reduced to fit on the page.) A B inches 1 2 3 4 5 READING MATH 6 7 In the figure, the distance between point A and point B is 5 inches. Of course, there is no need to place the zero of the ruler on point A. In the figure below, the 2-inch mark is on point A. In this case, AB, measured in inches, is |7 - 2| = |2 - 7| = 5, as before. A 2 AB denotes the distance between points A and B. If A and B are the _ endpoints of a segment ( AB ),_then AB denotes the length of AB . B 3 4 5 7 6 8 9 The number obtained as a measure of distance depends on the unit of measure. On many rulers one edge is marked in centimeters. Using the centimeter scale, the distance between the points A and B above is about 12.7 cm. 1. Determine the length of each segment in centimeters. MATH TERMS © 2010 College Board. All rights reserved. D a. DE = E F b. EF = The Ruler Postulate a. To every pair of points there corresponds a unique positive number called the distance between the points. c. DF = 2. Determine the length of each segment in centimeters. G H a. KH = b. HG = K c. GK = b. The points on a line can be matched with the real numbers so that the distance between any two points is the absolute value of the difference of their associated numbers. Unit 1 • Proof, Parallel and Perpendicular Lines 39 ACTIVITY 1.5 Segment and Angle Measurement continued It All Adds Up My Notes SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall, Create Representations 3. Using your results from Items 1 and 2, describe any patterns that you notice. Item 4 and your answer together form a statement of the Segment Addition Postulate. For each part of Item 5, make a sketch so that you can identify the parts of the segment. CONNECT TO AP You will frequently be asked to find the lengths of horizontal, vertical, and diagonal segments in the coordinate plane in AP Calculus. 40 4. Given that N is a point between endpoints__ M and P of line segment MP, describe how to determine the__ length of__ MP, without measuring, if you are given the lengths of MN and NP. 5. Use the Segment Addition Postulate and the given information to complete each statement. a. If B is between C and D, BC = 10 in., and BD = 3 in., then CD = _______. b. If Q is between R and T, RT = 24 cm, and QR = 6 cm, then QT = _______. c. If P is between L and A, PL = x + 4, PA = 2x - 1, and LA = 5x - 3, then x = _______ and LA = _______. SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. MATH TERMS Segment and Angle Measurement ACTIVITY 1.5 continued It All Adds Up SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall My Notes The midpoint of a segment is the point on the segment that divides__ it into two segments. For example, if B is the midpoint of AC , __ __ congruent then AB BC. A B 3 3 ACADEMIC VOCABULARY C The midpoint of a segment is a point on the segment that divides it into two congruent segments. __ 6. Given: M is the midpoint of RS. Complete each statement. a. If RS = 10, then SM = _______. Congruent segments are segments that have equal length. To indicate __ __ congruence, you can write AB BC. b. If RM = 12, then MS = _______, and RS = _______. You can use the definition of midpoint and properties of algebra to determine the length of a segment. EXAMPLE 1 __ If Q is the midpoint of PR , PQ = 4x - 5, and QR = 11 + 2x, __ determine the length of PQ. © 2010 College Board. All rights reserved. __ __ 1. PQ QR 1. Definition of midpoint 2. PQ = QR 2. Definition of congruent segments __ Use AB when you talk about segment AB. 3. 4x - 5 = 11 + 2x 3. Substitution Property 4. 2x - 5 = 11 4. Subtraction Property of Equality 5. 2x = 16 5. Addition Property of Equality 6. x = 8 6. Division Property of Equality 7. PQ = 4(8) - 5 = 27 7. Substitution Property Use AB when you talk about __ the measure, or length, of AB . Unit 1 • Proof, Parallel and Perpendicular Lines 41 ACTIVITY 1.5 Segment and Angle Measurement continued It All Adds Up SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share My Notes EXAMPLE 2 __ If Y is the midpoint __ of WZ, YZ = x + 3 and WZ = 3x - 4, determine the length of WZ. 1. 2. 3. 4. 5. 6. 7. 8. WZ = WY + YZ __ __ 1. 2. WY YZ 3. WY = YZ 4. WZ = YZ + YZ 3x - 4 = (x + 3) + (x + 3) 5. 6. x-4=6 7. x = 10 8. WZ = 3(10) - 4 = 26 TRY THESE A Segment Addition Postulate Definition of midpoint Definition of congruent segments Substitution Property Substitution Property Subtraction Property of Equality Addition Property of Equality Substitution Property __ Given: M is the midpoint of RS. Use the given information to find the missing values. a. RM = x + 3 and MS = 2x - 1 x = _____ and RM = _____ b. RM = x + 6 and RS = 5x + 3 x = _____ and SM = _____ © 2010 College Board. All rights reserved. You measure angles with a protractor. The number of degrees in an angle is called its measure. C 0 80 100 90 100 80 110 70 12 0 60 14 30 0 15 30 20 160 160 20 O 180 0 0 180 10 170 170 10 42 E 0 15 The astronomer Claudius Ptolemy (about 85–165 CE) based his observations of the solar system on a unit that resulted from dividing the distance around a circle into 360 parts. This later became known as a degree. 50 0 40 CONNECT TO ASTRONOMY B 13 0 0 12 50 0 13 70 110 14 D 40 60 A 7. Determine the measure of each angle. a. m∠AOB = 50° b. m∠BOC = ____ c. m∠AOC = ____ d. m∠EOD = ____ e. m∠BOD = ____ f. m∠BOE = ____ SpringBoard® Mathematics with Meaning™ Geometry Segment and Angle Measurement ACTIVITY 1.5 continued It All Adds Up SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share My Notes 8. Use a protractor to determine the measure of each angle. P T MATH TERMS The Protractor Postulate Q a. m∠TQP = b. m∠TQR = R c. m∠RQP = 9. Using your results from Items 7 and 8, describe any patterns that you notice. a. To each angle there corresponds a unique real number between 0 and 180 called the measure of the angle. b. The measure of an angle formed by a pair of rays is the absolute value of the difference of their associated numbers. 10. Given that point D is in the interior of ∠ABC, describe how to determine the measure of ∠ABC, without measuring, if you are given the measures of ∠ABD and ∠DBC. MATH TERMS © 2010 College Board. All rights reserved. 11. Use the angle addition postulate and the given information to complete each statement. Item 10 and your answer together form a statement of the Angle Addition Postulate. a. If P is in the interior of ∠XYZ, m∠XYP = 25°, and m∠PYZ = 50°, then m∠XYZ = . b. If M is in the interior of ∠RTD, m∠RTM = 40°, and m∠RTD = 65°, then m∠MTD = . c. If H is in the interior of ∠EFG, m∠EFH = 75°, and m∠HFG = (10x)°, and m∠EFG = (20x - 5)°, then x= and m∠HFG = . Unit 1 • Proof, Parallel and Perpendicular Lines 43 ACTIVITY 1.5 Segment and Angle Measurement continued It All Adds Up SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Create Representations My Notes 11d. Lines DB and EC intersect at point F. If m∠BFC = 44° and m∠AFB = 61°, then ACADEMIC VOCABULARY m∠AFC = angle bisector m∠AFE = A E D B F C m∠EFD = The bisector of an angle is a ray that divides the angle into two congruent adjacent angles. For example, in the figure to the left, if BD bisects ∠ABC, then ∠ABD ∠DBC. A 30° 30° D B 12. Given: AH bisects ∠MAT. Determine the missing measure. C a. m∠MAT = 70°, m∠MAH = b. m∠HAT = 80°, m∠MAT = Adjacent angles are coplanar angles that share a vertex and side, but no interior points. In the figure above, ∠ABD is adjacent to ∠DBC. Congruent angles are angles that have the same measure. To indicate congruence, you can either write m∠ABD = m∠DBC. or ∠ABD ∠DBC. 44 You can use definitions, postulates, theorems, and properties of algebra to determine the measures of angles. EXAMPLE 3 If QP bisects ∠DQL, m∠DQP = 5x - 7 and m∠PQL = 11 + 2x, determine the measure of ∠DQL. ∠DQP ∠PQL m∠DQP = m∠PQL 5x - 7 = 11 + 2x 3x - 7 = 11 3x = 18 x=6 m∠DQP = 5(6) - 7 = 23 m∠PQL = 11 + 2(6) = 23 m∠DQL = m∠DQP + m∠PQL 10. m∠DQL = 23 + 23 = 46° 1. 