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Home , Concave polygon, Kite (geometry)

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CHAPTER

5 Discovering and Proving Properties

Overview polygon is equiangular, each interior has 180(n 2) measure degrees, or 180 360 degrees. In this chapter, students discover properties of n n . In Lessons 5.1 and 5.2, they discover Properties of the various can be the sum of the angle measures in a polygon and seen from their . A is symmetric the sum of the measures of a set of exterior about the through its angles. From of a polygon. The exploration looks at patterns this fact it can be seen that this diagonal bisects in star polygons. Discovering kite and the vertex angles and the other diagonal, that the properties in Lesson 5.3 leads students to seven two are , and that the conjectures about these quadrilaterals. In nonvertex angles are congruent. Lesson 5.4, students discover two properties of An also has reflectional the midsegment of a and two properties symmetry, but over the through the midpoints of the midsegment of a trapezoid. In Lessons 5.5 of the two parallel sides. This symmetry reveals and 5.6, students investigate properties of parallel- that consecutive angles (on the bases) are ograms,,,and . congruent, as are the diagonals. Between these lessons, Using Your Algebra Skills 5 reviews linear equations. In Lesson 5.7, In contrast, a has 2-fold rotational students use paragraph and flowchart proofs to symmetry about the at which its diagonals support those conjectures with deductive intersect. Because of the rotation, its properties

reasoning. mainly concern opposites: Opposite sides and EDITION TEACHER’S angles are congruent. When the figure is rotated The Mathematics 180° about the intersection of the diagonals, each half of a diagonal is taken to the other half, so the This chapter moves down the hierarchy of poly- diagonals bisect each other. gons from the most general polygons and quadri- As students examine how polygons are related, laterals to the most specific quadrilaterals, squares. you need to lead the way in modeling careful use It begins by extending the angle sum properties of of language. For example, it’s easy to say “In a to polygons in general. Then it focuses on , adjacent sides are congruent.” That’s special kinds of quadrilaterals: , kites, true in some rectangles. But the statement is false, and . It continues by examining two because in mathematics a often means “any” or special kinds of parallelograms: rhombuses and “every” or “all.” Mathematical statements are rectangles (and squares, which are both). Under- understood to begin with one of the words All lying the progression through parallelograms is the (equivalently, Any or Every), Some (equivalently, theme that properties of one category are inherited At least one), or No. When we say “A rectangle is a by all subcategories. For example, the property of ,”we mean “All rectangles are quadri- parallelograms that their diagonals bisect each laterals.” Instead of saying “A rectangle is a other is true for all special kinds of parallelograms, ,”we should say “Some rectangles are such as rhombuses and rectangles. squares” or “At least one rectangle is a square.” All polygons of n sides share certain properties. Also, instead of saying “All rectangles are not The sum of the measures of their interior angles is darts,”which is ambiguous, we should say “No 180(n 2) degrees, and the sum of the measures rectangle is a dart.” of their exterior angles is 360°. Consequently, if a

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At this point in the course, many students are at Resources Materials van Hiele level 2. This chapter’s consideration of inheritance of properties challenges them without Discovering Geometry Resources • construction tools asking them to move to a new level. If students Teaching and Worksheet Masters •protractors propose different valid proofs of conjectures, you Lessons 5.1, 5.3–5.7, and Chapter 5 Review •calculators can begin to lay the groundwork for level 3 by Discovering Geometry with The Geometer’s legitimizing those proofs. You can also begin to • double-edged Sketchpad straightedges ask students whether the reasons they’re citing in Lessons 5.1–5.6 flowchart proofs are conjectures or definitions, • graph paper Using Your Algebra Skills 5 although it’s too early to expect most students to • scissors appreciate the differences very deeply. Assessment Resources • The Geometer’s Quiz 1 (Lessons 5.1 and 5.2) Sketchpad, optional Using This Chapter Quiz 2 (Lessons 5.3 and 5.4) Quiz 3 (Lessons 5.5–5.7) TEACHER’S EDITION Lessons 5.1 and 5.2 are both quick, single- Chapter 5 Test investigation lessons, but there are many follow- Chapter 5 Constructive Assessment Options ups that you can do. Use the extensions and the Practice Your Skills for Chapter 5 Take Another Look activities if you have class time. Cooperative Learning Using Jigsaw Condensed Lessons for Chapter 5 Jigsaw methods of cooperative group learning Other Resources involve assigning different problems, or pieces of a www.keypress.com/DG problem, to different groups. Later the groups report on their thoughts to the entire class or divide up so that each group can share its thinking with one other group. Jigsaw work has the advantage of mimicking a workplace situation, in which teams don’t replicate efforts. In addition, students learn to learn from and teach each other. The jigsaw method is suggested for the investiga- tions in Lessons 5.1 and 5.2 and the exercises in Lesson 5.7.

Pacing Guide day 1day 2day 3day 4day 5day 6day 7day 8day 9day 10 standard 5.1 5.2 quiz, 5.3 5.4 5.5 quiz, 5.6 5.7 quiz, review Algebra 5 review enriched 5.1 5.2 quiz, 5.3 5.4, project 5.5 quiz, Algebra 5.6 5.7 quiz, review Exploration 5, project block 5.1, 5.2 quiz, 5.3, 5.4, quiz, 5.5 Algebra 5, 5.7 quiz, review, assessment Exploration project 5.6 project day 11 day 12 day 13 day 14 day 15 day 16 day 17 day 18 day 19 day 20 standard assessment

enriched project, assessment, review TAL

257B CHAPTER 5 INTERLEAF Discovering and Proving Polygon Properties

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CHAPTER Discovering and 5 Proving Polygon CHAPTER 5 Properties OBJECTIVES

● Discover the sum of both the interior and the exterior angle measures in a polygon ● Explore angle measures of equiangular and star polygons ● Discover properties of kites, trapezoids, and various kinds of parallelograms ● Define and discover properties of mid- segments in triangles and trapezoids ● Practice writing flowchart and paragraph proofs ● Review graphing and writing linear equations ● Learn new vocabulary TEACHER’S EDITION TEACHER’S ● Practice construction skills The mathematicians may well nod their heads OBJECTIVES in a friendly and interested manner—I still am a In this chapter you will ● Develop reasoning, tinkerer to them. And the “artistic” ones are primarily ● study properties of convex problem-solving skills, irritated. Still, maybe I’m on the right track if I experience more polygons and cooperative behavior joy from my own little images than from the most beautiful ● discover relationships camera in the world . . .” among their angles, sides, and diagonals Still Life and Street, M. C. Escher, 1967–1968 ● learn about real-world ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. applications of special polygons

The woodcut by M. C. Escher shows many different polygons and near-polygons, as well as polygons changing shape as they recede into the background. [Ask] “What polygons do you see?” [rectangles (books, windows), squares (windows), triangles (fence on the roof ), parallelograms (tops of books seen in perspective), trapezoids (sides of buildings receding into background)]

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LESSON 5.1 LESSON Polygon Sum Conjecture There are many kinds of triangles, but 5.1 in Chapter 4, you discovered that the sum of their angle measures is always PLANNING 180°. In this lesson you’ll investigate I find that the harder I work, the sum of the angle measures in the more luck I seem to have. convex quadrilaterals, LESSON OUTLINE , and other THOMAS JEFFERSON polygons. Then One day: you’ll look for a 15 min Investigation pattern in the sum 10 min Sharing of the angle measures in any polygon. 5min Closing 15 min Exercises MATERIALS Investigation ⅷ construction tools Is There a Polygon Sum Formula?

ⅷ protractors You will need For this investigation each person in your group should draw a different version of ⅷ ● protractor the same polygon. For example, if your group is investigating , try to think TEACHER’S EDITION calculators of different ways you could draw a . ⅷ Window Frame (T) for One Step ⅷ Quadrilateral Sum Conjecture (W), optional ⅷ Sketchpad activity Polygon Sum Conjecture, optional Step 1 Draw the polygon. Carefully measure all the interior angles, then find the sum. Step 2 Share your results with your group. If you measured carefully, you should all TEACHING have the same sum! If your answers aren’t exactly the same, find the average.

Students find that all polygons Step 3 Copy the table below. Repeat Steps 1 and 2 with different polygons, or share results with other groups. Complete the table. with the same number of sides have the same angle measure Number of sides of polygon 345678…n sum. Conversely, knowing this Sum of measures of angles 180°360° 540° 720° 900° 1080° … sum allows students to find the 180°(n 2) number of sides. You can now make some conjectures. [ELL] Yo u m i g h t r e v i e w t h e t e r m s C-29 quadrilateral, , hexagon, Quadrilateral Sum Conjecture , and . The sum of the measures of the four interior angles of any quadrilateral is ? . 360°

C-30 Guiding the Investigation Pentagon Sum Conjecture The sum of the measures of the five interior angles of any pentagon is ? . 540° One Step Show the Window Frame trans- parency and pose this problem: “You need to build a window angle sum of any polygon. Encourage those who Step 2 Don’t worry too much if some groups aren’t frame for an octagonal window finish first to try to prove their conjectures. getting multiples of 180°; they’ll see a pattern when like this one. To make the frame, results are shared. you’ll cut identical trapezoidal Assign each group a polygon with a different pieces. What are the measures of number of sides. You need not assign a triangle, Step 3 Groups should share with each other, either the angles of the trapezoids?” As because students saw the Triangle Sum Conjecture formally or by sending representatives to visit other students work, encourage them to in Chapter 4. Remind students to look for patterns, groups. You might also have the table on the board think about the sum of angle as they did in Chapter 2. or overhead to be filled in by group representatives when each group has completed its investigation. measures of . Suggest Step 1 To get enough data, each student may need You might keep the class together for the remaining that they use inductive reasoning to draw several polygons. For advanced classes you steps. Students should copy the full table into their to make a conjecture about the might suggest that some of the polygons be concave. notebooks for future reference.

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If a polygon has n sides, it is called an n-gon. Step 4 If students are having Step 4 Look for a pattern in the completed table. Write a general formula for the sum of difficulty finding a pattern and a Step 5 The the angle measures of a polygon in terms of the number of sides, n. formula, you might have them number of triangles add a third row to their table formed is n 2. Step 5 Draw all the diagonals from one vertex of your polygon. How many triangles do the diagonals labeled “Number of triangles create? How many triangles do the diagonals formed by diagonals from one create in the at right? Try some vertex.” They can complete this other polygons. How do the number of row by using their own polygons triangles relate to n? and by sharing with other State your observations as a conjecture. groups. Developing Proof C-31 Polygon Sum Conjecture Because d, e, u, and v are The sum of the measures of the n interior angles of an n-gon is ? . 180°(n 2) measures of angles, it’s valid to write d e mD and u v mU. D Possible proof: q d u Developing Proof As a group, write a proof of the d e 180° and e a v 180° by Ⅲ Quadrilateral Sum Conjecture using the diagram at right. Q q a A the Triangle Sum Conjecture. Does the Polygon Sum Conjecture apply to concave q d e a v u 360° polygons? This question is left for you to explore as a u v by addition property of equality. Take Another Look activity. Therefore, the sum of the U measures of the angles of a EXERCISES You will need quadrilateral is 360°. Geometry software ᮣ 1. Use the Polygon Sum Conjecture to complete the table. for Exercise 19 SHARING IDEAS As a class, agree on the state- Number of sides of polygon 7 8 9 10 11 20 55 100 ments of the conjectures to be Sum of measures of angles 900° 1080° 1260° 1440° 1620° 3240° 9540° 17640° added to students’ notebooks. In discussing the inductive 2. What is the measure of each angle of an equiangular pentagon? An equiangular reasoning process, [Ask] “What hexagon? Complete the table. contributes to slight differences in Number of sides of 5 6 7 8 9 10 12 16 100 answers?” [measurement error] You might ask whether students Measure of each angle 4 1 2 108° 120° 128° 135° 140° 144° 150° 157° 176° EDITION TEACHER’S of equiangular polygon 7 2 5 can explain why the sum is what it is by using techniques similar to In Exercises 3–8, use your conjectures to calculate the measure of each lettered angle. those used for the Triangle Sum 3. a ? 122° 4. b ? 136° 5. e ? 108° Conjecture. For example, they ? might tear off angles and form f 36° 76 70 them around a point (see exten- b sion A on page 261). Or, if you 110 e imagined with your class the forward/backward walk around 72 116 a 68 f a triangle turning through the interior angles, students might imagine walking around the polygon. NCTM STANDARDS LESSON OBJECTIVES Follow through with a question CONTENT PROCESS ⅷ Discover the sum of the angle measures in a polygon about equiangular polygons. For example, what’s the measure of ߜ ⅷ Practice construction skills Number Problem Solving each angle of an equiangular ⅷ ߜ Algebra ߜ Reasoning Develop reasoning, problem-solving skills, and cooperative polygon of 12 sides? Students behavior need to realize that the number of ߜ Geometry ߜ Communication angles equals the number of sides. ߜ Measurement Connections Let students derive a formula for this measure if they wish, but ߜ Data/Probability Representation don’t insist that they do so.

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Closing the Lesson 6. c ? 108°7. g ? 105° 8. j ? 120° d ? 106°h ? 82° k ? 38° The Polygon Sum Conjecture can h be used to find the sum of angle 66 d k 38 measures for any polygon. More- c 130 117 108 over, students can use it to find j 30 the number of sides given the 44 78 g total angle measure. If students seem uncomfortable with these ideas, you might do one of Exer- 9. Developing Proof What’s wrong 10. Developing Proof What’s 11. Three regular polygons meet cises 3–8 before having the class with this picture? wrong with this picture? at point A. How many sides begin on the other exercises. The sum of the interior does the largest polygon have? angle measures of the 18 quadrilateral is 358°. It 154 82 should be 360°. 135 BUILDING 102 49 A UNDERSTANDING

76 The exercises give practice in applying the Polygon Sum Conjecture. 12. Developing Proof Trace the figure at right. Calculate each e p 36 lettered angle measure. Explain TEACHER’S EDITION ASSIGNING HOMEWORK 60 how you determined the 1 2 106 Essential 1–14 measures of angles d, e, and f. n m 116 Performance d a k 1 assessment 15 13. How many sides does a polygon have if the sum of its 98 bc Portfolio 12 angle measures is 2700°? 17 j 2 16 g Journal 14. How many sides does an 116 f equiangular polygon have if 138 122 Group 17 87 each interior angle measures h 77 Review 18–21 156°? 15 Algebra review 11, 13, 14, 17 15. Archaeologist Ertha Diggs has uncovered a piece of a ceramic plate. She measures it and finds MATERIALS that each side has the same length and each ⅷ Exercise 12 (T), optional angle has the same measure. She conjectures that the original plate was the shape of a . She knows | ᮣ that if the original plate was a regular Helping with the Exercises 16-gon, it was probably a ceremonial dish Exercises 3–8 If students are from the third century. If it was a regular having difficulty, suggest that 18-gon, it was probably a palace dinner plate from the twelfth century. they copy the diagrams and mark them with what they know. If each angle measures 160°, from what century did the plate likely originate? 10. The measures of the inte- the twelfth century rior angles shown sum to 554°. However, the figure is a pentagon, so the measures Exercise 12 Suggest that students use pair-share on 12. a 116°, b 64°, c 90°, d 82°, e 99°, of its interior angles should the angle-chase exercises. One student finds an f 88°, g 150°, h 56°, j 106°, k 74°, sum to 540°. answer and explains why; then the partner agrees m 136°, n 118°, p 99°; Possible explanation: Exercise 11 This problem fore- or disagrees, then finds another answer and The sum of the angles of a quadrilateral is 360°, shadows Chapter 7 and tilings. explains why. Partners take turns back and forth as so a b 98° d 360°. Substituting 116° for You might ask what other combi- they progress through the network. There are varia- a and 64° for b gives d 82°. Using the larger nations of polygons students can tions on this method that you can use. For example, quadrilateral, e p 64° 98° 360°. Substi- find that fit around a point. when student A finds an answer it is the task of tuting e for p, the equation simplifies to 2e 198°, student B to explain why; then student B finds the so e 99°. The sum of the angles of a pentagon is next answer and student A explains why. 540°, so e p f 138° 116° 540°. Substi- tuting 99° for e and p gives f 88°.