2. 3. 4. 5. 6. 7. 8. 9. SpringBoard® Mathematics with Meaning™ Geometry 1. 2. 3. 4. 5. 6. 7. 8. 9. Definition of Angle Bisector Definition of congruent angles Substitution Property Subtraction Property of Equality Addition Property of Equality Division Property of Equality Substitution Property Substitution Property Angle Addition Postulate 10. Substitution Property © 2010 College Board. All rights reserved. MATH TERMS Segment and Angle Measurement ACTIVITY 1.5 continued It All Adds Up SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Group Presentation My Notes GUIDED EXAMPLE Complete the missing statements and reasons. ∠A and ∠B are complementary, m∠A = 3x + 7, and m∠B = 6x + 11. Determine the measure of each angle. Statements 1. 2. 3. 4. 5. 6. 7. Reasons m∠A + m∠B = 90° + = 90° 9x + 18 = 90 9x = 72 x=8 m∠A = 3(8) + 7 = 31 m∠B = 6( ) +11 = 1. 2. 3. 4. 5. 6. 7. Definition of Substitution Property Substitution Property Property of Equality Property of Equality Substitution Property Substitution Property TRY THESE B Given: FL bisects ∠AFM. Determine each missing value. a. m∠LFM = 11x + 4 and m∠AFL = 12x - 2 x= , m∠LFM = , and m∠AFM = b. m∠AFM = 6x - 2 and m∠AFL = 4x - 10 © 2010 College Board. All rights reserved. x= and m∠LFM = In this diagram, lines AC and DB intersect as shown. Determine the missing values. A c. x = m∠AEB = m∠CEB = . D (5x + 68)° (9x)° E B C (Diagram is not drawn to scale.) d. ∠P and ∠Q are complementary. m∠P = 2x + 25 and m∠Q = 4x + 11. Determine the measure of each angle. Unit 1 • Proof, Parallel and Perpendicular Lines 45 ACTIVITY 1.5 Segment and Angle Measurement continued It All Adds Up CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show your work. Write paper. Show 7. Given: EA ⊥ ED , your work. EB bisects ∠AEC, m∠AEB = 4x + 1, 1. Given: Point K is between points H and J, and m∠CED = 3x. HK = x - 5, KJ = 5x - 12 and HJ = 25. a. x = Find the value of x. b. 2 c. 6 2. If K is the midpoint of HJ, HK = x + 6, and HJ = 5x - 6, then KJ = ? . a. 6 b. 3 c. 12 b. 23° c. 17° b. 29 c. 0 b. 20° c. 6° d. 2° a. x = 46 4 b. m∠4 = d. 14 6. Given: m∠1 = 4x + 30 and m∠3 = 2x + 48 c. m∠2 = 3 a. x = Use the given diagram and information to determine the missing measures. Show all work. b. m∠3 = 2 1 d. 77° 5. Ray QS bisects ∠PQR. If m∠PQS = 5x and m∠RQS = 2x + 6, then m∠PQR = ? . a. 10° D d. 9 4. If ∠P and ∠Q are complementary, m∠P = 5x + 3, and m∠Q = x + 3, then x= ? a. 30 E 8. Given: m∠1 = 2x + 8, m∠2 = x + 4, and m∠3 = 3x + 18 3. Point D is in the interior of ∠ABC, m∠ABC = 10x - 7, m∠ABD = 6x + 5, and m∠DBC = 36°. What is m∠ABD? a. 3° C b. m∠BEC = d. 9 ___ B 1 2 3 SpringBoard® Mathematics with Meaning™ Geometry c. Is ∠4 complementary to ∠2? Explain. 9. MATHEMATICAL How could you model two R E F L E C T I O N angles with a common vertex and a common side that are not adjacent angles? © 2010 College Board. All rights reserved. a. 7 A Parallel and Perpendicular Lines Patios by Madeline ACTIVITY 1.6 SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell My Notes ACADEMIC VOCABULARY Parallel lines are coplanar lines that do not intersect. Gas Line Matt works for Patios by Madeline. A new customer has asked him to design a brick patio and walkway. The customer requires a blueprint showing both the patio and the walkway. The customer would like the rows of bricks in the completed patio to be parallel to the walkway. To help create his blueprint, Matt will attach a string between two stakes on either side of the patio as shown in the diagram below, and continue the pattern with parallel strings. In his blueprint, Matt must include a line to indicate the path of the underground gas line, which is located below the patio. (Matt drew in the location of the gas line from information supplied by the local gas company in order to avoid accidents during construction of the patio.) Stake © 2010 College Board. All rights reserved. ing Str ay lkw Wa e Lin Stake House Unit 1 • Proof, Parallel and Perpendicular Lines 47 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Visualization, Create Representations My Notes CONNECT TO OTHER GEOMETRIES ___ 1. Follow these steps to draw a line parallel to XY. Step A Determine the measure of ∠HAX. Step B Locate the third ___› stake on the left edge of the patio at point P by drawing BP so that m∠HBP = m∠HAX. Step C Locate the fourth stake ___› at point J on the right edge of the patio by extending BP in the opposite direction. © 2010 College Board. All rights reserved. Euclidean geometry is based on five postulates proposed by Euclid (about 300 BCE). His fifth postulate, “Through a point outside a line, there is exactly one line parallel to the given line” was a source of contention for many mathematicians. Their thought was that this fifth postulate could be proven from the first four postulates and should therefore be called a theorem. Since Euclid could not prove it, he kept it as a postulate. Some mathematicians thought that since the postulate could not be proven they should be able to replace it with a contrary postulate. This created other “non-Euclidean” geometries. Continue designing the patio using the blueprint Matt has already begun. Matt’s beginning blueprint is on the next page. Add information to the blueprint as you work through this Activity. Matt has placed the gas line on the blueprint from the information he received from the gas company. To assure that the bricks will be parallel to the walkway, Matt extended the string line along the edge of the walkway and labeled points X, A, and Y. He also labeled additional points B, C, and D where the string lines will cross the gas line. 48 SpringBoard® Mathematics with Meaning™ Geometry Parallel and Perpendicular Lines ACTIVITY 1.6 continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Visualization; Create Representations My Notes Matt’s Blueprint D C Y B A X © 2010 College Board. All rights reserved. H Unit 1 • Proof, Parallel and Perpendicular Lines 49 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations My Notes MATH TERMS A transversal is a line that intersects two or more coplanar lines in different points. ∠HAX and ∠HBP are called corresponding angles because they are in corresponding positions on the same side of the transversal (gas line). According to the Corresponding Angles Postulate, when parallel lines are cut by a transversal, the corresponding angles will always have the same measure. 2. Postulates and theorems are often stated in if-then form. a. State the Corresponding Angles Postulate in if-then form. b. State the converse of the Corresponding Angles Postulate. d. State the inverse of the Corresponding Angles Postulate. e. Determine the validity of the inverse of the Corresponding Angles Postulate. Support your answer with illustrations. 50 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. c. How does the statement in part b ensure that Matt’s string lines are parallel? Parallel and Perpendicular Lines ACTIVITY 1.6 continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Self/Peer Revision My Notes 3. Use the diagram shown to complete the table. a. Fill in the corresponding angles column in the table below. l 2 1 4 3 6 5 8 7 9 10 11 12 Angle m n Corresponding Angles ∠1 ∠2 ∠3 ∠4 © 2010 College Board. All rights reserved. b. What conditions are necessary in order to apply the Corresponding Angles Postulate? c. What conditions are necessary in order to apply the inverse of the Corresponding Angles Postulate? Unit 1 • Proof, Parallel and Perpendicular Lines 51 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation My Notes 4. Other types of special angle pairs formed when lines are intersected by a transversal are alternate interior angles. In the diagram below, ∠3 and ∠6 are alternate interior angles. l 2 1 4 3 6 5 8 7 m a. Explain why you think ∠3 and ∠6 are alternate interior angles. c. Name the remaining pair of alternate interior angles in the diagram. What appears to be true about them? d. Using what you know about corresponding and vertical angles, write a convincing argument to prove your response to part b. 52 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. b. Lines l and m in the diagram are parallel. What appears to be true about the measures of ∠3 and ∠6? Parallel and Perpendicular Lines ACTIVITY 1.6 continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations My Notes 5. Use your results from Item 4 to state a theorem in if-then form about alternate interior angles. 6. State the inverse of the Alternate Interior Angles Theorem you wrote in Item 5. 7. Determine the validity of the inverse of the Alternate Interior Angles Theorem. Support your answer with illustrations. © 2010 College Board. All rights reserved. 8. State the converse of the Alternate Interior Angles Theorem. 9. Matt realizes that he is not limited to using corresponding angles to draw parallel lines. Use a protractor to draw another parallel string line through point C on the blueprint, using alternate interior angles. Mark the angles that were used to draw this new parallel line. 10. How does the converse of the Alternate Interior Angles Theorem ensure that Matt’s string lines are parallel? Unit 1 • Proof, Parallel and Perpendicular Lines 53 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Think/Pair/Share My Notes 11. Another angle pair relationship is same-side interior angles. In the diagram below, ∠3 and ∠5 are same-side interior angles. l 1 2 4 3 m 6 5 7 8 a. Explain why you think ∠3 and ∠5 are same-side interior angles. c. Name the remaining pair of same-side interior angles in the diagram. What appears to be true? d. Provide a convincing argument to prove your response to part b. 54 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. b. Lines l and m in the diagram are parallel. What appears to be true about the measures of ∠3 and ∠5? Parallel and Perpendicular Lines ACTIVITY 1.6 continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations My Notes 12. Use your results from Item 11 to state a theorem in if-then form, about same-side interior angles. 13. State the inverse of the Same-Side Interior Angles Theorem you wrote in Item 12. 14. Determine the validity of the inverse of the Same-Side Interior Angles Theorem. Support your answer with illustrations. © 2010 College Board. All rights reserved. 15. State the converse of the Same-Side Interior Angles Theorem. 16. Use a protractor to draw another parallel string line through point D on the blueprint using same-side interior angles. Mark the angles you used to draw this line. Unit 1 • Proof, Parallel and Perpendicular Lines 55 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge My Notes ACADEMIC VOCABULARY Perpendicular figures intersect to form right angles. A second customer of Madeline’s company has commissioned them to build a patio with bricks that are perpendicular to the walkway. To help plan this design, Matt will attach a string between two stakes on either side of the patio as he did for the previous diagram. Continue drawing the blueprint for the patio using the diagram below. To assure that the bricks will be perpendicular to the walkway, Matt extended the string line along the edge of the walkway and labeled points X and Y. He also attached a string to another stake labeled point W as shown below. W Y This theorem guarantees that the line you draw in Item 17 is perpendicular to line XY: In a given plane containing a line, there is exactly one line perpendicular to the line through a given point not on the line. X House ‹___› 18. Use your protractor and your knowledge of parallel lines to draw a ‹___› line, across the patio, parallel to WZ. Explain how you know the lines are parallel. 56 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. 17. Use a protractor to draw the line perpendicular to XY that passes ‹___› through W. Label the point at which the line intersects XY point Z. Parallel and Perpendicular Lines ACTIVITY 1.6 continued Patios by Madeline SUGGESTED LEARNING STRATEGIES: RAFT ‹___› My Notes 19. Describe the relationship between the newly drawn line and XY. 20. Complete the statement to form a true conditional statement. “If a transversal is perpendicular to one of two parallel lines, then it is to the other line also.’’ © 2010 College Board. All rights reserved. 21. Matt’s boss Madeline is so impressed with Matt’s work that she asks him to write an instruction guide for creating rows of bricks parallel to patio walkways. Madeline asks that the guide provide enough direction so that a bricklayer can use any pair of angles to determine the location of each pair of stakes since not all patios will allow for measuring the same pair of angles for each string line. She also requests that Matt include directions for creating rows of bricks that are perpendicular to the walkway. Use the space below to write Matt’s instruction guide, following Madeline’s directions. Unit 1 • Proof, Parallel and Perpendicular Lines 57 ACTIVITY 1.6 Parallel and Perpendicular Lines continued Patios by Madeline CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. Use this figure for Items 1-14. l m 6 5 8 7 9 10 11 12 c. ∠1 and ∠8 b. ∠3 and ∠5 d. ∠4 and ∠5 7. If l m, m∠4 = 15x − 7, and m∠6 = 20x + 12, then m∠8 = ? a. 112° c. 68° b. 5° d. 3° In the diagram, l m and m ∦ n. n Determine whether each statement is true or false. Justify your response with the appropriate postulate or theorem. 1. List all pairs of alternate interior angles. 8. ∠5 is supplementary to m∠3 2. List all pairs of same-side interior angles. 9. m∠9 = m∠6 3. If l m and m∠5 = 80°, then what is m∠2? 10. ∠3 ∠8 a. 100° c. 80° 11. ∠4 ∠10 b. 50° d. Not enough information 12. ∠4 ∠6 4. If l m and m∠6 = 110°, then what is m∠3? a. 70° c. 80° b. 110° d. Not enough information 5. If l m, m∠4 = 12x + 5, and m∠5 = 8x + 17, then what is m∠2? 58 a. ∠1 and ∠6 a. 139° c. 3° b. 7.9° d. 41° SpringBoard® Mathematics with Meaning™ Geometry 13. m∠8 + m∠10 = 180° 14. ∠4 is supplementary to ∠9. 15. MATHEMATICAL How could the types of R E F L E C T I O N angles you studied in this activity be useful to a city planner? © 2010 College Board. All rights reserved. 1 2 4 3 6. If l m, which two angles are supplementary? Two-Column Proofs Now I'm Convinced ACTIVITY 1.7 SUGGESTED LEARNING STRATEGIES: Close Reading, Activating Prior Knowledge, Think/Pair/Share My Notes A proof is an argument, a justification, or a reason that something is true. A proof is an answer to the question “why?” when the person asking wants an argument that is indisputable. There are three basic requirements for constructing a good proof. 1. Awareness and knowledge of the definitions of the terms related to what you are trying to prove. 