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16. Application You need to build a window frame for an octagonal Exercise 15 To extend this exercise, window like this one. To make the frame, you’ll cut identical suggestthatstudentsbringin trapezoidal pieces. What are the measures of the angles of the pieces of supposed “archaeological trapezoids? Explain how you found these measures. treasures” (fragments of regular 17. Developing Proof Restate your proof of the Quadrilateral Sum polygons they make from paper, Conjecture from the developing proof activity on page 259. Then use plastic, or other materials) and this diagram to show another way to prove the same conjecture. have other students determine the Q number of sides the unbroken h a Answers will vary; see the answer for Developing “artifact”would have had. Proof on page 259. Using the Triangle Sum b U c Conjecture, a b j c d k e f l Exercise 16 This exercise is the j g i g h i 4(180°), or 720°. The four angles one-step investigation. D k f l in the center sum to 360°, so j k l i 360°. Subtract to get a b c d e f 16. The angles of the trapezoid g h 360°. measure 67.5° and 112.5°; 67.5° d e is half the value of each angle of A a regular octagon, and 112.5° is half the value of 360° 135°. ᮣ Review x 135 60 2.5 18. This figure is a detail of one vertex of the tiling at the beginning of 67.5 this lesson. Find the missing angle measure x. x 120° Exercise 17 19. Technology Use geometry software to construct a quadrilateral and One alternative to locate the midpoints of its four sides. Construct segments drawing diagonals from one connecting the midpoints of opposite sides. Construct the point of intersection of vertex is to select a random point the two segments. Drag a vertex or a side so that the quadrilateral becomes concave. within the polygon and draw Observe these segments and make a conjecture.The segments joining P segments to the vertices. This the opposite midpoints of a quadrilateral always bisect each other. 20. Line is parallel to AB. As P moves to the right along , which creates n triangles, and the sum of these measures will always increase? D of the angle measures of the n A. The distance PA B. The of ABP triangles is 180°n.But you do not C. The measure of APB D. The measure of ABP want the angle sum around the ABpoint, so you subtract 360°. Thus 4.2 21. Draw a counterexample to show that this statement is false: If a Counterexample: The base angles of an the formula is 180°n 360°, or triangle is isosceles, then its base angles are not complementary. isosceles measure 45°; thus 180°(n 2). they are complementary. Exercise 20 As vertex P moves IMPROVING YOUR VISUAL THINKING SKILLS from left to right: The distance EDITION TEACHER’S PA decreases and then soon Net Puzzle increases, the perimeter of the The clear cube shown has the letters DOT printed triangle decreases before it on one face. When a light is shined on that face, increases, and the measure of the image of DOT appears on the opposite face. APB increases for a while but The image of DOT on the opposite face is then then decreases. painted. Copy the net of the cube and sketch the DOT

painted image of the word, DOT, on the DOT EXTENSIONS correct square and in the correct position. A. In addition to measuring angles of a quadrilateral or pentagon with a protractor, students can tear off the corners pentagon will overlap and will surround a point and arrange the angles of a 1 IMPROVING VISUAL THINKING SKILLS 12 times (for a sum of 540°).] polygon around a point, as B. Students could build window frames or picture they did when investigating frames based on different regular polygons. triangle sums. [The angles of a

DOT quadrilateral completely surround C. Use Take Another Look activities 1 and 2 on a point, so the sum of their

page 307. measures is 360°. The angles of a DOT

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LESSON LESSON Exterior Angles of 5.2 5.2 a Polygon In Lesson 5.1, you discovered a formula for PLANNING the sum of the measures of the interior If someone had told me I angles of any . In this lesson would be Pope someday, I you will discover a formula for the sum of LESSON OUTLINE Set of exterior would have studied harder. the measures of the exterior angles of a convex polygon. angles One day: POPE JOHN PAUL I 10 min Investigation Best known for her 10 min Sharing participation in the Dada Movement, German artist 10 min Extensions Hannah Hoch (1889–1978) painted Emerging Order in 5min Closing the Cubist style. Do you see any examples of exterior 10 min Exercises angles in the painting? MATERIALS

ⅷ construction tools

ⅷ

TEACHER’S EDITION protractors ⅷ calculators ⅷ Sketchpad demonstration Exterior Angles, optional ⅷ Sketchpad activity Exterior Angles of a Polygon, optional Investigation Is There an Exterior Angle Sum? TEACHING You will need Let’s use some inductive and deductive reasoning to find the exterior angle measures in a polygon. The sum of the exterior angles of ● a straightedge ● a protractor any polygon is constant and is Each person in your group should draw the same kind of polygon for Steps 1–5. closely related to the sum of the Step 1 Draw a large polygon. Extend its sides to form a set of exterior angles. interior angles. Step 2 Measure all the interior angles of the polygon except one. Use the Polygon Sum Conjecture to calculate the measure of the remaining interior angle. Check your Guiding the Investigation answer using your protractor. Step 3 Use the Linear Pair Conjecture to calculate the measure of each exterior angle. One Step Step 4 Calculate the sum of the measures of the exterior angles. Share your results with Ask students to draw any your group members. polygon and imagine walking Step 5 Exterior angle Step 5 Repeat Steps 1–4 with different kinds of polygons, or share results with other around it, always turning through measures add up to 360° groups. Make a table to keep track of the number of sides and the sum of the an exterior angle and keeping for any polygon. exterior angle measures for each kind of polygon. Find a formula for the sum of track of the number of degrees the measures of a polygon’s exterior angles. they’ve turned. Encourage them to experiment with a variety of polygons, keeping track of the number of sides and the total Step 1 As needed, introduce the concept of an exte- for their polygon and may try to generalize this measure of all the angles turned. rior angle of a polygon. Students should draw only finding by writing a formula that includes n. For one exterior angle at each vertex. You need not example, the pentagons group might write You might assign each group to specify at this time which angle they should draw. 180°(n 3). investigate the same kind of polygon for which they found Step 4 Students can check their findings by You might have groups check their angle sums by the sum of the interior angle measuring the exterior angles. cutting out the exterior angles and arranging their vertices around a point. measures in Lesson 5.1. Step 5 The book tells students to “find a formula” for the sum of the exterior angle measures. This is intentionally somewhat misleading. They will find that the sum of the exterior angle measures is 360°

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Step 7 If students have trouble, Exterior Angle Sum Conjecture C-32 remind them that the number For any polygon, the sum of the measures of a set of exterior angles is ? . 360° of angles equals the number of sides, so n can be used for either. Equiangular Polygon Conjecture Ask why the two expressions are equivalent algebraically. After 180°(n2) writing n , [Ask] “What does the order of operations tell us to do first?” [distribute] “The division symbol is a grouping symbol, so n should divide into [ᮣ For an interactive version of this sketch, see the Dynamic Geometry Exploration The Exterior Angle what?” [180n and 360] “To find ᮤ Sum of a Polygon at www.keymath.com/DG . ] the measure of an interior angle keymath.com/DG of an equiangular polygon, Step 6 Study the software construction above. Explain how it demonstrates the Exterior Angle Sum Conjecture. The exterior angles slide into place as angles around a point, would it be easier to use the sum which always add up to 360°. of exterior angles or the sum of Step 7 Using the Polygon Sum Conjecture, write a formula for the measure of each 180°(n 2) interior angles?” [Exterior: interior angle in an equiangular polygon. 360° n interior angle 180° n ; Step 8 Using the Exterior Angle Sum Conjecture, write the formula for the measure of interior: interior angle each exterior angle in an equiangular polygon. 360° 180°(n 2) n n . Exterior is usually easier.]

SHARING IDEAS Have students share a variety of formulas for the Exterior Angle Sum Conjecture. Let them discuss the discrepancies until Step 9 Using your results from Step 8, you can write the formula for an interior angle of an they agree that the sum is always equiangular polygon a different way. How do you find the measure of an interior 360°. Many students learn best angle if you know the measure of its exterior angle? Complete the next conjecture. through discussion, so try not to limit it. Students may object that C-33 a constant isn’t a formula. Say Equiangular Polygon Conjecture 360° 180°(n2) 180° n ; n that constant functions or EDITION TEACHER’S You can find the measure of each interior angle of an equiangular n-gon by expressions are very important in using either of these formulas: ? or ? . mathematics. Remind students of the equation for a horizontal line, for example, y 4. You might also make the point that it is important to consider a variety EXERCISES of examples before making a ᮣ 1. What is the sum of the measures of the exterior angles of a ? 360° conjecture. Help students reach consensus on the phrasing to 2. What is the measure of an exterior angle of an equiangular pentagon? 72°; 60° write in their notebooks. An equiangular hexagon? Ask if it matters which exterior angle they chose at each vertex. NCTM STANDARDS LESSON OBJECTIVES Help them see that, where a CONTENT PROCESS ⅷ Discover the sum of the measures of the exterior angles of a polygon is convex, the two exte- polygon rior angles are vertical angles Number Problem Solving and thus have the same measure. ⅷ Write formulas for the measure of the interior angle of an ߜ Algebra ߜ Reasoning equiangular polygon ߜ Geometry ߜ Communication ⅷ Practice construction skills ߜ Measurement Connections ⅷ Develop reasoning, problem-solving skills, and cooperative behavior Data/Probability Representation

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If you did not do the one-step 3. How many sides does a regular polygon have if each exterior angle measures 24°? 15 investigation, explore the conjec- 4. How many sides does a polygon have if the sum of its interior angle measures ture by walking around the is 7380°? 43 polygon and turning through In Exercises 5–10, use your new conjectures to calculate the measure of each lettered angle. the exterior angles. A walker will keep walking forward, 5. 6. 7. turning a total of once around. a 43 The Polygon Sum Conjecture 140 60 68 can be derived algebraically from 69 the Exterior Angle Sum Conjec- d c ture: The total of the measures 68 84 b 3 5 of the n interior plus n exterior a 108° 1 c 517°, d 1157° b 453° angles is 180°n, the interior 8. 9. 10. angles sum to 180°n 360°, 86 which factors to 180°(n 2). a g 39 d f 44 For the reverse situation, c c e [Ask] “How could the Exterior g d h Angle Sum Conjecture be derived e b a k h f b 56 94 18 algebraically from the Polygon e 72°, f 45°, g 117°, h 126° Sum Conjecture?” [The sum of a 30°, b 30°, c 106°, d 136° the exterior angle measures is 11. Developing Proof Complete this flowchart proofofthe Exterior Angle TEACHER’S EDITION 180°n 180°(n 2), which Sum Conjecture for a triangle. d simplifies to 360°.] Flowchart Proof c

Exterior Angles of a 1 a b 180 a e f For advanced students who ? 540° b Linear Pair Conjecture 4 explored concave polygons, a + b + c + d + e + f = ? remind them that the exterior 2 c d 180 360° Addition property 6 b + d + f = ? angle is the supplement of the Linear Pair Conjecture ? of equality interior angle. Define all exterior 180° Subtraction property 3 5 angles as having measure 180° e f 180 a + c + e = ? of equality minus the measure of the inte- Linear Pair Conjecture ? ? Triangle Sum Conjecture rior angle. If the interior angle’s 12. Is there a maximum number of obtuse exterior angles that any polygon can have? measure is more than 180°, the If so, what is the maximum? If not, why not? Is there a minimum number of acute exterior angle has negative interior angles that any polygon must have? If so, what is the minimum? If not, measure and is inside the figure. why not? Yes. The maximum is three. The minimum is zero. A polygon (It is an exterior angle only in a might have no acute interior angles. technical sense.) With that defi- nition, the Exterior Angle Sum ᮣ Review Conjecture is still true. This defi- N N f f nition will make sense for the 5.1 13. Developing Proof Prove the Pentagon Sum g o n g e h e analogy of “walking around” the Conjecture using either method discussed T I T m I h d i k l d figure: If you are walking in a in the last lesson. Use whichever diagram i at right corresponds with your method. If j clockwise direction, you have to c you need help, see page 261, Exercise 17. a b c a b turn counterclockwise at a Q U Q U nonconvex vertex. Assessing Progress Check students’ familiarity with Closing the Lesson 10. a 162°, b 83°, c 102°, d 39°, e 129°, the Polygon Sum Conjecture and f 51°, g 55°, h 97°, k 83° the Linear Pair Conjecture, with The fact that the sum of the measures of exterior angles of any polygon is 360° can be derived alge- 13. Answers will vary. Possible proof using various kinds of polygons and braically from, or lead algebraically to, the Polygon the diagram on the left: a b i 180°, c d with interior and exterior angles. Sum Conjecture that the sum of measures of the h 180°, and e f g 180° by the Triangle interior angles is 180°(n 2). In the case of Sum Conjecture. a b c d e f g h equiangular polygons, each interior angle has i 540° by the addition property of equality. Therefore, the sum of the measures of the angles measure 180° 360°. If students still need help n of a pentagon is 540°. To use the other diagram, in understanding these ideas, you might use students must remember to subtract 360° to Exercise 3 or 4 for a demonstration. account for angle measures k through o.

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Technology BUILDING UNDERSTANDING The aperture of a camera is an opening shaped like a regular polygon surrounded by thin sheets that form a set of exterior angles. These sheets move together or apart to close or open the aperture, limiting the amount of light As always, encourage students to passing through the camera’s lens. How does the sequence of closing apertures support their solutions by being shown below demonstrate the Exterior Angle Sum Conjecture? Does the ready to state the conjecture(s) number of sides make a difference in the opening and closing of the aperture? they have used to solve the exercises.

ASSIGNING HOMEWORK

Essential 1–11 Performance 5.1 14. Name the regular polygons that appear in 5.1 15. Name the regular polygons that appear in the assessment 9 the tiling below. Find the measures of the tiling below. Find the measures of the angles Portfolio 10 angles that surround point A in the tiling. that surround any vertex point in the tiling. Journal 12 regular polygons: regular polygons: square, regular Group 11, 13 and regular hexagon, and regular A 4, 13–17 dodecagon; angle dodecagon; angle Review measures: 60°, 150°, measures: 90°, 120°, Algebra review 3, 4, 11, 13 and 150° and 150°

| ᮣ Helping with the Exercises 4.6 16. Developing Proof RAC DCA, 4.6 17. Developing Proof DT RT,DA RA. Exercise 3 Students may need CD AR,AC DR. Is AD CR? Why? Is D R? Why? reminding that regular means RD D “equilateral and equiangular.” Ye s . RAC DCA by Ye s . DAT RAT by Exercise 12 As appropriate, SAS. AD CR by A T SSS. D R by CPCTC. CPCTC. remind students to think of A C R concave polygons. This exercise may require extra time and even scissors or patty paper. It can be used as a group activity to close

IMPROVING YOUR VISUAL THINKING SKILLS the day’s lesson or as a warm-up EDITION TEACHER’S activity the next day. Dissecting a Hexagon II Exercises 14, 15 These exercises Make six copies of the hexagon at right by tracing it onto your preview Chapter 7 on . paper. Find six different ways to divide a hexagon into twelve identical parts. EXTENSIONS A. Have students try to construct regular polygons with up to 20 sides using compass and straightedge. (They will not be able to construct all of them!) B. Use Take Another Look activity 3, 4, or 5 on page 308. For IMPROVING VISUAL THINKING SKILLS activity 4, on concave polygons, students should state a clear defini- Eleven possible answers are shown. tion of interior angle and exterior angle at points where the interior angle measure is greater than 180°.

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EXPLORATION

PLANNING

LESSON OUTLINE Star Polygons One day: If you arrange a set of points roughly around a C 45 min Activity or an oval, and then you connect each point to the next with segments, you should get a MATERIALS convex polygon like the one at right. What do D you get if you connect every second point with B ⅷ calculators segments? You get a like the ones shown in the activity below. ⅷ The Geometer’s Sketchpad In this activity you’ll investigate the angle measure sums of star polygons. E TEACHING A Star polygons have interesting Activity TEACHER’S EDITION angle sums and expand students’ understanding of what a polygon Exploring Star Polygons

can be. C

I

G uiding the Activity H

D The convention used to name B polygons by listing consecutive J vertices is not followed in the student book for star polygons. If G it makes more sense to you and K E your students to name star poly- A F gons by listing the vertices in 5-pointed star ABCDE 6-pointed star FGHIJK consecutive order, you might choose to follow that convention Step 1 Draw five points A through E in a circular path, clockwise. in your classroom and refer to the Step 2 Connect every second point, to get AC,CE,EB,BD, and DA. 5-pointed star as star ACEBD. The 6-pointed star has two sets of Step 3 Measure the five angles A through E at the star points. Use the calculator to find vertices and could be called star the sum of the angle measures. 180° FHJ,GIK to show that there is no Step 4 Drag each vertex of the star and observe what happens to the angle measures side between vertices J and G. and the calculated sum. Does the sum change? What is the sum? no; 180° You might do the first few steps Step 5 Copy the table on the next page. Use the Polygon Sum Conjecture to complete the first column. Then enter the angle sum for the 5-pointed star. as a class. Step 1 The points don’t actually need to be on a circle, as long as connecting them in order would LESSON OBJECTIVES NCTM STANDARDS make a convex polygon. ⅷ Explore angle measures of star polygons CONTENT PROCESS Step 3 Ask whether students ⅷ Develop skill at using technology to explore geometry ߜ Number ߜ Problem Solving could have found the sum by imagining walking around the ߜ Algebra ߜ Reasoning star. ߜ Geometry ߜ Communication Step 4 The sum will change if the ߜ Measurement Connections vertices are dragged out of order (so that every second point is no Data/Probability ߜ Representation longer connected).

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Step 6 Repeat Steps 1–5 for a 6-pointed star. Enter the angle sum in the table. Step 6 [Ask] “How was drawing a Complete the column for each n-pointed star with every second point five-pointed star different from connected. drawing a six-pointed star?” [The Step 7 What happens if you connect every third point to form a star? What would be 5-pointed star is a continuous the sum of the angle measures in this star? Complete the table column for every curve; the 6-pointed star has two third point. three intersecting lines; 0° unconnected pieces.] Different Step 8 Some Step 8 Use what you have learned to complete the table. What patterns do you notice? students in each group might patterns: 180° is a factor; Write the rules for n-pointed stars. take on different numbers of total measure decreases points, or jigsaw among groups. (to 0° or 180°) then Angle measure sums by how the star points are connected Students probably will need to increases; numbers are Number of Every Every Every Every Every repeated in the opposite star points point 2nd point 3rd point 4th point 5th point consider more than 7 points in order after the minimum. order to see a pattern. Students For n-pointed stars: if the might predict the sums before points of an n-gon are 5 calculating. They will need to connected every pth 540° 180° 180° 540° not a star point, then the angle extend their tables to allow for measure sum is the number of points in each star. 180°⏐n 2p⏐. 6 720° 360° 0° 360° 720° Step 7 [Ask] “For what number of points does connecting every 7 900° 540° 180° 180° 540° 900° third point result in a sum of zero?” [6] “Why?” [It does not form a star.] “For what number of points is the star for every Step 9 Ye s ; f o r Step 9 Let’s explore Step 4 a little further. Can you drag the vertices of each star even numbers of vertices polygon to make it convex? Describe the steps for turning each one into a second point the same as the the two overlapping poly- convex polygon, and then back into a star polygon again, in the fewest star for every third point?” [5] gons can be dragged to be steps possible. “Why?” [2 3 5] non-overlapping. For odd numbers the Step 10 In Step 9, how did the sum Step 8 Numbers diminish and star needs to be of the angle measure change then increase. It’s as if some uncrossed. The process when a polygon became convex? When did it change? quantity is decreasing but is can be reversed. being considered only as Step 10 Total angle nonnegative. [Ask] “How can measures are not changed you turn any number into a for those made up of nonnegative number?” [Take the overlapping polygons. For others the sum increases absolute value.] Students may

as each angle “turns write a short report explaining EDITION TEACHER’S inside out” (for example, why the angle measure sums are in ABCDE it changes what they are for any particular when A crosses DE and This blanket by Teresa Archuleta- polygon stars. the sides of D cross). Sagel is titled My Blue Vallero Heaven. Are these star polygons? The figures in the star quilt are Why? not star polygons because the eight vertices have not all been connected in the same way.