2. Knowledge and understanding of postulates and previous proven theorems related to what you are trying to prove. 3. Knowledge of the basic rules of logic. To write a proof, you must be able to justify statements. The statements in the Examples 1–5 are based on the diagram to the right in which lines AC, EG, and DF all intersect at point B. Each of the statements is justified using a property, postulate, or definition. F E A D 2 B 5 6 C 1 G © 2010 College Board. All rights reserved. EXAMPLE 1 Statement If ∠ABE is a right angle, then m∠ABE = 90°. Justification Definition of right angle EXAMPLE 2 Statement If ∠2 ∠1 and ∠1 ∠5, then ∠2 ∠5. Justification Transitive Property EXAMPLE 3 Statement ___ AC. Given: ___ B is the midpoint of ___ Prove: AB BC Justification Definition of midpoint Unit 1 • Proof, Parallel and Perpendicular Lines 59 ACTIVITY 1.7 Two-Column Proofs continued Now I’m Convinced SUGGESTED LEARNING STRATEGIES: Think/Pair/Share My Notes EXAMPLE 4 Statement m∠2 + m∠ABE = m∠DBE Justification Angle Addition Postulate EXAMPLE 5 Statement If ∠1 is supplementary to ∠FBG, then m∠1 + m∠FBG = 180°. F E A D 2 B 1 5 6 Justification Definition of supplementary angles TRY THESE A C Using the diagram from the previous page, reproduced to the left, name the property, postulate, or definition that justifies each statement. a. G Statement Justification EB + BG = EG b. Statement Justification c. d. e. Statement If m∠1 + m∠6 = 90, then ∠1 is complementary to ∠6. Justification Statement If m∠1 + m∠5 = m∠6 + m∠5, then m∠1 = m∠6. Justification Statement Justification ⊥ EG Given: AC Prove: ∠ABG is a right angle. 60 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. If ∠5 ∠6, then BF bisects ∠EBC. Two-Column Proofs ACTIVITY 1.7 continued Now I’m Convinced SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Vocabulary Organizer My Notes One way to learn how to write proofs is to start by studying completed proofs and then practicing simple proofs before moving on to more challenging proofs. EXAMPLE 6 Theorem: Vertical angles are congruent. 1 Given: ∠1 and ∠2 are vertical angles. Prove: ∠1 ∠2 1. 2. 3. 4. 5. Statements m∠1 + m∠3 = 180° m∠2 + m∠3 = 180° m∠1 + m∠3 = m∠2 + m∠3 m∠1 = m∠2 ∠1 ∠2 1. 2. 3. 4. 5. 3 2 Reasons Angle Addition Postulate Angle Addition Postulate Substitution Property Subtraction Property of Equality Definition of congruent angles EXAMPLE 7 D Theorem: The sum of the angles of a triangle is 180°. 5 A 1 4 Given: ABC Prove: m∠2 + m∠1 + m∠3 = 180° MATH TERMS © 2010 College Board. All rights reserved. B Statements , 1. Through point A, draw AD . || BC so that AD 2. m∠5 + m∠1 + m∠4 = 180° 3. ∠5 ∠2; ∠4 ∠3 4. m∠5 = m∠2; m∠4 = m∠3 5. m∠2 + m∠1 + m∠3 = 180° 2 3 C Reasons 1. Through any point not on a line, there is exactly one line || to the given line. 2. Angle Addition Postulate 3. If parallel lines are cut by a transversal, then alternate interior angles are congruent. 4. Definition of congruent angles 5. Substitution Property Line AD is an example of an auxiliary line. An auxiliary line is a line that is added to a figure to aid in the completion of a proof. Unit 1 • Proof, Parallel and Perpendicular Lines 61 ACTIVITY 1.7 Two-Column Proofs continued Now I’m Convinced SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer My Notes GUIDED EXAMPLE b Supply the missing statements and reasons. Theorem: In a plane, two lines perpendicular a to the same line are parallel. ∠1 and ∠2 are right angles. m∠1 = 90°; m∠2 = 90° m∠1 = m∠2 ∠1 ∠2 a || c Reasons 1. Given information 2. Definition of 3. Definition of 4. 5. 6. b TRY THESE B a a. Complete the proof. Given: a || b, c || d Prove: ∠2 ∠16 c d Statements 62 Reasons 1. c || d 1. 2. ∠2 ∠7 2. 3. a || b 3. 4. ∠7 ∠16 4. 5. ∠2 ∠16 5. SpringBoard® Mathematics with Meaning™ Geometry 12 9 2 10 11 1 14 3 4 15 13 7 16 6 8 5 © 2010 College Board. All rights reserved. 1. 2. 3. 4. 5. 6. 2 c Given: a ⊥ b, c ⊥ b Prove: a || c Statements 1 Two-Column Proofs ACTIVITY 1.7 continued Now I’m Convinced SUGGESTED LEARNING STRATEGIES: Group Presentation My Notes TRY THESE B (continued) b. Write a two-column proof for the following. b a c d 12 9 11 2 10 1 14 3 4 15 13 6 7 16 8 5 Given: a || b, c || d Prove: ∠1 ∠15 Reasons © 2010 College Board. All rights reserved. Statements Unit 1 • Proof, Parallel and Perpendicular Lines 63 ACTIVITY 1.7 Two-Column Proofs continued Now I’m Convinced CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show your work. Write paper. Show 9. Supply the Statements and Reasons. your work. Given: ∠1 is complementary to ∠2; BE bisects ∠DBC. Lines CF, DH, and EA intersect at point B. Use this Prove: ∠1 is complementary to ∠3. figure for Items 1–8. Write the definition, postulate, property, or theorem that justifies each statement. A C D 1 E B 6 2 3 D A 5 4 H B G 1 E 2 3 F C 1. ∠1 ∠5 Statements 2. If ∠2 is supplementary to ∠CBE, then m∠2 + m∠CBE = 180°. 1. BE bisects ∠DBC. 2. ∠2 ∠3 3. If ∠2 ∠3, then BF bisects ∠GBE. 6. If m∠3 = m∠6, then m∠3 + m∠2 = m∠6 + m∠2. ___ 7. If AB BE, then B is the midpoint of AE. 8. m∠2 + m∠3 = m∠EBG 3. Definition of congruent angles 4. ∠1 is complementary 4. to ∠2. 5. m∠1 + m∠2 = ____ 5. 6. m∠1 + m∠3 = 90° 6. 7. 7. Definition of complementary angles 10. MATHEMATICAL Choose one of the theorems R E F L E C T I O N about angles formed when parallel lines are cut by a transversal (Activity 1–6: Patios by Madeline). Explain how you would set up and prove the theorem in two-column format. 64 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. . ⊥ CF 5. If ∠DBF is a right angle, then HD ___ 1. 2. 3. 4. CB + BF = CF ___ Reasons Angles and Parallel Lines Embedded Assessment 2 THROUGH THE LOOKING GLASS Use after Activity 1.7. Light rays are bent as they pass through glass. Since a block of glass is a rectangular prism, the opposite sides are parallel and a ray is bent the same amount entering a piece of glass as exiting the glass. B G I R E F S A 1. Extend and through the block of glass. Label the points of intersection __IR __EF with SB and AG; X and Y, respectively. __ 2. Name a pair of same side interior angles formed by the transversal RE. © 2010 College Board. All rights reserved. 3. Identify the geometric term that describes the relationship between the pair of angles ∠IRE and ∠REF. 4. Explain why a ray exits the glass in a path parallel to the path in which it entered the glass. Unit 1 • Proof, Parallel and Perpendicular Lines 65 Embedded Assessment 2 Angles and Parallel Lines Use after Activity 1.7. THROUGH THE LOOKING GLASS Exemplary Proficient Emerging The student: • Gives the name of a pair of angles that are not same side interior angles. • Gives an incorrect geometric term for the relationship. The student: • Gives the correct name for the pair of angles. (2) • Gives the correct geometric term for the relationship between the pair of angles. (3) Representations #1 The student correctly extends the rays and labels the points of intersection correctly. (1) The student correctly extends the rays and labels some, but not all of the points of intersection correctly. The student extends the rays, but does not label the points of intersection correctly. Communication #4 The student gives a complete and correct explanation. (4) The student gives an incomplete explanation that does not contain mathematical errors. The student gives an explanation that is not correct. © 2010 College Board. All rights reserved. Math Knowledge #2, 3 66 SpringBoard® Mathematics with Meaning™ Geometry Equations