Step 8 Every Every Every Every Every Every Every Every Every Every Every 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th Points point point point point point point point point point point point 8 1080° 720° 360° 0° 360° 720° 1080° 9 1260° 900° 540° 180° 180° 540° 900° 1260° 10 1440° 1080° 720° 360° 0° 360° 720° 1080° 1440° 11 1620° 1260° 900° 540° 180° 180° 540° 900° 1260° 1620° 12 1800° 1440° 1080° 720° 360° 0° 360° 720° 1080° 1440° 1800°

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LESSON LESSON Kite and Trapezoid 5.3 5.3 Properties Recall that a kite is a quadrilateral with PLANNING exactly two distinct pairs of congruent Imagination is the highest consecutive sides. LESSON OUTLINE kite we fly. If you construct two different isosceles LAUREN BACALL triangles on opposite sides of a common One day: base and then remove the base, you have 20 min Investigation constructed a kite. In an , the vertex angle is the angle between the 10 min Sharing two congruent sides. Therefore, let’s call 5min Closing the two angles between each pair of congruent sides of a kite the vertex angles 10 min Exercises of the kite. Let’s call the other pair the nonvertex angles. MATERIALS Nonvertex angles ⅷ double-edged straightedges with parallel edges ⅷ construction tools TEACHER’S EDITION ⅷ protractors Vertex angles ⅷ Isosceles Trapezoid Diagonals Conjecture (W), optional [ᮣ For an interactive version of this sketch, see the Dynamic Geometry Exploration ⅷ Sketchpad activity Kite and Trapezoid ᮤ Properties of Kites at www.keymath.com/DG . ] keymath.com/DG Properties, optional A kite also has one line of reflectional symmetry, just like an isosceles triangle. You can use this property to discover other properties of kites. Let’s investigate. TEACHING

Angles and diagonals of kites Investigation 1 and isosceles trapezoids have What Are Some Properties of Kites? some interesting properties. You will need In this investigation you will look at angles and diagonals in a kite to see what One Step ● patty paper special properties they have. Ask students (with their books ● a straightedge Step 1 On patty paper, draw two connected segments of different lengths, as shown. closed) to discover all they can Fold through the endpoints and trace the two segments on the back of the about the angles and diagonals of patty paper. kites, trapezoids, and isosceles Step 2 Compare the size of each pair of trapezoids. They might be expe- opposite angles in your kite by folding rienced enough by now to come an angle onto the opposite angle. Are up with all seven of this lesson’s the vertex angles congruent? Are the nonvertex angles congruent? Share conjectures on their own. your observations with others near you and complete the conjecture. Step 1 Step 2 Guiding Investigation 1

[ELL] For the next several lessons, create a vocabulary wall that LESSON OBJECTIVES NCTM STANDARDS shows each quadrilateral with its ⅷ Discover properties of kites and trapezoids CONTENT PROCESS definition and related terms. ⅷ Learn new vocabulary ߜ After students complete an inves- Number Problem Solving ⅷ tigation, call on students to illus- Practice construction skills Algebra ߜ Reasoning trate the related properties on ߜ ߜ the wall. Make sure to include Geometry Communication drawings of quadrilaterals in ߜ Measurement ߜ Connections “nonstandard position,” such as a trapezoid with bases that are not Data/Probability Representation horizontal.

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Students can explore the properties Kite Angles Conjecture C-34 of kites and complete the conjec- The ? angles of a kite are ? . nonvertex, congruent tures in Investigation 1 using the Dynamic Geometry Exploration at www.keymath.com/DG.

Step 3 Draw the diagonals. How are the diagonals related? Share Step 2 Students may find it your observations with others in your group and complete convenient to label with words the conjecture. the vertex and nonvertex angles.

Kite Diagonals Conjecture C-35 The diagonals of a kite are ? . perpendicular

What else seems to be true about the diagonals of kites? Step 4 Compare the lengths of the segments on both diagonals. Does either diagonal bisect the other? Share your observations with others near you. Copy and complete the conjecture.

Kite Diagonal Bisector Conjecture C-36 The diagonal connecting the vertex angles of a kite is the ? of the other diagonal. perpendicular bisector

Step 5 Fold along both diagonals. Does either diagonal bisect any angles? Share your observations with others and complete the conjecture.

Kite Angle Bisector Conjecture C-37 The ? angles of a kite are ? by a ? . vertex, bisected, diagonal TEACHER’S EDITION TEACHER’S

You will prove the Kite Diagonal Bisector Conjecture and the Kite Pair of base angles Angle Bisector Conjecture as exercises after this lesson. Let’s move on to trapezoids. Recall that a trapezoid is a quadrilateral Bases with exactly one pair of parallel sides. In a trapezoid the parallel sides are called bases. A pair of angles that share a base as a common side are called base angles. Pair of base angles In the next investigation you will discover some properties of trapezoids.

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Guiding Investigation 2 Science

Step 2 Consecutive angles share A trapezium is a quadrilateral with no two sides parallel. a side—in this case, one of the The words trapezoid and trapezium come from the Greek word trapeza, meaning table. nonparallel sides of the trapezoid. There are bones in your wrists Step 4 Students may construct an that anatomists call trapezoid and trapezium because of their isosceles trapezoid with the help geometric shapes. Trapezium of its line of symmetry, which goes through the midpoints of the bases. Investigation 2 Developing Proof See page 839 for a solution. What Are Some Properties of Trapezoids? You will need SHARING IDEAS ● a double-edged straightedge Have students read aloud ● a protractor selected conjectures for critique. ● a compass Reach consensus about which statements students will record in their notebooks.

TEACHER’S EDITION [Ask] “What are consecutive sides This is a view inside a deflating hot-air in a polygon?” [sides that share a balloon. Notice the vertex] “What are consecutive trapezoidal panels angles in a polygon?” [angles that that make up the share a side] “Are consecutive balloon. angles the same as adjacent Step 1 Use the two edges of your straightedge to draw parallel segments of unequal angles?” [Adjacent angles share length. Draw two nonparallel sides connecting them to make a trapezoid. a side, but they also share a Step 2 Use your protractor to find the sum of the vertex—not possible for distinct measures of each pair of consecutive angles Find sum. angles in a polygon.] “What between the parallel bases. What do you notice symmetry does a kite have?” [It about this sum? Share your observations with has reflectional symmetry over your group. the diagonal through its vertex Step 3 Copy and complete the conjecture. angles.] “Can the properties of a kite be seen from that Trapezoid Consecutive Angles Conjecture C-38 symmetry?” [The line of symmetry bisects the vertex The consecutive angles between the bases of a trapezoid are ? . supplementary angles. Because each nonvertex angle is reflected to the other one through a line perpendicular to the line of symmetry, the diago- Recall from Chapter 3 that a trapezoid whose two nals are perpendicular. And, as nonparallel sides are the same length is called an reflections of each other, the isosceles trapezoid. Next, you will discover a few nonvertex angles are congruent.] properties of isosceles trapezoids. To preview properties of other figures, you might also ask whether the symmetry guaran- formed by any line cutting any pair of parallel lines. have more difficulty seeing that the diagonals are tees that consecutive angles, adja- Students can see that it does by considering linear also reflections of each other, because the two parts cent sides, or vertex angles are pairs in which one angle is an opposite interior of each diagonal are reflected in different direc- congruent or whether the diago- angle to another. tions. As with the kite, you might ask whether the nals are congruent or bisect each symmetry guarantees the properties that the figure other. [The kite has none of these [Ask] “What kind of symmetry does the isosceles doesn’t have. properties.] trapezoid have?” [reflectional symmetry over a line through the midpoints of the parallel sides] “What Wonder a loud w hether the properties of the figure can be seen from this observation about consecutive symmetry?” [The base angles are reflections of each angles of a trapezoid extends to other, so they’re congruent.] Some students may consecutive interior angles

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Like kites, isosceles trapezoids have one line of reflectional symmetry. Through Sharing Ideas (continued) what points does the line of symmetry pass? Ask whether students could Step 4 Use both edges of your straightedge to draw parallel lines. Using your compass, prove the Isosceles Trapezoid construct two congruent, nonparallel segments. Connect the four segments to Conjecture from the Isosceles make an isosceles trapezoid. Triangle Conjecture. They might Step 5 Measure each pair of base angles. What do you say that if they extend the Compare. notice about the pair of base angles in each nonparallel sides of the trapezoid trapezoid? Compare your observations with Compare. to their intersection point, they’ll others near you. have an isosceles triangle, so the Step 6 Copy and complete the conjecture. base angles (which are also the base angles of the trapezoid) will Isosceles Trapezoid Conjecture C-39 be congruent. Ask why the triangle is isosceles; students may The base angles of an isosceles trapezoid are ? . congruent say that it is because its base angles, or the base angles What other parts of an isosceles trapezoid are congruent? Let’s continue. of the triangle not including the Step 7 Draw both diagonals. Compare their lengths. trapezoid, are congruent. This Share your observations with others near you. discussion can provide a good Step 8 Copy and complete the conjecture. example of circular reasoning. [Ask] “What strategies did you Isosceles Trapezoid Diagonals Conjecture C-40 use in writing your proof of the Isosceles Trapezoid Diagonals The diagonals of an isosceles trapezoid are ? . congruent Conjecture?” [draw a labeled diagram, use previous conjec- tures, possibly work backward] Thinking backward will be Developing Proof As a group, write a flowchart TR proof that shows how the Isosceles Trapezoid explicitly introduced as a strategy Diagonals Conjecture follows logically from for writing proofs in Lesson 5.7, the Isosceles Trapezoid Conjecture. Use the but some students may already diagram at right and a method for making be using it. Students are asked to the triangles easier to see. If you need help, Ⅲ prove both Isosceles Trapezoid see page 232, Example B. P A Conjectures in the exercises of Lesson 5.7. XERCISES You will need E Assessing Progress EDITION TEACHER’S Construction tools In this lesson, you can assess ᮣ Use your new conjectures to find the missing measures. for Exercises 14–16 students’ understanding of kites, 1. ABCD is a kite. 2. x ? 21° 3. x ? 52° trapezoids,diagonals,and parallel perimeter ? 64 cm y ? 146° y ? and nonparallel lines. You can B 128° also monitor their ability to use a 12 cm 128 protractor and to copy a segment 146 C and compare lengths of segments A x 47 y y with a compass. 20 cm D x

Closing the Lesson are supplementary. You might also mention that the vertex angles of kites are not congruent, the diago- Reiterate the main conjectures of this lesson: nals of kites are not congruent and the diagonal Nonvertex angles of kites and base angles of between the nonvertex angles doesn’t bisect the trapezoids are congruent; diagonals of kites are other diagonal or the nonvertex angles, and the perpendicular, and the diagonal between the vertex diagonals of isosceles trapezoids don’t bisect angles angles bisects the other diagonal and the vertex or each other. Urge students to learn to draw and angles; diagonals of isosceles trapezoids are mark pictures from which they can quickly observe congruent; and consecutive angles of any trapezoid what is and isn’t true about these figures. (not just isosceles ones) between the parallel sides

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BUILDING 4. x ? 15 cm 5. x ? 72° 6. x ? 99° ? 61° ? UNDERSTANDING perimeter 85 cm y y 38 cm 18 perimeter 164 cm x y 12 cm The exercises include a proof 37 cm y of the Kite Angle Bisector x Conjecture. As always, encourage x y y 12 cm 29 students to support all solutions 18 cm by stating relevant conjectures. 81

ASSIGNING HOMEWORK 7. ARTP is an isosceles trapezoid with RA PT. 8. FLYE is a kite with FL LY. Find w, x, and y. Find w, x, and y. Essential 1–14 L P Performance x assessment 10, 17 3.4 cm 30 Portfolio 9 3.0 cm 1.6 cm R w Journal 16 T F Y 48 x Z Group 14–18 w Review 19 y y 45 Algebra review 6 A R E TEACHER’S EDITION w 120°, x 45°, y 30° w 1.6 cm, x 48°, y 42° MATERIALS 9. Copy and complete the flowchart to show how the Kite Angle Bisector Conjecture ⅷ Exercise 9 (T), optional follows logically from one of the triangle conjectures. Y ⅷ Exercise 19 (T), optional Given: Kite BENY with BE BY,EN YN Show: BN bisects B | 2 ᮣ BN bisects N B 4 N Helping with the Exercises 1 3 Flowchart Proof Exercise 10 Students who use the Kite Angle Bisector Conjecture 1 BE BY E Given can prove, through congruent triangles, that the angles in the BYN 2 BN bisects B, BN bisects N 2 4 ? 5 ? 6 center are congruent. To prove EN YN BEN 1 and ? and ? ? Given ? Congruence 3 4 perpendicularity, they will use Definition of BN BN shortcut ? CPCTC angle bisector the fact that congruent angles 3 ? ? SSS that form a linear pair are right Same segment angles. 10. Answers may vary. This 10. Write a paragraph proofor flowchart proofofthe Kite Diagonal Bisector Conjecture. Either show how it follows logically from the Kite Angle Bisector proof uses the Kite Angle Conjecture that you just proved, or how it follows logically from the Converse of Bisector Conjecture. the Perpendicular Bisector Conjecture. I Y 11. Sketch and label kite KITE with vertex angles K and T and KI TE. Which angles are congruent? possible answer: E I T K

2 B N 1 X E

E same segment. By SAS, BXY BXE.So by Exercises 11–13 These exercises ask students only to Given: Kite BENY with vertex CPCTC, XY XE.Because YXB and EXB sketch the figures. Students can thus visualize the angles B and N form a linear pair, they are supplementary, so concepts before they get to the next step of constructing some of the figures, in Show: Diagonal BN is the m YXB m EXB 180°. By CPCTC, YXB Exercises 14–16. perpendicular bisector of EXB, or m YXB m EXB, so by substitution, diagonal YE. 2m YXB 180°, or m YXB 90°. So mYXB mEXB 90°. Because XY XE and From the definition of kite, YXB and EXB are right angles, BN is the BE BY.From the Kite Angle perpendicular bisector of YE. Bisector Conjecture, 1 2. BX BX because they are the

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12. Sketch and label trapezoid QUIZ with one base QU. What is the other base? Name possible answer: 13. possible answer: the two pairs of base angles. The other base is ZI. Q and U are a pair of base ZI WO angles. Z and I are a pair of base angles. 13. Sketch and label isosceles trapezoid SHOW with one base SH. What is the other base? Name the two pairs of base angles. Name the two sides of equal length. QUSH In Exercises 14–16, use the properties of kites and trapezoids to construct each figure. OW is the other base. S and You may use either patty paper or a compass and a straightedge. H are a pair of base angles. 14. Construction Construct kite BENF given sides BE and EN and diagonal BN. How O and W are a pair of base many different kites are possible? Only one kite is possible because three sides determine a triangle. angles. SW HO. F B E E N Exercises 14–16 If students make N B N B sketches of these figures before beginning the constructions, E 15. Construction Given W, I, base WI, and nonparallel they’ll find it easier to visualize side IS, construct trapezoid WISH. the various parts being given

SH to them. Exercise 17 The project Building an Arch in Lesson 5.4 is an I W extension of this exercise. I IS W WI 16. Construction Construct a trapezoid BONE with BO NE. How many different trapezoids can you construct? infinitely many possible construction: BO BO N E

E Architecture N Keystone Voussoir

The Romans used the classical arch design in bridges, aqueducts, and Abutment buildings in the early centuries of the Common Era. The classical semicircular arch is really half of a regular polygon built with wedge-shaped blocks whose Rise faces are isosceles trapezoids. Each block supports the blocks surrounding it. Span TEACHER’S EDITION TEACHER’S 17. Application The inner of the arch in the diagram above right is half of a regular 18-gon. Calculate the measures of all the angles in the nine isosceles trapezoids making up the arch. Then use your geometry tools to accurately draw a nine- stone arch like the one shown. 80°, 80°, 100°, 100°

This carton is shaped like an isosceles trapezoid block, like the voussoirs used in the arch above.

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18. The figure below shows the path of light through a trapezoidal prism and how an image is inverted. For the prism to work as shown, the trapezoid must be isosceles, AGF must be congruent to BHE, and GF must be congruent to EH. Show that if these conditions are met, then AG will be congruent to BH. D C G H Because ABCD is an isosceles trapezoid, A B. AGF BHE by SAA. Thus, AG BH by CPCTC.

A E F B

Science

The magnifying lenses of binoculars invert the objects you view through them, so trapezoidal prisms are used to flip the inverted images right-side-up again.

ᮣ Review

19. a 80°, b 20°, c 160°, 19. Developing Proof Trace the figure below. Calculate the measure of each lettered angle.