of Parallel and Perpendicular Lines Skateboard Geometry ACTIVITY 1.8 SUGGESTED LEARNING STRATEGIES: Look for a Pattern My Notes The new ramp at the local skate park is shown above. In addition to the wooden ramp, an aluminum rail (not shown) is mounted to the edge of the ramp. While the image of the ramp may conjure thoughts of kickflips, nollies, and nose grinds, there are mathematical forces at work here as well. 1. Use the diagram of the ramp to complete the chart below: © 2010 College Board. All rights reserved. Describe two parts of the ramp that appear to be parallel. Describe two parts of the ramp that appear to be perpendicular. Describe two parts of the ramp that appear to be neither parallel nor perpendicular. Unit 1 • Proof, Parallel and Perpendicular Lines 67 ACTIVITY 1.8 Equations of Parallel and Perpendicular Lines continued Skateboard Geometry SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Work Backward My Notes y Y E A F –12 CONNECT TO ALGEBRA y y2 − y1 Slope = ____ = ______ x x2 − x1 V X W G –8 S T U R 8 O J I H 4 B P K L –4 C N 4 M D 8 x 2. The diagram above shows a cross-section of the skate ramp transposed onto a coordinate grid. Find the slopes of the following segments. __ a. AB __ b. XT __ c. WG _ d. TJ __ 3. Compare and contrast the slopes of the segments. Make conjectures about how the slopes of these segments relate to the positions of the segments on the graph. 68 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. e. AD Equations of Parallel and Perpendicular Lines ACTIVITY 1.8 continued Skateboard Geometry SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Predict and Confirm, Look for a Pattern, Identify a Subtask My Notes __ 2 x + 4. 4. The equation of the line containing BD is y = −__ 3 Identify the slope and the y-intercept of this line. CONNECT TO ALGEBRA 5. Based on your conjectures in Item 3, what would be the slope of the __ line containing JM? Use the formula for slope to verify your answer or provide an explanation of the method you used to determine the slope. The equation of a line in slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept of the line. __ 6. Identify the y-intercept of the line containing JM. Use the slope and the y-intercept to write the equation of this line. __ _ 7. If JM ⊥ RJ, identify _ the slope, y-intercept, and the equation for the line containing RJ. __ _ © 2010 College Board. All rights reserved. 8. Is BD⊥RJ? Explain your reasoning. Unit 1 • Proof, Parallel and Perpendicular Lines 69 ACTIVITY 1.8 Equations of Parallel and Perpendicular Lines continued Skateboard Geometry CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show your work. Write paper. Show 5. Consider the lines described below: your work. 3x - 2 line a: a line with the equation y = __ 5 1. What is the slope of a line parallel to the line line b: a line containing the points (-6, 1) 4 x - 7? with equation y = __ and (3, -14) 5 2. What is the slope of a line perpendicular to line c: a line containing the points (-1, 8) the line with equation y = -2x + 1? and (4, 11) 3. Find the equation of the line that has a y-intercept of -4 and is parallel to the line with equation y = 9x + 11. Use the vocabulary from the lesson to write a paragraph that describes the relationships among lines a, b, c, and d. 6. MATHEMATICAL What is the slope of a line R E F L E C T I O N parallel to the line with equation y = -3? What is the slope of a line perpendicular to the line with equation y = -3? © 2010 College Board. All rights reserved. 4. Triangle ABC has vertices at the points A(-2, 3), B (7, 4), and C(5, 22). What kind of triangle is ABC? Explain your answer. line d: a line with the equation -5x + 3y = 6 70 SpringBoard® Mathematics with Meaning™ Geometry Distance and Midpoint We Descartes ACTIVITY 1.9 SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Visualization, Simplify the Problem My Notes If mathematics concept popularity contest were ever held, the Pythagorean Theorem might be the runaway victor. With well over 300 known proofs, people from Plato to U. S. President James A. Garfield have spent at least part of their lives studying various proofs and applications of the theorem. While some of these proofs involve a level of complexity comparable to figuring out how to land a remote control car on the surface of Mars, the following is a very simple example of how the Pythagorean Theorem actually works. c MATH TERMS The Pythagorean Theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs of the right triangle. a b © 2010 College Board. All rights reserved. 1. Suppose you painted the large square extending from the hypotenuse of the triangle above. If it required exactly one can of paint to cover the area of the large square, make a conjecture as to how many cans you would need to paint the other two squares. Explain your reasoning. Unit 1 • Proof, Parallel and Perpendicular Lines 71 ACTIVITY 1.9 Distance and Midpoint continued We Descartes SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Identify a Subtask, Create Representations My Notes You can use a common Pythagorean triple to explore the idea of the painted squares further. Pythagorean triples are sets of whole numbers that represent the side lengths of a right triangle. Some common Pythagorean triples are (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17). 5 cm 3 cm 4 cm a=3 b=4 c=5 Area = a2 = 32 = 9 Area = b2 = 42 = 16 Area = c2 = 52 = 25 25 = 9 + 16 c2 = a2 + b2 Therefore, 52 = 32 + 42 72 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. From the information in the figure above, you can derive that the area of the large square is (5 cm)2 = 25 cm2, and that the areas of the smaller squares are (3 cm)2 = 9 cm2 and (4 cm)2 = 16 cm2, respectively. Distance and Midpoint ACTIVITY 1.9 continued We Descartes SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Simplify the Problem My Notes You can use the Pythagorean theorem to determine the distance between two points on a coordinate grid by drawing auxiliary lines to create right triangles. EXAMPLE 1 What is the distance between the points with coordinates (1, 4) and (13, 9)? Step 1: Start by plotting the points and connecting them. MATH TERMS The Cartesian Coordinate System was developed by the French philosopher and mathematician René Descartes in 1637 as a way to specify the position of a point or object on a plane. y (13, 9) (1, 4) x Step 2: Add some auxiliary line segments to create a right triangle. y © 2010 College Board. All rights reserved. (13, 9) (1, 4) x Unit 1 • Proof, Parallel and Perpendicular Lines 73 ACTIVITY 1.9 Distance and Midpoint continued We Descartes SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Simplify the Problem My Notes EXAMPLE 1 (continued) Step 3: Count units along the horizontal and vertical legs to determine the coordinates of the vertex of the right angle. The coordinates of the vertex of the right angle are (13, 4). y (13, 9) 5 units (1, 4) 12 units x Step 4: Use the Pythagorean theorem to find the distance between the two points (the endpoints of the hypotenuse of the right triangle). hypotenuse 2 = leg 2 + leg 2 c2 = 52 + 122 __ ________ __ ____ √c2 = √25 + 144 √c2 = √169 Solution: The distance between the points with coordinates (1, 4) and (13, 9) is 13 units. TRY THESE A Plot each pair of points on grid paper, and demonstrate how you can use the Pythagorean Theorem to find the distance between the points. a. (-3, 6) and (3, 14) 74 SpringBoard® Mathematics with Meaning™ Geometry b. (-8, 5) and (7, -3) © 2010 College Board. All rights reserved. c = 13 Distance and Midpoint ACTIVITY 1.9 continued We Descartes SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Simplify the Problem, Identify a Subtask My Notes While this method for finding the distance between two points will always work, it may not be practical to plot points on a coordinate grid and draw auxiliary lines each time you want to find the distance between them. In Example 1, you determined that the lengths of the legs of the right triangle were 5 and 12 units, respectively. You could also have determined 5. that the slope of the segment between the points (1, 4) and (13, 9) is ___ 12 To understand why getting the numbers 5 and 12 is more than a coincidence, recall some ways in which slope can be defined. Δy y2 - y1 Slope = rate of change = ___ = ______ Δx x2 - x1 CONNECT TO SCIENCE To find the length of the horizontal leg, you simply counted to get 12 units. However, if you think of the x-coordinates of the points as x-values on a horizontal number line, the distance between x-values = (Δx) = 13 − 1 = 12. 1 13 The Greek letter Δ (delta) is used in mathematical formulas to denote a change in quantities such as value, magnitude, or position. The expression Δ x would represent a change in x. Similarly, in Physics, the expression Δv (where v = velocity) would represent a change in velocity. In other words, the distance between the x-values is the same as the length of the horizontal leg of the right triangle. Likewise, the distance between the y-values is the same as the length of the vertical leg of the right triangle: (Δy) = 9 − 4 = 5 9 © 2010 College Board. All rights reserved. 4 Remember that distances are always measured in positive units. Unit 1 • Proof, Parallel and Perpendicular Lines 75 ACTIVITY 1.9 Distance and Midpoint continued We Descartes SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem, Create Representations My Notes You can use algebraic methods to derive a formula for determining the distance between two points. From Example 1, the Pythagorean equation looked like this: hypotenuse 2 = leg 2 + leg 2 c2 = 52 + 122 c2 = 25 + 144 __ ________ __ ____ √c2 = √25 + 144 √c2 = √169 13 = c Now, redefine the variables based on exploration of slope on the previous page. a = horizontal leg = Δx = x2 - x1 b = vertical leg = Δy = y2 - y1 c = hypotenuse = distance between two points = d Use substitution: hypotenuse 2 = leg 2 + leg 2 c2 = a2 + b2 distance2 = (Δx)2 + (Δy)2 d 2 = (x2 - x1)2 + (y2 - y1)2 CONNECT TO AP The coordinate geometry formulas introduced in this activity are used frequently in AP Calculus in a wide variety of applications. 76 3. In the space to the right, create a Venn diagram that compares and contrasts the mathematical operations performed in the Pythagorean theorem and the distance formula. SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. 2. Finish simplifying the equation to find the formula for determining the distance d between two points. Distance and Midpoint ACTIVITY 1.9 continued We Descartes SUGGESTED LEARNING STRATEGIES: Guess and Check, Create Representations, Work Backward, Visualization My Notes TRY THESE B Use the distance formula to find the distance between the points with the coordinates shown. a. (-4, 6) and (15, 6) b. (12, -5) and (-3, 7) c. (8, 1) and (14, -2) In Activity 1–5, you defined midpoint as the point on the segment that divides it into two congruent segments. 4. Explore various methods, and explain how you could determine the midpoint of the following segments: a. © 2010 College Board. All rights reserved. b. c. –3 5 17 0 12 y (9, 12) (0, 0) x Unit 1 • Proof, Parallel and Perpendicular Lines 77 ACTIVITY 1.9 Distance and Midpoint continued We Descartes SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem, Create Representations My Notes 5. Consider the right triangle below: y x a. Using the definition of midpoint, identify and label the midpoints of the legs of the triangle b. Draw an auxiliary line vertically from the midpoint of the horizontal leg through the hypotenuse d. Identify the coordinates of the point of intersection of the two auxiliary lines. This point represents the midpoint of the hypotenuse! TRY THESE C Find the midpoint of the segment whose endpoints have the given coordinates. a. (2, 3) and (-5, 11) b. (-8, -1) and (-7, -5) c. (6, -11) and (22, 16) 78 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. c. Draw an auxiliary line horizontally from the midpoint of the vertical leg through the hypotenuse. Distance and Midpoint ACTIVITY 1.9 continued We Descartes SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem My Notes As was the case with the distance formula, it is not always practical to plot the points on a coordinate grid and use auxiliary lines to find the midpoint of a segment. Mathematically, the coordinates of the midpoint of a segment are simply the averages of the x- and y-coordinates of the endpoints. EXAMPLE 2 __ What are the coordinates of the midpoint of AC with endpoints A(1,4) and C(11,10)? Step 1: Find the average of the x-coordinates: x-coordinate of point A = 1 Step 2: x-coordinate of C = 11 1 + 11 = 6 Average = ______ 2 Find the average of the y-coordinates. y-coordinate of point A = 4 y-coordinate of C = 10 4 + 10 = 7 Average = ______ 2 ___ Solution: The coordinates of the midpoint of AC are (6, 7). TRY THESE D __ a. Find the midpoint of QR with endpoints Q(-3, 14) and R(7, 5). © 2010 College Board. All rights reserved. b. Find the midpoint of S(13, 7) and T(-2, -7) c. Find the midpoint of each side of the right triangle with vertices D(-1, 2), E(8, 2), and F(8, 14). 6. Let (x1, y1) and (x2, y2) represent the coordinates of two points. Write a formula for finding the coordinates of midpoint M of the segment connecting those two points. Unit 1 • Proof, Parallel and Perpendicular Lines 79 ACTIVITY 1.9 Distance and Midpoint continued We Descartes CHECK YOUR UNDERSTANDING Writeyour youranswers answersononnotebook notebook paper. Show your work. Write paper or grid 5. Find and explain the errors that were made paper. Show your work. in the following calculation of midpoint. Then fix the errors, and determine the 1. Find the distance between the points correct answer. A(-8, -6) and B(4, 10). Find the midpoint of the segment with 2. Find the distance between the points C(5, 14) endpoints R(-2, 3) and S(13, -7). and D(-3, -9) -2 + 13 , _____ 3-7 ________ 3. Use your Venn diagram from Item 3 to write 2 2 a RAFT . -15 , ___ -4 = ____ 2 2 Role: Teacher = (-7.5, -2) Audience: A classmate who was absent for the 6. The vertices of XYZ are X(-3, -6), lesson on distance between two points Y(21, -6) and Z(21, 4). Format: Personal note a. Find the length of each side of the triangle. Topic: Explain the mathematical similarities b. Find the coordinates of the midpoint of and differences between the Pythagorean each side of the triangle. theorem and the distance formula. 7. MATHEMATICAL Segment RS is graphed on a 4. Find the midpoint of the segment with R E F L E C T I O N coordinate grid. Explain endpoints E(4, 11) and F(-11, -5). how you would determine the coordinates of the point on the segment that is one fourth of the distance from R to S. ( ) ) © 2010 College Board. All rights reserved. ( 80 SpringBoard® Mathematics with Meaning™ Geometry Slope, Distance, and Midpoint Embedded Assessment 3 GRAPH OF STEEL Use after Activity 1.9. The first hill of the Steel Dragon 2000 roller coaster in Nagashima, Japan, drops riders from a height of 318 ft. A portion of this first hill has been transposed onto a coordinate plane and is shown to the right. Write your answers on notebook paper or grid paper. Show your work. y 1–6: See below. 