TEACHER’S EDITION d 20°, e 80°, f 80°, Explain how you determined measures e and g. g 110°, h 70°, m 110°, n 100°; Possible explanation: Because d forms a linear pair m b a 100 with e and its congruent f c adjacent angle, d 2e 180°. Substituting d 20° gives e g d n 2e 160°, so e 80°. Using the h Vertical Angles Conjecture and d 20°, the unlabeled angle in the small right triangle measures 20°, which means h 70°. Because g and h are a linear pair, they are supplemen- IMPROVING YOUR REASONING SKILLS tary, so g 110°. How Did the Farmer Get to the Other Side? A farmer was taking her pet rabbit, a basket of prize-winning baby carrots, and her small—but hungry—rabbit-chasing dog to town. She came to a river and realized she had a problem. The little boat she found tied to the pier was big enough to carry only herself and one of the three possessions. She couldn’t leave her dog on the bank with the little rabbit (the dog would frighten the poor rabbit), and she couldn’t leave the rabbit alone with the carrots (the rabbit would eat all the carrots). But she still had to figure out how to cross the river safely with one possession at a time. How could she move back and forth across the river to get the three possessions safely to the other side?

IMPROVING REASONING SKILLS If students are having difficulty, encourage them to think about which pairs of items can be together. One solution: Take the rabbit across and leave it there. Go back. Take the carrots across and return with the rabbit. Leave the rabbit on the original side. Take the dog across, leave it there, and go back. Finally, bring the rabbit across again.

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LESSON LESSON Properties 5.4 of Midsegments 5.4 As you learned in Chapter 3, the segment connecting the midpoints of two sides of a PLANNING Research is formalized triangle is a midsegment of the triangle. The segment connecting the midpoints of the two curiosity. It is poking and LESSON OUTLINE prying with a purpose. nonparallel sides of a trapezoid is also called the midsegment of the trapezoid. ZORA NEALE HURSTON One day: In this lesson you will discover special 25 min Investigation properties of midsegments. 5min Sharing Investigation 1 5min Closing Triangle Midsegment Properties 10 min Exercises You will need In this investigation you will discover two properties of triangle midsegments. Each MATERIALS ● patty paper person in your group can investigate a different triangle. ⅷ construction tools ● a straightedge ⅷ protractors ⅷ Sketchpad activity Properties of Midsegments, optional

TEACHING Step 1 Step 2 Step 3

Step 1 Drawatriangleonapieceofpattypaper.Pinchthepattypapertolocatemidpoints A midsegment of a triangle or a of the sides. Draw the midsegments.You should now have four small triangles. trapezoid is closely related to the side or sides it parallels. The Step 2 Place a second piece of patty paper over the first and copy one of the four triangles. one-step investigation replaces Step 3 Compare all four triangles by sliding the copy of one small triangle over the both investigations. other three triangles. Compare your results with the results of your group. Copy and complete the conjecture. Guiding Investigation 1 C-41

Three Midsegments Conjecture EDITION TEACHER’S The three midsegments of a triangle divide it into ? . four congruent triangles Step 1 [Alert] Students may have forgotten what a midsegment is.

Step 4 Each Step 4 Mark all the congruent angles on the original patty paper. Step 3 To see that all four midsegment is parallel If you find it too cluttered, redraw the original triangle Third side triangles are congruent, students to the third side. on regular paper with just one midsegment, as in the will rotate as well as translate the diagram at right, and then mark all the congruent sliding triangle. angles. Using the Corresponding Angles Midsegment Conjecture or its converse, what conclusions Step 5 Students can use patty can you make about a midsegment and the paper or a compass to see that large triangle’s third side? the midsegment fits twice on the large triangle’s third side. One Step NCTM STANDARDS LESSON OBJECTIVES Pose this problem: “You are trying to predict how long a ⅷ CONTENT PROCESS Define and discover properties of midsegments in triangles road tunnel going east-west and trapezoids Number ߜ Problem Solving through a hill will be. You’re at ⅷ Practice construction skills the west end of where the tunnel ߜ ߜ Algebra Reasoning will be dug, and your friend is at ߜ Geometry ߜ Communication the east end. You drive along a straight road until you reach an ߜ ߜ Measurement Connections intersection.Meanwhile,your Data/Probability Representation friend drives one mile north and two miles west to meet you at

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One Step (continued) Step 5 Compare the length of the midsegment to the large triangle’s third side. How do the intersection. You both drive they relate? Copy and complete the conjecture. back to your starting points, and C-42 then you drive along the same Triangle Midsegment Conjecture parallel, half, the third side road in the opposite direction for A midsegment of a triangle is ? to the third side and ? the length of ? . the same distance. Your friend drives one mile south and three miles west to meet you. Is this enough information for you to determine the length of the In the next investigation you will discover two properties of the midsegment of a trapezoid. tunnel?” As you circulate, be sure each group has a sketch of the trapezoidal figure; ask that each group member check it before Investigation 2 beginning work. Encourage students to generalize their guess Trapezoid Midsegment Properties to a conjecture about all trape- You will need Each person in your group can investigate a different trapezoid. Make sure you zoids. As groups finish, encourage ● patty paper draw the two bases perfectly parallel. them to think about what would ● a straightedge happen if one of the trapezoid’s parallel sides shrank to a point, 34 34 12 TEACHER’S EDITION so that the figure became a 12 triangle. During Sharing, extend the problem to drawing all midsegments of the triangle and Step 1 Step 2 Step 3 deriving the Three Midsegments Step 1 Draw a small trapezoid on the left side of a piece of patty paper. Pinch the paper Conjecture. to locate the midpoints of the nonparallel sides. Draw the midsegment. Step 2 Label the angles as shown. Place a second piece of patty paper over the first and Guiding Investigation 2 copy the trapezoid and its midsegment. Step 3 Compare the trapezoid’s base angles with the corresponding angles at the Define the midsegment of a midsegment by sliding the copy up over the original. trapezoid and ask student groups Step 4 Ye s ; t h e Step 4 Are the corresponding angles congruent? What can you conclude about the to discover and conjecture about midsegment is parallel midsegment and the bases? Compare your results with the results of other two special properties of a to the base. students. midsegment of a trapezoid. Circulate to check on the correct- ness of each group’s Trapezoid Midsegment Conjecture. The midsegment of a triangle is half the length of the third side. How does the length of the midsegment of a trapezoid compare to the lengths of the two bases? Step 1 You might emphasize that Let’s investigate. students should leave plenty of Step 5 On the original trapezoid, extend the longer base to the right by at least the room on the right side of the length of the shorter base. patty paper. In Step 5, they’ll need Step 6 Slide the second patty paper under the first. Show the sum of the lengths of the that space to add the two bases. two bases by marking a point on the extension of the longer base. Step 8 Students may say that the length of the midsegment is “half the sum of the bases” or “the average of the bases.” For encourage them to use the word average in their be said that the length of a triangle’s midsegment is students careful about detail, you conjectures, and lead the class to agree on the the average of the lengths of the two bases?” [Yes, if might mention during Sharing wording to be written in their notebooks. Point out one of the bases is considered to have length 0. In that the conjecture should refer that although every triangle has three midsegments, both the triangle and the trapezoid, the midsegment to the lengths of the bases rather any trapezoid has only one. is parallel to a side.] “What can be said about the than the bases themselves. change in angles as the trapezoid’s base shrinks to a [Ask] “Are triangles trapezoids?” [They’re not.] Yet point?” [The corresponding angles remain congruent their midsegments seem to have similar properties. SHARING IDEAS to each other even while changing measure.] Elicit the idea that if one of the parallel edges of the Have students share the variety trapezoid is shrunk to a point and the height is [Ask] “The term we are using for the of conjecture statements they’ve kept the same length, the trapezoid shrinks to a that connects the midpoints of the nonparallel sides come up with. As needed, triangle that shares the midsegment. [Ask] “Can it of a trapezoid is midsegment. This segment is also

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Sharing Ideas (continued) called the median of a trapezoid. 34 34 34 Which do you think is a better 12 12 12 Sum name for the segment, and why?” Each group might write a letter about its views to be read aloud Step 5 Step 6 Step 7 to the class.

Step 7 twice; half Step 7 How many times does the midsegment fit onto the segment representing the Assessing Progress the sum of the sum of the lengths of the two bases? What do you notice about the length of the Assess students’ understanding of lengths of the bases midsegment and the sum of the lengths of the two bases? trapezoid, midpoint, midsegments Step 8 Combine your conclusions from Steps 4 and 7 and complete this conjecture. of triangles, congruent triangles, and average of lengths. You can Trapezoid Midsegment Conjecture C-43 also see how well students under- the average of the lengths of the bases stand corresponding angles The midsegment of a trapezoid is ? to the bases and is equal in length to ? . parallel formed by parallellinesanda transversal and the Converse of the Parallel Lines Conjecture. Also check how well they can What happens if one base of the trapezoid shrinks to a point? Then the trapezoid construct the midpoint of a line collapses into a triangle, the midsegment of the trapezoid becomes a midsegment segment on patty paper. of the triangle, and the Trapezoid Midsegment Conjecture becomes the Triangle Midsegment Conjecture. Do both of your midsegment conjectures work for the last figure? Closing the Lesson The midsegment of any trapezoid is parallel to the bases, and its T E T E T length is the average of their lengths. If one of the bases shrinks MD M D MD to a point to make a triangle, the midsegment remains parallel to A A A R R R the remaining base, and its length becomes half the length of that side of the triangle. The three [ᮣ For an interactive version of this sketch, see the Properties of the Dynamic Geometry Exploration midsegments of a triangle divide keymath.com/DG Midsegment of a Trapezoid atwww.keymath.com/DG . ᮤ] it into four congruent triangles. TEACHER’S EDITION TEACHER’S EXERCISES You will need Construction tools ᮣ 1. How many midsegments does a triangle have? A trapezoid have? for Exercises 9 and 18 three; one 2. What is the perimeter 3. x ? 60° 4. z ? 65° of TOP? 28 y ? 140° T 65 x y P O 8 10 60 40 z 42 RA20

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BUILDING 5. What is the perimeter 6. m ? 129° 7. q ? 35 of TEN? 23 n ? 73° UNDERSTANDING p ? 42 cm U 36 cm 13 The exercises include applica- 6 tions and extensions of the T N n 24 p m conjectures in this lesson about 8 midsegments. As always, 73 51 q encourage students to support P 9 E A 48 cm their solutions by being ready to 8. Developing Proof Copy and complete the flowchart to show that LN RD. F N state the conjecture(s) they have A Given: Midsegment LN in FOA used to solve the exercises. L Midsegment RD in IOA Show: LN RD O ASSIGNING HOMEWORK D Flowchart Proof Essential 1–9 R 1 FOA with 3 LN OA Performance midsegment LN I assessment 10 ? Given LN RD Triangle Midsegment Conjecture 5 Portfolio 9 ? OA RD 2 Two lines parallel to the Journal 11 IOA with 4 ? same line are parallel Group 8 midsegment RD Triangle Midsegment TEACHER’S EDITION Given Review 12–18 Conjecture Algebra review 15–17 MATERIALS 9. Construction When you connected the midpoints of the three sides of a triangle in Investigation 1, you created four congruent triangles. Draw a quadrilateral on patty ⅷ Exercises 11 and 14 (T), optional paper and pinch the paper to locate the midpoints of the four sides. Connect the midpoints to form a quadrilateral. What special type of quadrilateral do you get | when you connect the midpoints? Use the Triangle Midsegment Conjecture to ᮣ Helping with the Exercises explain your answer. Exercise 9 If students are having 10. Deep in a tropical rain forest, archaeologist difficulty, ask how they might Ertha Diggs and her assistant researchers make triangles so that they can have uncovered a square-based truncated pyramid (a square pyramid with the top apply the Triangle Midsegment part removed). The four lateral faces are Conjecture. [Draw one diagonal isosceles trapezoids. A line of darker to see two triangles.] mortar runs along the midsegment of each lateral face. Ertha and her co-workers make 9. Parallelogram. Draw a diag- some measurements and find that one of onal of the original quadrilateral. these midsegments measures 41 meters, and The diagonal forms two tri- each bottom base measures 52 meters. angles. Each of the two midseg- Now that they have this information, Ertha ments is parallel to the diagonal, and her team can calculate the length of and thus the midsegments are the top base without having to climb up parallel to each other. Now draw and measure it. Can you? What is the length of the top edge? How do you know? the other diagonal of the original The length of the edge of the top base measures 30 m. quadrilateral. By the same We know this by the Trapezoid Midsegment Conjecture. reasoning, the second pair of midsegments is parallel. There- fore, the quadrilateral formed by joining the midpoints is a parallelogram. Exercise 10 Students may use the 52 x equation 2 41, or they may simply realize that the other edge is 11 meters shorter than the midsegment, just as 41 is 11 less than 52.

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11. Ladie and Casey pride themselves on their estimation Exercise 12 Some students may skills and take turns estimating distances. Casey claims not realize that the volume of in that two large redwood trees visible from where they Cab liquid is unchanged when the are sitting are 180 feet apart, and Ladie says they are 275 feet apart. Cabin container rests on a different face. One solution method is to find The problem is, they can’t measure the distance to see whose estimate is better, because their cabin is located the volume of the liquid in the between the trees. All of a sudden, Ladie recalls her geometry: “Oh yeah, the original position, and then use Triangle Midsegment Conjecture!” She collects a tape measure, a hammer, and some that volume to solve for the wooden stakes. What is she going to do? Ladie drives a stake into the ground to create a triangle for height of the liquid in the new which the trees are the other two vertices. She finds the midpoint from the stake to each tree. The distance between these midpoints is half the distance between the trees. position. Another approach is to use the fact that the height of the ᮣ Review liquid is three-fourths the height 40 cm of the container. This will be 1.8 12. The 40-by-60-by-80 cm sealed rectangular container true no matter which face the shown at right is resting on its largest face. It is filled 30 cm container is resting on. So the with a liquid to a height of 30 cm. Sketch the container 60 cm height of the liquid in the new resting on its smallest face. Show the height of the 80 cm 3 liquid in this new position. Explain your method. position must be 4(80), or 60 cm. 12. Explanations will vary. 5.3 13. Developing Proof Write the converse of this statement: If exactly one diagonal bisects a Exercise 13 According to the pair of opposite angles of a quadrilateral, then the quadrilateral is a kite. Is the exclusivedefinitionusedinthis converse true? Is the original statement true? If either conjecture is not true, sketch a book, a kite cannot be a . 80 counterexample. If a quadrilateral is a kite, then exactly one diagonal bisects a 60 cm If your students use an inclusive pair of opposite angles. Both the original and converse statements are true. definition, so that a rhombus is a 14. Developing Proof Trace the figure below. Calculate the measure of each lettered angle. special kite, then the converse of Explain how you determined the measures h and k. 40 60 the statement is false.

1 2 14. a 54°, b 72°, c 108°, d 72°, e 162°, f 18°, a b n m k 1 g 81°, h 49.5°, i 130.5°, k 49.5°, m 162°, n 99°; g Possible explanation: The third f 54 c e i d 2 angle of the triangle containing h f and g measures 81°, so using the Vertical Angles Conjecture, the vertex angle of the triangle

containing h also measures 81°. EDITION TEACHER’S 5.315. CART is an isosceles trapezoid. What are 5.3 16. HRSE is a kite. What are the coordinates the coordinates of point T?ofpoint(3, 8) R? (0, 8) Subtract 81°from 180°and divide by 2 to get h 49.5°. The y y other base angle must also E (0, 8) T (?, ?) R (12, 8) measure 49.5°. By the Corre- sponding Angles Conjecture, H (5, 0) S (10, 0) x k 49.5°.

x EXTENSION C (0, 0) A (15, 0) R (?, ?) Challenge students to use coordi- nate geometry to confirm that the midsegment of a triangle is parallel to the third side and that the midsegment of a trapezoid is parallel to the two bases.

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18. UYAS 3 17. Find the coordinates of midpoints E and Z. Show that the slope of y the line containing midsegment EZ is equal to the slope of the line R 3 R (4, 7) containing YT. coordinates: E(2, 3.5), Z(6, 5); the slope of EZ 8, 3 and the slope of YT 8 Z (?, ?) 5.3 18. Construction Use the kite properties you discovered in Lesson 5.3 to F N E (?, ?) construct kite FRNK given diagonals RK and FN and side NK. Is T (8, 3) there only one solution? x Y (0, 0) R K

F NN K

K

There is only one kite, but more than one way to construct it. BUILDING AN ARCH In this project you’ll design and build your own Roman arch. Arches can have a simple semicircular shape, or a pointed “broken arch” shape.

TEACHER’S EDITION Horseshoe Arch Basket Arch Tudor Arch Lancet Arch In arch construction, a wooden support holds the voussoirs in place until the keystone is placed (see arch diagram on page 273). It’s said that when the Romans made an arch, they would make the architect stand under it while the wooden support was removed. That was one way to be sure architects carefully designed arches that wouldn’t fall! What size arch would you like to build? Decide the of the opening, the thickness of the arch, and the The arches in this Roman aqueduct, above the Gard River in number of voussoirs. Decide on the France, are typical of arches you can find throughout regions materials you will use. You should have that were once part of the Roman Empire. An arch can carry a lot of weight, yet it also provides an opening. The abutments your trapezoid and your materials on the sides of the arch keep the arch from spreading out and approved by your group or your teacher falling down. before you begin construction. c Your project should include b ᮣ A scale diagram that shows the exact b a b c size and angle of the voussoirs and the keystone. ᮣ A template for your voussoirs. ᮣ Your arch.

Supporting the OUTCOMES ᮣ The project shows knowledge of the ⅷ There is an estimate of the rise and the Students may use folded cardboard, wood, properties of isosceles trapezoids. span. (To actually calculate the rise and foam, or take-out containers (purchased ᮣ The diagram includes the measure of the span of an arch requires from a restaurant supply store) to make one voussoir and the number in the trigonometry.) trapezoidal prisms for the voussoirs. arch. [Language] Vou ss oir is pronounced ᮣ Measurement of each angle is deter- [vü-'swär]. mined from the number of sides of the circle that contains the arch’s arc.