1. The structure of the supports for the hill consists of steel beams that run parallel and perpendicular to one another. The endpoints of the longer of the two support beams highlighted in Quadrant I are (0, 150) and (120, 0). If the endpoints of the other highlighted support beam are (0, 125) and (100, 0), verify and explain why the two beams are parallel. 2. Determine the equations of the lines containing the beams from Item 1, and explain how the equations of the lines can help you determine that the beams are parallel. 3. The equation of a line containing another support beam is 4 x + 150. Determine whether this beam is parallel or y = __ 5 perpendicular to the other two beams, and explain your reasoning. x 4. A linear portion of the first drop is also highlighted in the photo and has endpoints of (62, 258) and (110, 132). To the nearest foot, determine the distance between the endpoints of the linear section of the track. Justify your result by showing your work. © 2010 College Board. All rights reserved. 5. A camera is being installed at the midpoint of the linear portion of the track described in Item 4. Determine the coordinates where this camera should be placed. 6. Explain how you could use the distance formula to verify that the coordinates you determined for the midpoint are correct. Unit 1 • Proof, Parallel and Perpendicular Lines 81 Slope, Distance, and Midpoint Use after Activity 1.9. GRAPH OF STEEL 82 Exemplary Proficient Emerging Math Knowledge #2, 3, 4, 5 The student: • Gives the correct equations for the parallel lines. (2) • Determines the correct relationship among the beams. (3) • Determines the correct distance between the points. (4) • Correctly determines the coordinates of the midpoint of the linear section. (5) The student: • Gives the correct equation for one, but not both, of the parallel lines. • Uses a correct method to determine the distance but makes a computational error. • Uses a correct method to determine the coordinates of the midpoint but makes a computational error. The student: • Gives incorrect linear equations. • Does not determine the correct relationship among the beams. • Uses an incorrect method to determine the distance. • Uses an incorrect method to determine the coordinates of the midpoint. Representations #2 The student gives the correct equations for the parallel lines. (2) The student gives the correct equation for only one of the lines. The student gives the correct equation for neither of the lines. Communication #1, 2, 3, 4, 6 The student: • Verifies and writes a correct explanation for the reason the two beams are parallel. (1) • Gives a correct explanation for the connection between the equations and the parallel relationship of the lines. (2) • Gives a complete explanation for the parallelism and perpendicularity of the beams. (3) • Shows work that is complete. (4) • Gives a complete explanation of how the distance formula could be used to verify the coordinates of the midpoint. (6) The student: • Verifies and writes an incomplete explanation for the reason the two beams are parallel. • Gives an incomplete explanation for the connection between the equations and the parallel relationship of the lines. • Gives a complete explanation for the parallelism or the perpendicularity of the beams but not both. • Shows incomplete work that contains no mathematical errors. • Gives an incomplete explanation of how the distance formula could be used for verification. The student: • Says the beams are parallel, but does not give an explanation. • Gives an incorrect explanation for the connection. • Gives an incorrect explanation for the relationship among the beams. • Shows incorrect work • Gives an incorrect explanation for the use of the distance formula. SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. Embedded Assessment 3 Practice UNIT 1 Use this circle Q for Items 4–7. ACTIVITY 1.1 1. Which is the correct name for this line? U G E W R M ____ a. G c. MG b. GM d. ME Q 2. Use the diagram to name each of the following. T L M J N 4. Name the radii of circle Q. 5. Name the diameter(s) in circle Q. P Q 6. Name the chord(s) in circle Q. R 7. Which statement below must be true about circle Q? a. parallel lines b. perpendicular lines 3. In this diagram, m∠SUT = 25°. R a. The distance from U to W is the same as the distance from R to T. b. The distance from U to W is the same as the distance from Q to J. c. The distance from R to T is half the distance from Q to R. d. The distance from R to T is twice the distance from Q to J. © 2010 College Board. All rights reserved. Q ACTIVITY 1.2 P U S T a. Name another angle that has measure 25°. b. Name a pair of complementary angles. c. Name a pair of supplementary angles. 8. Use inductive reasoning to determine the next two terms in the sequence. a. 1, 3, 7, 15, 31, … b. 3, -6, 12, -24, 48, … 9. Write the first five terms of two different sequences for which 24 is the third term. 10. Generate a sequence using the description: the first term in the sequence is 2 and the terms increased by consecutive odd numbers beginning with 3. Unit 1 • Proof, Parallel and Perpendicular Lines 83 UNIT 1 Practice 11. Use this picture pattern. ACTIVITY 1.4 Construct a truth table for each of the following compound statements. 12. Use expressions for odd integers to confirm the conjecture that the product of two odd integers is an odd integer. ACTIVITY 1.3 13. Identify the property that justifies the statement: 5(x – 3) = 5x – 15 a. multiplication c. subtraction b. transitive d. distributive 23. p ∧ ˜q 24. ˜(˜p ∨ q) 25. p ∨ (q → p) ACTIVITY 1.5 26. If Q is between A and M and MQ = 7.3 and AM = 8.5, then QA = a. 5.8 Given: 5(x – 2) = 2x – 4 Prove: 15. Write the statement in if-then form. The sum of two even numbers is even. Use this statement for Items 16–19. If today is Thursday, then tomorrow is Friday. 16. State the conclusion of the statement. 17. Write the converse of the statement. 18. Write the inverse of the statement. 19. Write the contrapositive of the statement. 20. Given the false conditional statement, “If a vehicle is built to fly, then it is an airplane”, write a counter example. 21. Give an example of a statement that is false and logically equivalent to its inverse. 84 SpringBoard® Mathematics with Meaning™ Geometry c. 7.3 d. 14.6 27. Given: K is between H and J, HK = 2x - 5, KJ = 3x + 4, and HJ = 24. What is the value of x? a. 9 b. 5 c. 19 d. 3 __ 28. If K is the midpoint of HJ , HK = x + 6, and HJ = 4x - 6, then KJ = ? . a. 15 14. Complete the prove statement and write a two column proof for the equation: b. 1.2 b. 9 c. 4 d. 10 29. P lies in the interior of ∠RST. m∠RSP = 40° and m∠TSP = 10°. m∠RST = ? a. 100° b. 50° c. 30° d. 10° 30. QS bisects ∠PQR. If m∠PQS = 3x and m∠RQS = 2x + 6, then m∠PQR = ? . a. 18° b. 36° c. 30° d. 6° 31. ∠P and ∠Q are supplementary. m∠P = 5x + 3 and m∠Q = x + 3. x = ? a. 14 b. 0 c. 29 d. 30 © 2010 College Board. All rights reserved. a. Draw the next shape in the pattern, b. Write a sequence of numbers that could be used to express the pattern, c. Verbally describe the pattern of the sequence. 