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LESSON LESSON Properties of 5.5 Parallelograms 5.5 In this lesson you will discover some special properties of parallelograms. A parallelogram is a quadrilateral whose opposite sides are parallel. PLANNING If there is an opinion, facts Rhombuses, rectangles, and squares all fit this definition as well. Therefore, any will be found to support it. properties you discover for parallelograms will also apply to these other shapes. LESSON OUTLINE JUDY SPROLES However, to be sure that your conjectures will apply to any parallelogram, you should investigate parallelograms that don’t have any other special properties such One day: as right angles, all congruent angles, or all congruent sides. 25 min Investigation 10 min Sharing Investigation 5min Closing Four Parallelogram Properties 5min Exercises You will need First you’ll create a parallelogram. MATERIALS ● graph paper ⅷ ● patty paper or a double-edged straightedges

compass ⅷ E V compasses or patty paper ● a double-edged straightedge ⅷ protractors ● a protractor LO ⅷ graph paper ⅷ Avoiding Hot Water (T) for One Step

Step 1 Step 2 ⅷ Sketchpad activity Properties of Parallelograms, optional Step 1 Using the lines on a piece of graph paper as a guide, draw a pair of parallel lines that are at least 6 cm apart. Using the parallel edges of your double-edged straightedge, make a parallelogram. Label your parallelogram LOVE. TEACHING Step 2 Let’s look at the opposite angles. Measure the angles of parallelogram LOVE. Compare a pair of opposite angles using patty paper or your protractor. In this lesson students discover Compare results with your group. Copy and complete the conjecture. four conjectures about parallel- ograms. The one-step investiga- C-44 tion is on page 282.

Parallelogram Opposite Angles Conjecture EDITION TEACHER’S Students can also explore The opposite angles of a parallelogram are ? . congruent parallelograms using the Dynamic Geometry Exploration at www.keymath.com/DG.

Two angles that share a common side in a polygon are consecutive angles. In parallelogram LOVE, LOV and EVO are a pair of consecutive angles. The Guiding the Investigation consecutive angles of a parallelogram are also related. Step 3 Find the sum of the measures of each pair of consecutive angles in Step 1 To draw the parallelogram parallelogram LOVE. 180° without using a double-edged straightedge, students may draw lines between points on the graph paper if they are careful NCTM STANDARDS LESSON OBJECTIVES to apply what they know about slope triangles and slopes of ⅷ CONTENT PROCESS Discover properties of parallelograms parallel lines. Encourage them to Number ߜ Problem Solving ⅷ Learn new vocabulary related to vectors look beyond the special case of

ⅷ rectangles. If possible, help them ߜ Practice construction skills Algebra Reasoning draw the parallelograms without ⅷ Develop inductive reasoning and cooperative behavior ߜ Geometry ߜ Communication using the fact that the opposite sides have the same length. ߜ Measurement ߜ Connections Step 3 As needed, review the Data/Probability ߜ Representation term supplementary.

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Step 7 Students may conjecture Share your observations with your group. Copy and complete the conjecture. that the diagonals are congruent, or are perpendicular, or intersect Parallelogram Consecutive Angles Conjecture C-45 in the middle of the parallelo- The consecutive angles of a parallelogram are ? . supplementary gram, or go through their midpoints, or bisect each other, or are bisectors. Have them Step 4 The opposite Step 4 Describe how to use the two conjectures you just made to find all the angles of a report their conjectures during angle will have the same parallelogram with only one angle measure given. measure and the two Sharing, when the class can consecutive angles Step 5 Next let’s look at the opposite sides of a parallelogram. With your compass or critique them. will have measure 180° patty paper, compare the lengths of the opposite sides of the parallelogram minus the measure of you made. One Step the angle. Share your results with your group. Copy and complete the conjecture. Show the Avoiding Hot Water transparency and pose this Parallelogram Opposite Sides Conjecture C-46 problem: “Your supervisor wants you to design a mechanism that The opposite sides of a parallelogram are ? . congruent allows workers to raise and lower pans of water from a heater. The Step 6 Finally, let’s consider the diagonals of a parallelogram. Construct the diagonals supervisor has rejected the idea LV and EO, as shown below. Label the point where the two diagonals intersect of a cranelike device, because a point M. tilting pan might cause dangerous Step 7 Measure LM and VM.What can you conclude E V sloshing. You suggest a linkage as about point M? Is this conclusion also true for

TEACHER’S EDITION M in the picture, but your super- diagonal EO? How do the diagonals relate? visor is skeptical, saying that the Share your results with your group. Copy and nonhorizontal sides would have complete the conjecture. L O to change lengths as the pan moved up and down. The super- Parallelogram Diagonals Conjecture C-47 visor also suggests putting in a The diagonals of a parallelogram ? . bisect each other diagonal piece for more stability. Do you agree?” If students have difficulty, encourage them to cut pieces out of paper or use [ᮣ For an interactive version of this investigation, see the Dynamic Geometry Exploration Properties of ᮤ wooden sticks to replicate the Parallelograms at www.keymath.com/DG . ] keymath.com/DG device. Encourage students to Parallelograms are used in vector diagrams, which have generalize their claims to conjec- many applications in science. A vector is a quantity that tures about the opposite sides has both magnitude and direction. and diagonals of parallelograms. Vectors describe quantities in physics, such Then ask them to investigate as velocity, acceleration, and force. You can represent a vector by drawing an Engine relationships among various velocity 560 mi/h angles of the changing arrow. The length and direction of the arrow represent the magnitude and parallelogram. direction of the vector. For example, a [Link] Students will encounter velocity vector tells you an airplane’s speed and direction. The lengths of vectors in physics and in future vectors in a diagram are proportional to algebra and calculus classes. The Wind the quantities they represent. velocity 80 mi/h symbol notation for a vector is a half arrow. For example, wind velocity can be written Vw. [Language] Magnitude means SHARING IDEAS intersection of its diagonals] “How does the “size” or “amount.” It is expressed symmetry show the properties of a parallelogram?” as a number with its dimensions. As students are making conjectures about a parallelo- [Because the symmetry is rotational, opposite sides In a velocity vector, the magni- gram, urge them to justify their responses. For are congruent, opposite angles are congruent, and tude is the speed, which might example, they might justify their conjecture about the diagonals bisect each other.] be the number of miles per hour consecutive angles using the Parallel Lines Conjec- or meters per second. A student ture. Your goal is to help them understand the conjec- Also ask what false conjectures someone might who has seen vectors in science tures, not to achieve some particular answer. Ask for make. For example, the diagonals of a parallelogram courses may be able to describe explanations when you think anyone in the class are not congruent and do not bisect the opposite the parallelogram rule for might not understand the concepts deeply enough. angles. Nothing about the rotational symmetry of the finding resultant vectors. [Ask] “What symmetry does a parallelogram have?” figure guarantees these properties. [2-fold rotational symmetry about the “middle,”the 282 CHAPTER 5 Discovering and Proving Polygon Properties

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In many physics problems, you combine vector quantities acting on the same object. Sharing Ideas (continued) For example, the wind current and engine thrust vectors determine the velocity of an [Ask] “Can you make a conjec- airplane. The resultant vector of these vectors is a single vector ture about consecutive sides of a that has the same effect. It can parallelogram? They’re not also be called a vector sum. V e To find a resultant vector, Vector V e represents always congruent, but can they make a parallelogram engine velocity. be?” [If so, the figure would be a The resultant vector with the vectors as sides. represents the actual rhombus.] “Can kites be parallel- The resultant vector is the speed and direction ograms?” [No; kite is defined as diagonal of the of the plane. having exactly two pairs of parallelogram from congruent consecutive sides.] the two vectors’ tails to Vw the opposite vertex. Vector V w represents Every trapezoid has only one pair wind velocity. In the diagram at right, the of opposite sides parallel; wonder resultant vector shows that aloud whether trapezoids have the the wind will speed up the consecutive angles property. [They plane, and will also blow it do.] Do trapezoids have the other slightly off course. properties of parallelograms [ᮣ For an interactive version of this diagram, see the Dynamic Geometry Exploration Resultant Vector at found in this lesson? keymath.com/DG www.keymath.com/DG . ᮤ] Assessing Progress You can assess students’ under- You will need EXERCISES standing of congruent segments, Construction tools supplementary angles, and diag- ᮣ Use your new conjectures in the following exercises. for Exercises 7 and 8 In Exercises 1–6, each figure is a parallelogram. onals and their ability to compare lengths of segments with a compass and to measure angles 1. c ? 34 cm 2. a ? 132° 3. g ? 16 in. with a protractor. d ? 27 cm b ? 48° h ? 14 in. Closing the Lesson 34 cm R U b h Reiterate the four main conjec- d M 27 cm g 16 in. tures about parallelograms: 48 a 14 in. Opposite angles are congruent, c F O consecutive angles are supple- 4. VF 36 m 5. What is the perimeter? 80 6. e ? 63° mentary, opposite sides are EF 24 m f ? 78° congruent, and diagonals bisect

EI 42 m x 3 each other. If students aren’t EDITION TEACHER’S What is the perimeter e f comfortable with these conjec- of NVI? 63 m x 3 17 tures, you might discuss E V 63 Exercise 4 before they begin 78 N working on the exercises.

F I 7. Construction Given side LA, side AS, and L, construct parallelogram LAST.

L A L A S

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BUILDING 8. Construction Given side DR and diagonals DO and PR, construct parallelogram DROP.

UNDERSTANDING D R

D O The exercises give students prac- tice in applying the four conjec- P R tures about parallelograms. In Exercises 9 and 10, copy the vector diagram and draw the resultant vector.

ASSIGNING HOMEWORK 9. 10. Vh Essential 1–8 V Performance Vb assessment 12 Portfolio 8 Journal 14 Group 9–11, 13 Review 15–22 Vcc Algebra review 5, 11

V MATERIALS w TEACHER’S EDITION ⅷ scissors (Exercise 12) y ⅷ Exercises 9 and 10 (T), optional 11. Find the coordinates of point M in parallelogram PRAM. (b a, c) ⅷ Exercise 13 (T), optional 12. Draw a quadrilateral. Make a copy of it. Draw a diagonal in M (?, ?) A (b, c) the first quadrilateral. Draw the other diagonal in the duplicate

| quadrilateral. Cut each quadrilateral into two triangles along ᮣ x Helping with the Exercises the diagonals. Arrange the four triangles into a parallelogram. P R (a, 0) Make a sketch showing how you did it. Exercises 7, 8 Encourage students to make and label sketches. 13. Developing Proof Copy and complete the flowchart to show how EA the Parallelogram Diagonals Conjecture follows logically from 7. T S other conjectures. T Given: LEAN is a parallelogram L N Show: EN and LA bisect each other Flowchart Proof L A 1 LEAN is a 3 AEN LNE parallelogram 7 8. AIA Conjecture ET NT EN and LA bisect Given ? EAL NLA AET LNT CPCTC each other 2 4 6 9 R EA LN ? ? ? D O ? AIA Conjecture ASA AT LT Definition of Definition of AE LN 8 P ? segment bisector 5 parallelogram ? CPCTC Opposite sides P R congruent

Exercise 11 Suggest to any Exercise 12 If students are having difficulty, 12. possible answer: students having difficulty that encourage them to label the sides, diagonals, and a they think about how much the angles of both quadrilaterals and then match up d right side of the parallelogram b c edges with the same label. [Ask] “Which four has moved horizontally and angles will fit around a point?” [the four angles of b vertically between points R and d the quadrilateral] c a A. To be parallel, the left side must make the same amount of change to move the same amount.

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Exercise 14 As needed, have Inner wheel traces a smaller circle, so it must be turned students think about the conjec- at a greater angle. tures made in this lesson. A B Front axle Center of 14. The parallelogram linkage turning circle is used for the sewing box so that the drawers remain parallel O Tie rod to each other (and to the Trapezoid linkage (top view) ground) so that the contents Technology cannot fall out.

Quadrilateral linkages are used in mechanical design, robotics, the automotive industry, and toy making. In cars, they are used to turn each front wheel the right amount for a smooth turn.

14. Study the sewing box pictured here. Sketch the box as viewed 21. Parallelogram. Because the from the side, and explain why a parallelogram linkage is used. triangles are congruent by SAS, 1 2. So, the segments are ᮣ parallel. Use a similar argument Review to show that the other pair of opposite sides is parallel. 5.1 15. Find the measures of the lettered angles 5.1 16. Trace the figure below. Calculate the measure 1 in this tiling of regular polygons. of each lettered angle. a 120°, b 108°, c 90°, a 135°, b 90° d 42°, e 69°

2 e c Exercise 22 [Ask] “When is the d figure a dart?” [when the line a a b b segment between the intersection points, the chord, is not between the two centers] “Can the figure be a triangle?” [yes] “When is the figure a rhombus?” [when the 5.3 17. Find x and y. Explain. 5.3 18. What is the measure of each angle in the are congruent and the x 104°, y 98°. The isosceles trapezoid face of a voussoir in this chord is between the centers] EDITION TEACHER’S quadrilaterals on the left 15-stone arch? a 84°, b 96° and right sides are kites. a ? 22. Kite or dart. Radii of the Nonvertex angles are same circle are congruent. If congruent. The quadrilateral at the bottom is an isosceles the circles have equal radii, a trapezoid. Base angles are rhombus is formed. y congruent, and consecutive 154 x 160 angles between the bases are supplementary. b ?

78

EXTENSIONS A. Are the converses of the parallelogram conjectures true? Have students choose a tool and devise an investigation to decide. B. Have students use coordinate geometry to verify one of the parallelogram conjectures.

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Parametric Equations 4.6 19. Developing Proof Is XYW WYZ? 1.8 20. Sketch the section formed when this pyramid Explain. is sliced by the plane. Graphing calculators are used No. The congruent mostly for graphing equations Z angles and side do in which y is a function of x;but W 83 not correspond. 58 y is not a function of x in a graph 83 that’s a circle or a polygon, because most values of x in the 39 domain correspond to more than 58 39 X Y one value of y. Therefore, to 21. Developing Proof graph a circle or a polygon we use 4.4 Construct two segments that bisect each other. Connect their endpoints. What type of quadrilateral is this? Draw a diagram and explain why. See page 285. a third variable, usually called t, and write both x and y as func- 5.3 22. Developing Proof Construct two intersecting circles. Connect the two centers and the tions of t. Students can think of t two points of intersection to form a quadrilateral. What type of quadrilateral is this? as giving a time and of x and y as Draw a diagram and explain why. See page 285. giving the positions of a bug crawling on the screen. These functions are called parametric equations. DRAWING REGULAR POLYGONS Central [Link] Students will learn more You can draw a regular polygon’s central angle by extending angle about parametric equations and segments from the center of the polygon to its consecutive vertices. 60 trigonometric functions in a later For example, the measure of each central angle of a hexagon is 60°. TEACHER’S EDITION math course. Using central angles, you can draw regular polygons on a graphing calculator. This is done with parametric equations, which give the The focus of the project is on x- and y-coordinates of a point in terms of a third variable, or the central angles of a regular parameter, t. polygon. Help students keep the Set your calculator’s mode to degrees and parametric. Set a friendly mathematical ideas in mind even window with an x-range of 4.7 to 4.7 and a y-range of 3.1 to while they are immersed in the 3.1. Set a t-range of 0 to 360, and t-step of 60. Enter the equations details of graphing. x 3cost and y 3sint, and graph them. You should get a hexagon. Friendly Windows To help students underst and The equations you graphed are actually the parametric equations for a circle. By using a t-step of 60 for t-values from 0 to 360, friendly calculator windows, you tell the calculator to compute only six points for the circle. explain that pixels are dots on the screen. Pictures are made by 1. Choose different t-steps to draw different regular polygons. Find the measure of each turning selected pixels dark and central angle for at least three different n-gons. leaving the rest light. If a window 2. What happens to the measure of each central angle as you draw polygons with more and more sides? is 94 pixels wide and you select 3. the x-window to be [4.7, 4.7], Experiment with rotating your polygons by choosing different t-min and t-max values. For example, set a t-range of 45 to 315, and then draw a square. then every pixel represents 9.4 4. Explain how to draw star polygons on your calculator (see p. 266). adistance across of 94 0.1 unit. Or, if a screen is 62 pixels Your project should include tall, you could select a y-window ᮣ Your answers to at least three of the questions above. of [31, 31] to give a vertical ᮣ A summary of your findings. value of 1 unit per pixel. If you set the window so that each pixel represents a different number vertically than horizontally, figures will appear distorted. Avoiding this distortion may not Supporting the OUTCOMES be important when you are ᮣ The polygons are regular. (A friendly window graphing functions. But if you If the steps were infinitely small, the two para- was used.) want a regular hexagon to look metric equations would generate a circle. ᮣ At least three questions are answered regular, you need to use a Considering this case can help students see how correctly. friendly window, that is, one in the calculator is using the central angle to draw ⅷ All questions are answered correctly. which each pixel represents the a polygon. ⅷ The measure of the central angle of a polygon same number of units vertically is given as a formula. as horizontally.

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ALGEBRA SKILLSUSING 1 ● USINGYOUR YOUR ALGEBRA ALGEBRA SSKILLSKILLS 1 ●5USING YOUR ALGEBRA SKILLS 5 ● USING YO USING YOUR ALGEBRA SKILLS 5 Writing Linear Equations

A linear equation is an equation whose graph is a straight line. Linear equations PLANNING are useful in science, business, and many other . For example, the linear 9 equation f 32 5c gives the rule for converting a temperature from degrees LESSON OUTLINE 9 Celsius, c, to degrees Fahrenheit, f. The numbers 32 and 5 f determine the graph of the equation. One day: (100, 212) 212 Understanding how the numbers in a linear equation 15 min Examples determine the graph can help you write a linear equation 30 min Exercises based on information about a graph. The y-coordinate at which a graph crosses the y-axis is called MATERIALS the y-intercept. The measure of steepness is called the slope. 32 ⅷ (0, 32) Sketchpad demonstration Equations c Below are the graphs of four equations. The table gives the 100 equation, slope, and y-intercept for each graph. How do the of Lines, optional numbers in each equation relate to the slope and y-intercept? ⅷ Sketchpad activity Writing Linear Equations, optional y 9 Equation Slope y-intercept y 2 3x 32 TEACHING y 2x 1 y 2x 12 1 y 3x 4 Students review equations of y 3x 4 34 lines and applications to perpen- 3 3 y 5 x 5 dicular lines and segment bisec- 2 2 y 2 3x tors in coordinate geometry. x 6 [Language] The term y-intercept 3 is used for the y-coordinate of y 5 _ x 2 the point where the graph crosses the y-axis. Sometimes the term may also be used to refer to the point (0, y), where

the line crosses the y-axis. EDITION TEACHER’S Assessing Progress In each case, the slope of the line is the coefficient of x in the equation. The You can assess students’ under- y-intercept is the constant that is added to the x term. standing of slope and y-intercept as students work through the In your algebra class, you may have learned about one of these forms of a linear equation in slope-intercept form: exercises. y mx b, where m is the slope and b is the y-intercept y a bx, where a is the y-intercept and b is the slope

NCTM STANDARDS LESSON OBJECTIVES

CONTENT PROCESS ⅷ Review the relationship between the constant term in a ߜ linear equation of the form y a bx (or y mx b) Number Problem Solving and the y-intercept of the equation’s graph ߜ Algebra Reasoning ⅷ Review the relationship between the coefficient of x and the ߜ slope of the graph of a linear equation in intercept (or slope- Geometry Communication intercept) form ߜ Measurement Connections ⅷ Write a linear equation from the graph of a line Data/Probability ߜ Representation

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ᮣ EXAMPLE A ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING Y Yo u m i g h t d i r e c t s t u d e n t s t o look first for the y-intercept and then for the slope of the line. The only difference between these two forms is the order of the x term and the constant term. For example, the equation of a line with slope 3 and y-intercept 1 [Alert] Many students reverse can be written as y 3x 1 or y 1 3x. the definition of slope to be the change in x over the change in y. Let’s look at a few examples that show how you can apply what you have learned Remind them that larger positive about the relationship between a linear equation and its graph. slopes describe lines that are steeper, that is, lines that have EXAMPLE A Find the equation of AB from its graph. y more change vertically for the Run 4 same horizontal change. ᮣ Solution AB has y-intercept 2 and slope 3,so B (4, 1) 4 x the equation is Rise 3 ᮣ EXAMPLE B 3 y x 2 A (0, –2) Another approach is to use the 4 powerful mathematical technique EXAMPLE B Given points C(4, 6) and D( 2, 3), find the y of expressing the same thing in equation of CD. two different ways. In this case, C (4, 6) the slope between any point (x, y) ᮣ Solution Calculate the slope. D (–2, 3) and one of the given points, y y 1 y 6 1 2 1 3 6 3 1 say (4, 6), is 2.That is,x 4 2. slope of CD x TEACHER’S EDITION x x 2 4 6 2 1 2 1 Solve for y to get y 2x 4. Students will see that slope is rise Substitute the slope into the slope-intercept form of the line, y mx b. over run and can be calculated 1 y x b asthechangein y over the change 2 y2 y1 in x: . [Alert] Students x2 x1 Use either given point, say (4, 6), to form an equation with one variable and solve. may forget that the same point 1 must be used first in the 6 2(4) b numerator and the denominator; y1 y2 6 2 b the expression also gives x1 x2 4 b the slope. 1 Thus, the equation for CD is y 2x 4. y EXAMPLE C Find the equation of the perpendicular (2, 9) bisector of the segment with endpoints (2, 9) and (4, 3).

ᮣ Solution The perpendicular bisector passes through the midpoint of the segment. x 2 (4) 2 x 2 2 1 (–4, –3) 9 (3) 6 y 2 2 3 midpoint is (1, 3)

ᮣ EXAMPLE C Another approach is to use the point-slope form, Make sure students know the relationship between y y1 m x x1 ,and substitute the point 1 the slope of a line and the slope of a perpendicular ( 1, 3) and the slope 2: 1 to the line: One slope is the opposite reciprocal of y 3 (x 1) the other. [Language] To find a reciprocal, write the 2 1 1 number as a fraction and interchange the numerator y 3 x and the denominator. [Language] The negative of a 2 2 1 1 1 1 number is the number with the opposite sign, often y 2 x, or y x 2 called the opposite of the number; the negative of a 2 2 2 2 negative number is a positive number. [Ask] “What 1 is the opposite reciprocal of 5?” 5

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ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING YO Closing the Lesson Review the major points: Calculate the slope of the segment. The slope-intercept form is a 3 9 12 convenient way to express the slope of segment 2 4 2 6 equation of a line. Given a graph The slope of the perpendicular bisector is the opposite reciprocal of the slope of of a line or two points on a line, the segment. you can find the equation of the 1 slope of perpendicular bisector line in slope-intercept form. 2 Given a line and a point, you can To f ind t he y-intercept of the perpendicular bisector, either use the method in Example B or the slope formula. The slope between any point (x, y) on the write the equation of a perpen- 1 dicular line through that point. If perpendicular bisector and the calculated midpoint ( 1, 3) must be 2. you showed students how to y 3 1 x (1) 2 solve Example C using point- Solve this equation for y to find the equation of the perpendicular bisector. slope form, you might ask them to express some of their answers y 3 1 x 1 2 to Exercises 4–14 in point-slope form for additional practice. y 3 1 (x 1)x 1 (x 1) 2 x 1 y 3 2 BUILDING x 1 UNDERSTANDING 2(y 3) 2 2 2y 6 x 1 The exercises have students move 2y x 5 back and forth between graphs and equations of lines. Even if 1 1 y 2x 22 the focus of an exercise is on the equation, encourage students to draw a graph to help them detect EXERCISES any calculation errors they might have made. ᮣ In Exercises 1–3, graph each linear equation. 4 1. 2. 3. y 1 2x y 3x 4 2y 3x 12 ASSIGNING HOMEWORK Write an equation for each line in Exercises 4 and 5. Essential 1–10

6 74 EDITION TEACHER’S 4.y y x 2 5. y y 13 x 13 Performance 4 assessment 11 2 (0, 2) (–5, 8) x Portfolio 7, 9 4 6 –2 Group 12–14 –4 (8, 2)

x In Exercises 6–8, write an equation for the line through each pair of points. 6. (1, 2), (3, 4) y x 1 7. (1, 2), (3, 4) 8. (1, 2), (6, 4) 2 8 y 3x 5 y 5 x 5

1.y 2. y 3. y (3, 8) (2, 9) (0, 1) x (0, 6) (1, –1) (0, 4)

x x

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| ● ● ● ᮣ Helping with the Exercises ALGEBRA SKILLS 5 USING YOUR ALGEBRA SKILLS 5 USING YOUR ALGEBRA SKILLS 5 USING YO Exercise 9 You might use this exercise to review other terms 9. The math club is ordering printed T-shirts to sell for a fundraiser. The T-shirt company charges $80 for the set-up fee and $4 for each printed T-shirt. Using x for the number and concepts from algebra, such of shirts the club orders, write an equation for the total cost of the T-shirts. y 80 4x as domain, range, discrete, and continuous. [Ask] “A r e a l l v a l u e s 10. Write an equation for the line with slope 3 that passes through the midpoint of the segment with endpoints (3, 4) and (11, 6). for x possible? What is the y 3x 26 domain?” [No; the domain is 11. Write an equation for the line that is perpendicular to the line y 4x 5 and that passes through the point (0, 3). 1 restricted to positive integer y 4 x 3 values.] “If you graphed the For Exercises 12–14, the coordinates of the vertices of WHY are W(0, 0), H(8, 3), equation, would you connect the and Y(2, 9). points?” [A discrete graph would 12. Find the equation of 13. Find the equation of the 14. Find the equation of better represent this situation.] the line containing perpendicular bisector the line containing 6 2 43 median WO.ofsidey x HY.altitudey x 1 HT. y x Exercises 12–14 Students may 5 9 9 need to be reminded of the defi- nitions of median, perpendicular bisector, and altitude. The coordi- IMPROVING YOUR REASONING SKILLS nates of point O must be found before finding the equation in Lunar Lockout Exercise 12, but in Exercise 14 The goal of this puzzle is to move the red piece into the center square. All of the pieces TEACHER’S EDITION the coordinates of point T don’t move according to these two rules: need to be determined. ᮣ A piece can move only horizontally or vertically, not diagonally. Improving Your Reasoning Skills ᮣ Apiececontinuestomoveuntilitspathisblockedbyanotherpiece. [Alert] Students may miss the A piece can’t move in a direction where there is no other piece to rule that markers must move block its path. In the board at right, for example, you can move the blue piece up until it is stopped by the red piece, but you can’t move until they are blocked by another it in any other direction. One possible solution to this puzzle is this marker. Markers will not move to sequence of moves: green right, green up, red down, red left. an edge of the large square and Try the puzzles below. stop. Students must plan a path for the red marker to follow. Having actual markers, even slips of paper, will help. Answers are not unique, but they’re not so numerous as students may think initially.

Beginner Puzzle Intermediate Puzzle Advanced Puzzle Find links to online versions of Lunar Lockout and other games and puzzles by ThinkFun™ at www.keymath.com/DG .

IMPROVING REASONING SKILLS Beginner Puzzle: green left, red up, red right, red down Intermediate Puzzle: orange down, orange right, red up, red right, red down Advanced Puzzle: orange down, orange left, red up, red right, yellow down, yellow left, red down

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LESSON LESSON Properties of Special 5.6 Parallelograms 5.6 The legs of the lifting platforms shown at right form rhombuses. Can you visualize PLANNING You must know a great deal how this lift would work differently if the legs formed parallelograms that weren’t about a subject to know how LESSON OUTLINE little is known about it. rhombuses?

LEO ROSTEN In this lesson you will discover some One day: properties of rhombuses, rectangles, and 20 min Investigation squares. What you discover about the diagonals of these special parallelograms 15 min Sharing will help you understand why these lifts 5min Closing work the way they do. 5min Exercises

Investigation 1 MATERIALS What Can You Draw ⅷ construction tools You will need with the Double-Edged Straightedge? ⅷ double-edged straightedges ● patty paper In this investigation you will discover the special parallelogram that you can draw ⅷ protractors ● a double-edged using just the parallel edges of a straightedge. straightedge ⅷ graph paper ⅷ Sketchpad activity Properties of Special Parallelograms, optional

TEACHING

Step 1 Step 2 Step 3 In this lesson students study diagonals of rhombuses and Step 1 On a piece of patty paper, use a double-edged straightedge to draw two pairs of rectangles. parallel lines that intersect each other. Step 2 Assuming that the two edges of your straightedge are parallel, you have drawn a You can replace or extend Inves- parallelogram. Place a second patty paper over the first and copy one of the tigations 1–3 with the Dynamic TEACHER’S EDITION TEACHER’S sides of the parallelogram. Geometry Exploration at Step 3 Compare the length of the side on the second patty paper with the lengths of www.keymath.com/DG. the other three sides of the parallelogram. How do they compare? Share your results with your group. Copy and complete the conjecture. Guiding Investigation 1 C-48 Double-Edged Straightedge Conjecture If you do not do the one-step If two parallel lines are intersected by a second pair of parallel lines that are rhombus investigation (page 292), you the same distance apart as the first pair, then the parallelogram formed is a ? . might use Investigation 1 as a follow-along activity. Be sure students are following along, not just watching.

NCTM STANDARDS LESSON OBJECTIVES

CONTENT PROCESS ⅷ Discover properties of rectangles, rhombuses, and squares Number ߜ Problem Solving ⅷ Practice construction skills Algebra ߜ Reasoning ߜ Geometry ߜ Communication ߜ Measurement Connections Data/Probability Representation

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One Step Recall that a rhombus is a parallelogram with four congruent sides, or an Direct students’ attention to the equilateral parallelogram. In Chapter 3, you learned how to construct a rhombus photograph of platforms at the using a compass and straightedge, or using patty paper. Now you know a quicker and easier way, using a double-edged straightedge. To construct a parallelogram beginning of the lesson. Then that is not a rhombus, you need two double-edged staightedges of different widths. pose this problem from Exercise 22: “What properties of Parallelogram geometric figures guarantee that Rhombus the platforms remain hori- zontal?” In their discussions, students may mention the rhom- Now let’s investigate some properties of rhombuses. buses of the figure. You might ask what properties the diagonals of a rhombus have, though their Investigation 2 perpendicularity is not relevant Do Rhombus Diagonals Have Special to the horizontal aspect of the platforms. As needed, wonder You will need Properties? aloud whether parallelograms ● patty paper other than rhombuses are hidden ● a straightedge in the picture. As students see ● a protractor (optional) diagonals of rectangles, ask whether the diagonals of all the

TEACHER’S EDITION rectangles are the same length. While students’ attention is on diagonals, ask how the diagonals relate to the angles of the paral- Step 1 Step 2 lelograms under consideration. Step 1 Draw in both diagonals of the rhombus you created in Investigation 1. Urge students to phrase their ideas as conjectures about the Step 2 Use the corner of a patty paper or a protractor to measure the angles formed by general figures. the intersection of the two diagonals. Are the diagonals perpendicular? Compare your results with your group. Also, recall that a rhombus is a parallelogram and that the diagonals of a parallelogram bisect each other. Guiding Investigation 2 Combine these two ideas into your next conjecture.

If students found this result in Rhombus Diagonals Conjecture C-49 Lesson 3.7, Exercise 15, you perpendicular, bisect each other The diagonals of a rhombus are ? , and they ? . might skip the first two steps.

Step 3 The diagonals and the sides of the rhombus form two angles at each vertex. Fold your patty paper to compare each pair of angles. What do you observe? Compare your results with your group. Copy and complete the conjecture.

Rhombus Angles Conjecture C-50 The ? of a rhombus ? the angles of the rhombus. diagonals, bisect

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So far you’ve made conjectures about a quadrilateral with four congruent sides. Guiding Investigation 3 Now let’s look at quadrilaterals with four congruent angles. What special properties do they have? Step 1 As needed, encourage Recall that a rectangle is a parallelogram with four students to draw rectangles that congruent angles, or an equiangular parallelogram. aren’t squares. Here is a thought experiment. Why is each angle of a rectangle a right angle? The Quadrilateral Sum SHARING IDEAS Conjecture says all four angles add up to 360°. The definition of a rectangle says they’re congruent. After groups share a variety of So each angle must be 90°, or a right angle. conjectures and agree on the wording for the five conjectures in this lesson, [Ask] “Which of Investigation 3 these properties of rhombuses, rectangles, and squares hold for Do Rectangle Diagonals Have Special all parallelograms?” [none of You will need Properties? them] “What special features of ● graph paper Now let’s look at the diagonals of rectangles. these figures result in their ● a compass having the special properties?” [Some of them come from equal angles, some from equal sides.] [Ask] “When we draw a rhombus with a double-edged straightedge, is the constant distance between Step 1 Step 2 the parallel edges of the straight- Step 1 Draw a large rectangle using the lines on a piece of graph paper as a guide. edge the length of each side of the rhombus?” Help students see Step 2 Draw in both diagonals. With your compass, compare the lengths of the two that the distance across the diagonals. straightedge is perpendicular to Compare results with your group. In addition, recall that a rectangle is also a the opposite edges of the parallelogram. So its diagonals also have the properties of a parallelogram’s diagonals. Combine these ideas to complete the conjecture. rhombus. Ask whether that distance is ever the length of one Rectangle Diagonals Conjecture C-51 side of the rhombus. [It is for a square.] Is every square a The diagonals of a rectangle are ? and ? . congruent, bisect each other rhombus then?