22. p → ˜q Practice , m∠BHC = 20°, and ⊥ CG In this figure, AE m∠DHE = 60°. Use this figure determine the angle measures for Items 32–35. UNIT 1 37. In this figure, m∠1 = 4x + 50 and m∠3 = 2x + 66. Show all work for parts a, b, and c. A F B 1 G H 2 C 3 D E 32. m∠CHD 33. m∠AHF 34. m∠GHF 35. m∠DHA 36. In this figure, m∠3 = x + 18, m∠4 = x + 15, and m∠5 = 4x + 3. Show all work for parts a, b, and c. a. What is the value of x? b. What is the measure of m∠3? c. What is the measure of m∠2? 38. Given: AE ⊥ AB , AD bisects ∠EAC, m∠CAB = 2x - 4 and m∠CAE = 3x + 14. Show all work for parts a and b. E © 2010 College Board. All rights reserved. 1 D C 5 3 4 A B a. What is the value of x? b. What is the measure of m∠DAC? a. What is the value of x? b. What is the measure of ∠1 c. Is ∠3 complementary to ∠1? Explain. Unit 1 • Proof, Parallel and Perpendicular Lines 85 Practice UNIT 1 ACTIVITY 1.6 49. If l n and m ⊥ n, then m∠3 = ? Use this diagram for Items 39–50. 50. If l n and m ⊥ l, then m∠13 = ? . ACTIVITY 1.7 n Use this diagram to identify the property, postulate, or theorem that justifies each statement in Items 51–55. l 12 11 10 16 9 1 13 2 14 3 5 P D 4 2 15 8 A 4 Q E 6 7 m t R 39. List all angles in the diagram that form a corresponding angle pair with ∠5. 51. PQ + QR = PR 41. List all angles in the diagram that form a same side interior angle pair with ∠14. 42. Given l n. If m∠6 = 120°, then m∠16 = ? . 43. Given l n. If m∠12 = 5x +10 and m∠14 = 6x - 4, then x = ? and m∠4 = ? . 44. Given l n. If m∠14 = 75°, then m∠2 = ? . ? . 47. Given l n. If m∠16 = 5x + 18 and m∠15 = 3x+ 2, then x = ? and m∠8 = ? . 48. If m∠10 = 135°, m∠12 = 75°, and l n determine the measure of each angle. Justify each answer. a. m∠16 c. m∠8 e. m∠3 Angle addition postulate Addition property Definition of congruent segments Segment addition postulate ___ ___ b. m∠15 d. m∠14 SpringBoard® Mathematics with Meaning™ Geometry ___ 52. If Q is the midpoint of PR, then PQ QR. a. b. c. d. Definition of midpoint Definition of congruent segments Definition of segment bisector Segment addition postulate 53. ∠3 ∠4 45. Given l n. If m∠9 = 3x + 12 and m∠15 = 2x + 27, then x = ? and m∠10 = ? . 46. Given l n. If m∠3 = 100°, then m∠2 = a. b. c. d. a. b. c. d. Definition of linear pair Definition of congruent angles Vertical angles are congruent. Definition of angle bisector 54. If ∠1 is complementary to ∠2, then m∠1 + m∠2 = 90 a. b. c. d. Angle Addition Postulate Addition property Definition of perpendicular Definition of complementary © 2010 College Board. All rights reserved. 40. List all angles in the diagram that form an alternate interior angle pair with ∠14. 86 3 1 Practice 55. If m∠3 = 90, then ∠3 is a right angle. ACTIVITY 1.9 a. Definition of perpendicular b. Definition of complementary angles c. Definition of right angle d. Vertical angles are congruent. 56. Write a two-column proof. 61. Calculate the distance between the points A(-4, 2) and B(15, 6). Prove: ∠5 ∠12 b a 9 1 2 4 62. Calculate the distance between the points R(1.5, 7) and S(-2.3, -8). 63. Kevin and Clarice both live on a street that runs through the center of town. If the police station marks the midpoint between their houses, at what point is the police station on the number line? Given: a b, c d 12 10 -11 11 15 3 6 7 5 14 16 13 8 c d ACTIVITY 1.8 57. Determine the slope of the line that contains the points with coordinates (1, 5) and (-2, 7). © 2010 College Board. All rights reserved. UNIT 1 58. Which of the following is NOT an equation for a 1 x - 6? line parallel to y = __ 2 2 __ a. y = x + 6 b. y = 0.5x - 3 4 1x + 1 d. y = 2x - 4 c. y = __ 2 59. Determine the slope of a line perpendicular to the line with equation 6x - 4y = 30. Kevin’s House 1 Clarice’s House 64. Determine the coordinates of the midpoint of the segment with endpoints R(3, 16) and S(7, -6). 65. Determine the coordinates of the midpoint of the segment with endpoints W(-5, 10.2) and X(12, 4.5). 66. Two explorers on an expedition to the Arctic Circle have radioed their coordinates to base camp. Explorer A is at coordinates (-26, -15). Explorer B is at coordinates (13, 21). The base camp is located at the origin. a. Determine the linear distance between the two explorers. b. Determine the midpoint between the two explorers. c. Determine the distance between the midpoint of the explorers and the base camp. 60. Line m contains the points with coordinates (-4, 1) and (5, 8), and line n contains the points with coordinates (6, -2) and (10, 7). Are the lines parallel, perpendicular, or neither? Justify your answer. Unit 1 • Proof, Parallel and Perpendicular Lines 87 UNIT 1 Reflection An important aspect of growing as a learner is to take the time to reflect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning. Essential Questions 1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specific examples from concepts and activities in the unit. Why are properties, postulates, and theorems important in mathematics? How are angles and parallel and perpendicular lines used in real-world settings? Academic Vocabulary 2. Look at the following academic vocabulary words: angle bisector counterexample perpendicular complementary angles deductive reasoning postulate conditional statement inductive reasoning proof congruent midpoint of a segment supplementary angles conjecture parallel theorem Choose three words and explain your understanding of each word and why each is important in your study of math. Self-Evaluation 3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each. Is Your Understanding Strong (S) or Weak (W)? Concept 1 Concept 2 Concept 3 a. What will you do to address each weakness? b. What strategies or class activities were particularly helpful in learning the concepts you identified as strengths? Give examples to explain. 4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world? 88 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. All rights reserved. Unit Concepts Math Standards Review Unit 1 1. Keisha walked from her house to Doris’ house. This can best be represented by which of the following: A. a ray C. a line B. a segment D. a point 2. Sam wanted to go fishing. He took his fishing pole and walked 6 blocks due south to pick up his friend. Together, they walked 8 blocks due east before they reached the lake. There is a path that goes directly from Sam’s house to the lake. If he had taken that path, how many blocks would he have walked? 1. Ⓐ Ⓑ Ⓒ Ⓓ 2. ‒ ○ ⊘⊘⊘⊘ • ○ • ○ • ○ • ○ • ○ • ○ ⓪⓪⓪ ⓪ ⓪ ⓪ ①①① ① ① ① ②②② ② ② ② ③③③ ③ ③ ③ ④④④ ④ ④ ④ ⑤⑤⑤ ⑤ ⑤ ⑤ ⑥⑥⑥ ⑥ ⑥ ⑥ ⑦⑦⑦ ⑦ ⑦ ⑦ ⑧⑧⑧ ⑧ ⑧ ⑧ ⑨⑨⑨ ⑨ ⑨ ⑨ 3. To amuse her brother, Shirka was making triangular formations 3. out of rocks. She made the following pattern: ‒ ○ ⊘⊘⊘⊘ © 2010 College Board. All rights reserved. • ○ • ○ • ○ • ○ • ○ • ○ How many rocks will be in the next triangular formation in the pattern? ⓪⓪⓪ ⓪ ⓪ ⓪ ①①① ① ① ① ②②② ② ② ② ③③③ ③ ③ ③ ④④④ ④ ④ ④ ⑤⑤⑤ ⑤ ⑤ ⑤ ⑥⑥⑥ ⑥ ⑥ ⑥ ⑦⑦⑦ ⑦ ⑦ ⑦ ⑧⑧⑧ ⑧ ⑧ ⑧ ⑨⑨⑨ ⑨ ⑨ ⑨ Unit 1 • Proof, Parallel and Perpendicular Lines 89 Math Standards Review Unit 1 (continued) 4. When camping, Sudi and Raul did not want to get lost. They made a coordinate grid for their camp site. Camp was at the Solve point (0, 0). On their grid, Sudi went to the point (-3, 10) and Explain Raul went to the point (0, 6). Read Part A: On the grid below, draw a diagram of the camp site and label the campsite location, Sudi’s location, and Raul’s location. y 12 10 8 6 4 2 –12 –10 –8 –6 –4 –2 2 4 6 8 10 12 x –2 –4 –6 –8 –10 Part B: Find the distance between Sudi and Raul. Part C: Find the midpoint between Sudi and Raul. 90 SpringBoard® Mathematics with Meaning™ Geometry © 2010 College Board. 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