Refer students to Lesson 1.6 and EDITION TEACHER’S the definition “A square is a 4-sided figure that has all sides Career congruent and all angles measuring 90°.” The book promises better definitions later. A tailor uses a button spacer to mark the locations of the buttons. The tool opens and closes, but the tips always remain equally spaced. What Later is now. To encourage quadrilateral properties make this tool work correctly? students to grapple with the ideas, ask them to critique the two definitions of square proposed in this lesson. Then ask what these definitions allow them to conclude about squares from Sharing Ideas (continued) Assessing Progress the conjectures of this lesson. “any”or “every”or “each.”For Exercise 1,the state- See how well students voluntarily use the terms Working through Exercises 1–10 ment is true only in the special case of an equi- parallel,rhombus,rectangle,diagonal,equiangular, as a class can help you see how angular parallelogram, or a rectangle. So, while the equilateral, perpendicular, segment bisector, and well students are understanding statement is “sometimes true” in this context, it angle bisector. You can also assess students’ ability the ideas. Make sure students would be considered false if given as a strictly to compare lengths using patty paper and how well understand that in true/false true/false question. [Alert] If students are having they understand that properties of a larger class questions, the word true means difficulty, suggest that they look back at their draw- (rectangles or rhombuses) are inherited by “always true” and that in mathe- ings from this lesson and the previous lesson. members of a subclass (squares). matics, the word a usually means

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Closing the Lesson What happens if you combine the properties of a rectangle and a rhombus? We call the shape a square, and you can think of it as a regular quadrilateral. So you can Reiterate that the diagonals of define it in two different ways. any rhombus are perpendicular A square is an equiangular rhombus. bisectors of each other and bisect Or opposite angles but are not necessarily congruent to each A square is an equilateral rectangle. other. On the other hand, the A square is a parallelogram, as well as both a rectangle and a diagonals of any rectangle are rhombus. Use what you know about the properties of these congruent and bisect each other three quadrilaterals to copy and complete this conjecture. but are not necessarily perpen- C-52 dicular and are not necessarily Square Diagonals Conjecture congruent, perpendicular, angle bisectors. The diagonals The diagonals of a square are ? , ? ,and ? . bisect each other have all four properties in the case of a rhombus that is also a [ᮣ For an interactive version of all the investigations in this lesson, see the Dynamic Geometry Exploration rectangle—that is, the diagonals keymath.com/DG Properties of Special Parallelogram at www.keymath.com/DG . ᮤ] of a square are congruent, are You will need perpendicular bisectors of each EXERCISES other, and bisect opposite angles. Construction tools ᮣ Developing Proof For Exercises 1–10, state whether each statement is always for Exercises 17–19, More important than remem- true, sometimes true, or never true. Use sketches or explanations to 24, 25, and 32 bering all this is being able to support your answers. figure it out from quick sketches. TEACHER’S EDITION If students are struggling, you 1. The diagonals of a parallelogram are congruent. might work Exercise 12 or 13 2. The consecutive angles of a rectangle are congruent and with them. supplementary. 3. The diagonals of a rectangle bisect each other. Always true by the Rectangle Diagonals Conjecture. BUILDING 4. The diagonals of a rectangle bisect the angles. UNDERSTANDING 5. The diagonals of a square are perpendicular bisectors of each other. Always true by the Square Diagonals Conjecture. 6. A rhombus is a square. The exercises involve using all the conjectures from this lesson. 7. A square is a rectangle. Always true; all squares fit the definition of rectangle. 8. A diagonal divides a square into two isosceles right triangles. ASSIGNING HOMEWORK 9. Opposite angles in a parallelogram are congruent. Always true by the Parallelogram Opposite Angles Conjecture. Essential 2–20 evens 10. Consecutive angles in a parallelogram are congruent. Portfolio 18, 27 11. WREK is a rectangle. 12. PARL is a parallelogram. 13. SQRE is a square. Journal 21, 22 CR 10 y ? 37° x ? 45° WE ? 20 y ? 90° Group 1–19 odds, 23–25 K E ER Review 28–32 L R Algebra review 14–16, 29–31 y C 10 y 95 MATERIALS 48 x W R P A S Q ⅷ Exercise 26 (T), optional ⅷ Exercise 28 (T), optional 1. Sometimes true; it is true only if the parallelo- 4. Sometimes true; it is true only if the rectangle | ᮣ Helping with the Exercises gram is a rectangle. is a square. Exercises 1–5, 8–10 [Language] In false these exercises, the word a means true false “any.” For the statement to be true always true, it must be true for 2. Always true; by the definition of rectangle, all any parallelogram, rectangle, or the angles are congruent. By the Quadrilateral square. Sum Conjecture and division, they all measure 90°, so any two angles in a rectangle, including consecutive angles, are supplementary.

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Developing Proof For Exercises 14–16, use deductive reasoning to explain your answers. 6. Sometimes true; it is true 14. Is DIAM a rhombus? Why? 15. Is BOXY a rectangle? Why? 16. Is TILE a parallelogram? Why? only if the rhombus is equian- gular. y y y x –9 O A 9 T (0, 18) false I B E (10, 0) x X I (–10, 0) M true D –9 Y L (0, –18) x 9 8. Always true; all sides of a square are congruent and form 17. Construction Given the diagonal LV, construct square LOVE. 17. right angles, so the sides EVbecome the legs of the isosceles LV right triangle and the diagonal is the hypotenuse. 18. Construction Given diagonal BK and B, construct 10. Sometimes true; it is true rhombus BAKE. LO only if the parallelogram is a rectangle. Consecutive angles of a parallelogram are always

B supplementary, but are B K congruent only if they are right angles. 19. Construction Given side PS and diagonal PE, construct rectangle PIES. one possible I E construction: Exercises 14–16 Be sure students P S explain their reasoning on these P E exercises.

20. Developing Proof Write the converse of the Rectangle Diagonals Conjecture. Is it true? PS14. DIAM is not a rhombus Prove it or show a counterexample. because it is not equilateral and opposite sides are not parallel. 21. To make sure that a room is rectangular, builders check the two diagonals of the room. Explain what they check about the diagonals, and why this works. 15. BOXY is a rectangle If the diagonals are because its adjacent sides are congruent and bisect each perpendicular. other, then the room is rectangular (converse of 16. Ye s . TILE is a rhombus, and the Rectangle Diagonals a rhombus is a parallelogram. EDITION TEACHER’S Conjecture). Exercise 17 If students are having difficulty, suggest that they think about what properties diagonals of squares have. In this and later exercises, they might make a 22. The platforms shown at the beginning of this lesson lift objects straight sketch first to help with the up. The platform also stays parallel to the floor. You can clearly see rhombuses in the picture, but you can also visualize the frame as the construction. diagonals of three rectangles. Explain why the diagonals of a rectangle guarantee this vertical movement. The platform stays parallel to the floor because opposite sides of a rectangle are parallel (a rect- angle is a parallelogram).

Exercise 18 As needed, wonder aloud how a Exercise 19 Students having difficulty might find it rhombus’s diagonal is related to one of its angles. useful to start with perpendicular lines and draw the diagonal from the other end of the given side. 18. Constructions will vary. A Exercise 22 This exercise is the one-step K A K investigation.

B B E E

See pages 839–840 for answer to Exercise 20.

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23. The crosswalks form a 23. At the street intersection shown at right, one of the streets is parallelogram: The streets are of wider than the other. Do the crosswalks form a rhombus or different widths, so the cross- a parallelogram? Explain. What would have to be true about the streets if the crosswalks formed a rectangle? A square? walks are of different lengths. The streets would have to meet In Exercises 24 and 25, use only the two parallel edges of your at right angles for the cross- double-edged straightedge. You may not fold the paper or use walks to form a rectangle. The any marks on the straightedge. corners would have to be right 24. Construction Draw an angle on your paper. Use your angles and the streets would double-edged straightedge to construct the bisector of also have to be of the same the angle. width for the crosswalk to form 25. Construction Draw a segment on your paper. Use your double-edged straightedge to a square. construct the perpendicular bisector of the segment. Exercises 24, 25 As needed, ask 26. Developing Proof Complete the flowchart proof below to demonstrate DA students what kind of figure they logically that if a quadrilateral has four congruent sides then it is a 3 2 must construct so that a diagonal rhombus. One possible proof for this argument has been started for you. of that figure solves the problem. Given: Quadrilateral QUAD has QU UA AD DQ with diagonal DU Show: QUAD is a rhombus 1 4 24. Place one side of the ruler QU along one side of the angle. Flowchart Proof Rotate your ruler so that each endpoint of the segment barely shows on each side of the ruler.Draw the parallel lines on each side of Draw a line with the other side 1 QU AD your ruler. Now rotate your ruler the other way and repeat the process to get a of the ruler. Repeat with the rhombus. The original segment is one diagonal of the rhombus. The other Given

TEACHER’S EDITION other side of the angle. Draw a diagonal will be the perpendicular bisector of the original segment. line from the vertex of the angle 2 4 5 6 7 to the point where the two lines QD AU QUD ADU 1 2 QU AD QUAD is a 3 4 QD AU parallelogram meet. ? Given ? SSS ? CPCTC Converse of the Definition of Exercise 26 If in Chapter 1 Parallel Lines parallelogram 3 students defined a rhombus as DU DU Conjecture 9 an equilateral quadrilateral Same segment 8 QUAD is a QU UA AD DQ (rather than an equilateral paral- rhombus Given lelogram), there is nothing to ? Definition prove here. of rhombus 27. Developing Proof In the last exercise, you proved that if the four sides of a Exercise 27 If students defined a quadrilateral are congruent, then the quadrilateral is a rhombus. So, when we rectangle as an equiangular defined rhombus, we did not need the added condition of it being a parallelogram. quadrilateral, there is nothing to We only needed to say that it is a quadrilateral with all four sides congruent. Is prove here. However, if students this true for rectangles? Your conjecture would be, “If a quadrilateral has all four angles congruent, it must be a rectangle.” Can you find a counterexample that defined a rectangle as an equian- proves it false? gular parallelogram, as in the text, then in this exercise they If not, prove this conjecture by showing that A D ABCD in the diagram at right is a parallelogram. 1 45 will show that any quadrilateral Note the auxilliary line. with four equal angles is a rectangle.

2 3 B C

27. Ye s , i t i s t r u e f o r r e c t a n g l e s . solve to get m5 90°. By definition of Given: 1 2 3 4 congruent angles, 5 3, and 5 and 3 are alternate interior angles, so AD BC by the Show: ABCD is a rectangle Converse of the Parallel Lines Conjecture. Simi- larly, 1 and 5 are congruent corresponding By the Quadrilateral Sum Conjecture, m 1 m2 m3 m4 360°. It is given that all angles, so AB CD by the Converse of the Parallel four angles are congruent, so each angle measures Lines Conjecture. Thus, ABCD is a parallelogram 90°. Because 4 and 5 form a linear pair, by the definition of parallelogram. Because it is an m4 m5 180°. Substitute 90° for m4 and equiangular parallelogram, ABCD is a rectangle.

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8 86 30. y 9x 9 or ᮣ Review 8x 9y 86 28. Trace the figure below. Calculate the measure of each lettered angle. 7 12 31. y x or a 54°, b 36°, c 72°, 10 5 d 108°, e 36°, 7x 10y 24 54 f 144°, g 18°, e d h 48°, j 48°, k 84° h c j k f a g b

UYAS 5 29. Find the coordinates of three more points that lie on the line 2 mi/h passing through the points (2, 1) and ( 3, 4). 2 mi/h possible answers: (1, 0); (0, 1); (1, 2); (2, 3) UYAS 5 30. Write the equation of the perpendicular bisector of the segment 60 with endpoints (12, 15) and (4, 3). 60

UYAS 5 31. ABC has vertices A(0, 0), B(4, 2), and C(8, 8). What is the equation of the median to side AB? 5.5 32. Construction Oran Boatwright is rowing at a 60° angle from the 1.5 mi/h upstream direction as shown. Use a ruler and a protractor to draw 1.5 mi/h the vector diagram. Draw the resultant vector and measure it to find his actual velocity and direction. velocity 1.8 mi/h; angle of path 106.1° clockwise from the north

IMPROVING YOUR VISUAL THINKING SKILLS A Puzzle Quilt Fourth-grade students at Public School 95, the Bronx, New York, made the puzzle quilt at right A with the help of artist Paula Nadelstern. Each square has a twin made of exactly the same B

shaped pieces. Only the colors, EDITION TEACHER’S chosen from traditional Amish colors, are different. For example, square A1 is the twin of C square B3. Match each square with its twin.

D

1234

IMPROVING VISUAL THINKING SKILLS Matching pairs: (A1, B3)(B1, D3)(A2, C2)(B2, C4)(A3, B4) (C1, C3)(A4, D2)(D1, D4)

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LESSON LESSON Proving Quadrilateral 5.7 5.7 Properties Most of the paragraph proofs and flowchart proofs you have done so far have PLANNING been set up for you to complete. Creating your own proofs requires good reasoning “For instance” is not a “proof.” strategies and planning. One excellent reasoning strategy is “thinking backward.” If JEWISH SAYING you know where you are headed but are unsure where to start, start at the end of LESSON OUTLINE the problem and work your way back to the beginning one step at a time. One day: The firefighter below asks another firefighter to turn on one of the water hydrants. 15 min Example But which one? A mistake could mean disaster—a nozzle flying around loose under all that pressure. Which hydrant should the firefighter turn on? 10 min Investigation 5min Closing 15 min Exercises

MATERIALS

ⅷ Finding the Square Route (T), optional ⅷ Japanese Puzzle Quilts (W) for Project TEACHER’S EDITION TEACHING

This lesson gives students prac- tice in thinking backward to Did you “think backward” to solve the puzzle? Thinking backward is a useful reasoning write a flowchart proof. strategytousewhenyouwriteproofswhichyoumayhavealreadybeenusing. One Step To help plan a proof and visualize the flow of reasoning, you can make a flowchart. As you think backward through a proof, you draw a flowchart backward to show Before saying anything about the steps in your thinking. working backward, give students the puzzle in the investigation. If Work with a partner when you first try planning your geometry proof. Think some groups solve it working backward to make your plan: Start with the conclusion and reason back to the given. Let’s look at an example. forward, have groups share their approaches so that they all A concave kite is sometimes called a dart. C realize the power of backward thinking. Then have them work EXAMPLE Given: Dart ADBC with AC BC,AD BD on the example and exercises. Show: CD bisects ACB D A B ᮣ EXAMPLE ᮣ The underlying question students Solution Plan: Begin by drawing a diagram and marking the given information on it. will be asking themselves as they Next, construct your proof by reasoning backward and using other reasoning strategies. Then convert this reasoning into a flowchart. Your flowchart should work backward is “What could I start with boxes containing the given information and end with what you are use to prove this?” As they trying to demonstrate. The arrows indicate the flow of your logical argument. answer that question, they have Your thinking might go something like this: a reason for the corresponding step in the proof. LESSON OBJECTIVES NCTM STANDARDS

ⅷ Practice writing flowchart and paragraph proofs CONTENT PROCESS ⅷ Develop deductive reasoning skills Number ߜ Problem Solving

ⅷ Review properties of quadrilaterals Algebra ߜ Reasoning ߜ Geometry ߜ Communication Measurement Connections Data/Probability ߜ Representation

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“I can show CD is the bisector of ACB if I can show ACD BCD.”

ACD BCD CD is the bisector of ACB

“I can show ACD BCD if they are corresponding angles in congruent triangles.”

ADC BDC ACD BCD CD is the bisector of ACB

“Can I show ADC BDC? Yes, I can, by SSS, because it is given that AC BC and AD BD, and CD CD because it is the same segment in both triangles.”

AD BD

AC BC ADC BDC ACD BCD CD is the bisector of ACB

CD CD By adding the reason for each statement below each box in your flowchart, you can make the flowchart into a complete flowchart proof. 1 AD BD Given

2 AC BC 4 ADC BDC 5 ACD BCD 6 CD is the bisector of ACB Given SSS CPCTC Definition of angle bisector 3 CD CD Same segment Some students prefer to write their proofs in a flowchart format, and others prefer to write out the proof as an explanation in paragraph form. By reversing the reasoning in your plan, you can make the plan into a complete paragraph proof. “It is given that AC BC and AD BD.CD CD because it is the same

segment in both triangles. So, ADC BDC by the SSS Congruence EDITION TEACHER’S Conjecture, and ACD BCD by the definition of congruent triangles (CPCTC). Therefore, by the definition of angle bisectors, CD is the bisector of ACB. Q.E.D.”

Language

The abbreviation Q.E.D. at the end of a proof stands for the Latin phrase quod erat demonstrandum, meaning “which was to be demonstrated.”This is a translation of the Greek phrase that Euclid used at the end of each of his proofs in the Elements.

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Guiding the Investigation Investigation Finding the Square Route If students work backward from 100, they’ll find that there’s a Here is a puzzle for you to solve that has single path. If they work forward, nothing to do with square roots. In the puzzle grid at right, the goal is to find a they may have to track one or Step 1 From 1, you can route—a path—that starts at 1 in the upper multiply by 5 and move two paths until they’re stuck left and ends at 100 in the lower right. before finding the successful to 5, multiply by 2 and move to 2, or add 2 and path. Unless they track all Yo u c a n m o v e t o a n a d j a c e n t s q u a r e move to 3. horizontally, vertically, or diagonally. In possible paths, they won’t know order to move, you must be able to add, for sure that the result is unique. Step 2 work backward: subtract, multiply, or divide the number in 100 5; 95 5; the square you occupy by 2 or 5 to get the Assessing Progress 19 5; 14 2; 28 2; number in the new square. For example, if Most of your assessment oppor- 30 5; 6 2; you happen to be in square 11, you could tunities in this lesson will come 8 5; 3 2; 1 move to square 9 by subtracting 2, or to through observation of student square 55 by multiplying by 5. 1 30 work on the exercises. You can 3 6 28 Step 1 Using this puzzle’s rule for moving, explain why there are three possible first assess students’ understanding 8 14 19 moves. of many major mathematical 95 Step 2 Solve the puzzle—which route will take you from 1 to 100? Show it with arrows. concepts encountered in this 100 course so far. Step 3 Think about any problem-solving strategies that were particularly helpful for solving this puzzle. How can these strategies help you in developing proofs?

TEACHER’S EDITION Working backward is a helpful strategy for solving this problem and for Closing the Lesson writing proofs. The main point of this lesson is that proofs are often written by working backward. If you think EXERCISES students will have difficulty with ᮣ Developing Proof In Exercises 1–3, each conjecture has also been stated as a “given” and a the exercises, you may want to “show.”Any necessary auxiliary lines have been included. Complete a flowchart proof or work through Exercise 1 with write a paragraph proof. them. 1. Prove the conjecture: The diagonal of a parallelogram divides the K A 4 parallelogram into two congruent triangles. 2 1 BUILDING Given: Parallelogram SOAK with diagonal SA 3 SO UNDERSTANDING Show: SOA AKS Flowchart Proof 2 SO KA 4 3 4 The exercises allow students to practice thinking backward to Definition of AIA parallelogram develop proofs of some proper- SK ties of various quadrilaterals. 1 3 5 7 SOAK is a OA ? 1 2 SOA ? AKS Urge students to copy all parallelogram ? ? AIA ? ASA diagrams and to mark and label Given Definition of them accordingly. parallelogram 6 SA ? SA ASSIGNING HOMEWORK ? Same segment

Essential 1–9 odds Performance assessment 4 MATERIALS Portfolio 6 ⅷ Exercise 1 (T), optional Journal 10 ⅷ Exercise 2 (T), optional Group 2–8 evens ⅷ Exercise 3 (T), optional Review 11–16 Algebra review 13, 14

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| 2. Prove the conjecture: The opposite angles of a parallelogram are congruent. H T ᮣ Given: Parallelogram BATH with diagonals BT and HA Helping with the Exercises Show: HBA ATH and BAT THB Many students will be unsure of Flowchart Proof B A whether a reason they want to cite is a definition or a conjecture. 1 2 3 Parallelogram BATH BAT THB BAT THB Begin to encourage them to think with diagonal BT Conjecture proved in CPCTC about the identification question. Given Exercise 1 THA 4 5 6 You may wish to jigsaw, giving Parallelogram BATH BAH = ? HBA ? ATH with diagonal HA different exercises to different ? Conjecture proved ? CPCTC groups, after all groups have ? Given in Exercise 1 completed Exercise 1.

3. Prove the conjecture: If the opposite sides of a quadrilateral are RTExercise 2 The given proof 2 congruent, then the quadrilateral is a parallelogram. 3 depends on the conjecture in Given: Quadrilateral WATR, with WA RT and WR AT , Exercise 1. and diagonal WT Show: WAT R is a parallelogram 4 1 Exercise 3 Students may list given W A Flowchart Proof information in a different order. 1 Exercise 6 ? ? WA RT Students may use the RT WA by Converse of the Parallel ? 2 fact that every rectangle is a Given 5 6 Lines Conjecture WRT TAW 1 ? ? ? parallelogram so they can use WR AT ? CPCTC ? 9 WATR is a the property that opposite sides 2 ? ? 4 ? ? parallelogram are congruent. Students may 3 RW TA ? Given ? SSS have difficulty realizing that the 7 4 ? 8 ? ? ? Definition of WT WT parallelogram two triangles they want to prove ? ? 3 CPCTC Converse of the ? ? Parallel Lines congruent are overlapping. ? Same segment Conjecture

4. Developing Proof Write a flowchart proof to demonstrate that P A quadrilateral SOAP is a parallelogram. 1 3 Given: Quadrilateral SOAP with SP OA and SP OA Show: SOAP is a parallelogram 4 2 5. The results of the proof in Exercise 4 can now be stated as a S O proved conjecture. Complete this statement beneath your proof: TEACHER’S EDITION TEACHER’S “If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a ? .” parallelogram Developing Proof For Exercises 6–9, prove the conjecture.

6. Conjecture: The diagonals of a rectangle are congruent. G O Given: Rectangle YOGI with diagonals YG and OI Show: YG OI

I Y

4. 6. sample flowchart proof: 1 2 5 SP OA SP OA ᭝SOP ᭝APO 1 IY GO 2 ЄYOG ЄOYI 3 YO OY

Given Given SAS Opposite sides of Definition of rectangle Same segment 3 Є Є 6 rectangle are congruent 1 2 Є3 Є4 4 Є ᭝ AIA Conjecture CPCTC YOG OYI 4 SAS PO PO 7 PA SO Same segment Converse of 5 AIA Conjecture YG IO CPCTC 8 SOAP is a parallelogram

Definition of parallelogram

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Exercise 7 Students may question 7. Conjecture: If the diagonals of a parallelogram are congruent, A R what they need to prove because then the parallelogram is a rectangle. they’re accustomed to thinking Given: Parallelogram BEAR,with diagonals BA ER about rectangles as having many Show: BEAR is a rectangle different properties. They need to E show that the figure satisfies the B book’s definition of rectangle 8. Isosceles Trapezoid Conjecture: The base angles of an P Z A as a parallelogram with four 1 3 isosceles trapezoid are congruent. 2 congruent angles. Given: Isosceles trapezoid PART with PA TR,PT AR, Exercise 8 This is the Isosceles and TZ constructed parallel to RA T R Trapezoid Conjecture of Show: TPA RAP Lesson 5.3. Students may miss a 9. Isosceles Trapezoid Diagonals Conjecture: The diagonals of an R H subtle aspect of this proof: that isosceles trapezoid are congruent. the auxiliary line segment TZ is Given: Isosceles trapezoid GTHR with GR TH and different from the trapezoid’s diagonals GH and TR G T side PT. If point Z could be Show: GH TR concurrent with point T, the same proof would apply to 10. You have discovered that triangles are rigid, but parallelograms Warp threads (vertical) are not. This property shows up in the making of fabric, which has showing the congruence of the Bias warp threads and weft threads. Fabric is constructed by weaving base angles of a parallelogram, a thread at right angles, creating a grid of rectangles. What happens false conclusion. when you pull the fabric along the warp or weft? What happens

TEACHER’S EDITION Weft threads when you pull the fabric along a diagonal (the bias)? (horizontal) 8. Because AR is parallel to ZT, Ifthefabricispulledalongthewarportheweft,nothing corresponding 3 and 2 are happens.However,ifthefabricispulledalongthebias,itcanbe congruent. Opposite sides of stretched because the rectangles are pulled into parallelograms. parallelogram ZART are ᮣ congruent so AR TZ. Because Review the trapezoid is isosceles, 5.2 11. Find the measure of the acute angles AR PT, and substituting gives in the 4-pointed star in the Islamic ZT PT making PTZ tiling shown at right. The polygons isosceles and 1 and 2 are squares and regular hexagons. congruent. By substitution, 1 Find the measure of the acute angles and 3 are congruent. in the 6-pointed star in the Islamic tiling on the far right. The 6-pointed star design is created by Exercise 9 Yo u m i g h t s k i p t h i s arranging six squares. Are the angles in both stars the same? problem if students proved the 30° angles in 4-pointed star, 30° angles in 6-pointed star; yes 12. Isosceles Trapezoid Diagonals 2.6 A contractor tacked one end of a string to each vertical edge of a window. He then handed a protractor to his apprentice and said, Conjecture in Lesson 5.3. “Here, find out if the vertical edges are parallel.”What should the 13. ES: y 2x 3; apprentice do? Help him.Heshouldmeasurethealternateinterior 1 angles to see whether they’re congruent. If they are, the edges are parallel. 13. QI: y 2x 2 UYAS 5 Find the equations of the lines containing the diagonals of rhombus EQSI with vertices E(3, 3), Q(2, 3), S(1, 1), and I(6, 1).

2 UYAS 5 14. Find the point of intersection of the lines y 3x 1 and 3x 4y 8. (12, 7)

9. Sample flowchart proof:

2 GR Х TH Given

1 Isosceles trapezoid 3 Х 5 ᭝ ᭝ 6 Х GTHR GT GT RGT HTG GH TR Same segment SAS CPCTC Given

4 ЄRGT ЄHTG Isosceles Trapezoid Conjecture See page 840 for answer to Exercise 7.

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15. The last bus stops at the school some time between 4:45 and 5:00. What is the Exercise 15 The exercise reviews 1 probability that you will miss the bus if you arrive at the bus stop at 4:50? 3 the Exploration Geometric 1.8 16. The 3-by-9-by-12-inch clear plastic sealed container shown is resting on its Probability I, on page 88. smallest face. It is partially filled with a liquid to a height of 8 inches. Sketch the 12 16. container resting on its middle-sized face. What will be the height of the liquid in the container in this position? 9 9 3 6

12 3

6 inches

JAPANESE PUZZLE QUILTS Project Question Answers 1. At least two colors are When experienced quilters first see Japanese puzzle needed. quilts, they are often amazed (or puzzled?) because the straight rows of blocks so common to block quilts do 2. In a quilt of four colors, not seem to exist. The sewing lines between apparent around any pseudoblock of blocks seem jagged. At first glance, Japanese puzzle quilts one color there can be six look like American crazy quilts that must be handsewn different arrangements of and that take forever to make! pseudoblocks of the other However, Japanese puzzle quilts do contain straight sewing three colors (not counting lines. Study the Japanese puzzle quilt at right. Can you rotations); three of these find the basic quilt block? What shape is it? Mabry Benson designed this puzzle quilt, arrangements require four An actual pre-sewn block The puzzle quilt shown Red and Blue Puzzle (1994). Can you find any different quilt blocks. If four above is made of blocks rhombic blocks that are the same? How colors are used with no of four different-color many different types of fabric were used? kites sewn into rhombuses. touching, then the quilt can The rhombic blocks are sewn together with straight be made with two different sewing lines as shown in the diagram at left. quilt blocks, each with kites of all four colors. If the colors Now for your project: You will need copies of the Japanese puzzle quilt grid, color pencils or markers, and of one block are considered A pseudoblock color paper or fabrics. Plan, design, and create a Japanese in clockwise order beginning puzzle quilt out of paper or fabric. with a kite in the smaller 1. To produce the zigzag effect of a Japanese puzzle quilt, you need to avoid angle of the rhombus, then pseudoblocks of the same color sharing an edge. How many different colors or the other block will have the fabrics do you need in order to make a puzzle quilt? same colors in counterclock- 2. How many different types of pre-sewn rhombic blocks do you need for a four-color wise order, beginning with a EDITION TEACHER’S Japanese puzzle quilt? What if you want no two pseudoblocks of the same color to kite in the larger angle of the touch at either an edge or a vertex? rhombus. 3. Can you create a four-color Japanese puzzle quilt that requires more than four different color combinations in the rhombic blocks? 3. None of the four-color quilts Your project should include requires more than four different quilt blocks. ᮣ Your answers to at least two of the questions above. ᮣ Your Japanese puzzle quilt made of paper or fabric.

Supporting the OUTCOMES ᮣ At least two of the project questions are Each student needs eight to ten copies of answered. the worksheet. [Language] The pseudoblocks ᮣ A two- or four-color quilt is correctly are false, jagged blocks of one color, not the assembled. rhombic building blocks of two or four colors. ⅷ A quilt is designed with five or six colors. ⅷ All project questions are answered, and examples or explanations are given.

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CHAPTER REVIEW 5 ● CHAPTERCHAPTER 11 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTE In this chapter you extended your knowledge of triangles to other 5 polygons. You discovered the interior and exterior angle sums for REVIEW all polygons. You investigated the midsegments of triangles and PLANNING trapezoids and the properties of parallelograms. You learned what distinguishes various quadrilaterals and what properties apply to LESSON OUTLINE each class of quadrilaterals. First day: Along the way you practiced proving conjectures with flowcharts and paragraph proofs. Be sure you’ve added the new conjectures to 15 min Reviewing your list. Include diagrams for clarity. 30 min Exercises How has your knowledge of triangles helped you make discoveries Second day: about other polygons? 30 min Exercises 15 min Student self-assessment EXERCISES You will need Construction tools ᮣ 1. How do you find the measure of one exterior angle of a regular for Exercises 19–24 REVIEWING polygon? 360° divided by the number of sides 2. How can you find the number of sides of an equiangular polygon by measuring one Ask each student to draw a

TEACHER’S EDITION of its interior angles? By measuring one of its exterior angles? concept map relating different 3. kinds of quadrilaterals. Have 3. How do you construct a rhombus by using only a ruler or double-edged students check each other’s maps straightedge? Trace both sides of the ruler as shown at right. to be sure that they reflect the 4. How do you bisect an angle by using only a ruler or double-edged straightedge? hierarchy of the student book: The three main kinds of special 5. How can you use the converse of the Rectangle Diagonals Conjecture to determine if the corners of a room are right angles? Sample answer: Measure the diagonals with quadrilaterals are trapezoids, string to see if they are congruent and bisect each other. kites, and parallelograms; the 6. How can you use the Triangle Midsegment Conjecture to find a distance between two main kinds of special paral- two points that you can’t measure directly? lelograms are rhombuses and 7. Find x and y. x 10°, y 40° 8. The perimeter is 266 cm. 9. Find a and c. a 116°, c 64° rectangles; and every square is a Find x. 50 x 60 cm rhombus and a rectangle. x 116 94 cm c Quadrilateral a 80 y x Kite Trapezoid 52 cm

Parallelogram 10. MS is a midsegment. Find 11. Find x. x 38 cm 12. Find y and z. Rhombus Rectangle the perimeter of MOIS. 100 y 34 cm, z 51 cm I G Square 17 cm S x – 12 18 20 32 cm x Then ask students what proper- y ties can be assigned to each cate- O M 26 T D z gory but to none containing it. For example, all sides of a rhombus are congruent, but that is not true for parallelograms in to figure out how to fit in the other conjectures. 2. Sample answers: Using an interior angle, set general. Keep stressing that any They may want to add “isosceles trapezoids” as the interior angle measure formula equal to the property of a category is inher- a subcategory of trapezoids in order to include angle and solve for n. Using an exterior angle, ited by more specific quadrilat- conjectures related to those quadrilaterals, and divide into 360°. Or find the interior angle erals; for example, the fact that to add a category for polygons (including all measure and go from there. the diagonals of a parallelogram quadrilaterals) to fit in the earliest conjectures bisect each other is inherited by of the chapter. rhombuses, rectangles, and squares. In this way you can review most of the conjectures of this chapter. Try to get students

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If you have skipped any lessons of this chapter, be selective as 13. Copy and complete the table below by placing a yes (to mean always) or a no (to mean not always) in each empty space. Use what you know about special quadrilaterals. to which exercises you assign. Students will get a good review if Isosceles they do odds or evens in groups Kite trapezoid Parallelogram Rhombus Rectangle Square and the others individually. Opposite sides No No Ye s Ye s Ye s Ye s are parallel MATERIALS Opposite sides No No Ye s Ye s Ye s Ye s are congruent ⅷ Exercises 13 and 15 (T), optional Opposite angles are congruent No No Ye s Ye s Ye s Ye s | Diagonals bisect ᮣ No No each other Ye s Ye s Ye s Ye s Helping with the Exercises Diagonals are Ye s No No Ye s No Ye s Exercise 13 [Ask] “What can you perpendicular say about a quadrilateral if it has Diagonals are No congruent No Ye s No Ye s Ye s perpendicular diagonals?” [It Exactly one line could be a kite, a rhombus, a of symmetry Ye s Ye s No No No No square, or a general quadrilat- Exactly two lines eral.] “...exactly one line of No No Ye s of symmetry No Ye s No symmetry?” [It is a kite or isosceles trapezoid.] “...perpen- 14. Application A 2-inch-wide frame is to be built around the dicular diagonals and exactly one regular decagonal window shown. At what angles a and b line of symmetry?” [It is a kite.] 2 in. should the corners of each piece be cut? a 72°, b 108° “...exactly two lines of 15. Developing Proof Find the measure of each lettered angle. symmetry?” [rhombus or rect- Explain how you determined measures e, f, and g. b angle] “...perpendicular diago- a nals and two lines of symmetry?” k [rhombus] “...perpendicular k p diagonals and four lines of a symmetry?” [square] b m c g d 15. a 120°, b 60°, c 60°, e d 120°, e 60°, f 30°, g

EDITION TEACHER’S f 108°, m 24°, p 84°; Possible explanation: Because c 60°, the angle that forms a linear pair with e and its congruent adjacent angle measures 60°. So 16. Archaeologist Ertha Diggs has uncovered one stone that appears to be a voussoir from a semicircular 60° 2e 180°, and e 60°. stone arch. On each isosceles trapezoidal face, the The triangle containing f has a obtuse angles measure 96°. Assuming all the stones 60° angle. The other angle is a were identical, how many stones were in the right angle because it forms a original arch? 15 stones linear pair with a right angle. So f 30° by the Triangle Sum Conjecture. Because g is an inte- rior angle in an equiangular 4. Make a rhombus using the double-edged 6. pentagon, divide 540° by 5 to Sample answer: Draw a third point and straightedge, and draw a diagonal connecting the connect it with each of the two points to form two get g 108°. angle vertex to the opposite vertex. sides of a triangle. Find the midpoints of the two sides and connect them to construct the midseg- ment. The distance between the two points is twice the length of the midsegment.

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18. When the swing is motion- EW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPT less,the seat,the bar at the top, and the chains form a rectangle. When you swing left to right, 17. Kite ABCD has vertices A(3, 2), B(2, 2), C(3, 1), and D(0, 2). the rectangle changes to a paral- Find the coordinates of the point of intersection of the diagonals. (1, 0) lelogram. The opposite sides stay equal in length, so they stay 18. When you swing left to right on a swing, the seat stays parallel to the ground. Explain why. parallel. The seat and the bar at the top are also parallel to the 19. Construction The tiling of congruent pentagons shown below is created ground. from a honeycomb grid (tiling of regular hexagons). What is the measure of each lettered angle? Re-create the design with compass 20. and straightedge. a = 60°, b = 120° Resultant vector

900 km/h

a b

50 km/h TEACHER’S EDITION Speed: 901.4 km/h. Direction: 20. Construction An airplane is heading north at 900 km/h. However, a 50 km/h wind is slightly west of north. Figure is blowing from the east. Use a ruler and a protractor to make a scale drawing of these approximate. vectors. Measure to find the approximate resultant velocity, both speed and direction (measured from north). Exercise 21 Students may use the Construction In Exercises 21–24, use the given segments and angles to construct each diagonals in a different order and figure. Use either patty paper or a compass and a straightedge. The small letter above have a figure with different labels. each segment represents the length of the segment. 21. x E y

R 1x z S 2 1x 2

Q 21. Construct rhombus SQRE with SR y and QE x. 22. possible answers: 22. Construct kite FLYR given R R F, L, and FL x. Y Y 23. Given bases LP with length z F L F F L x L and EN with length y, nonparallel side LN with length x, and L, construct trapezoid PENL. 23. 24. Given F, FR x, and YD z, construct two trapezoids FRYD that are not NEy congruent to each other.

x

Lz P 24. possible answers: R x F Y

R z x F Y D

z

D

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● ● ● ● | CHAPTER 5 REVIEW CHAPTER 5 REVIEW CHAPTER 5 REVIEW CHAPTER 5 REVIEW CHAPTER 5 ᮣ Take Another Look 1. This activity extends to 25. Three regular polygons meet at point B.Only 26. Find x. 12 cm general polygons the proof for four sides of the third polygon are visible. quadrilaterals in Lesson 5.1. How many sides does this polygon have? 20 sides Sample answer: There are n 2 x triangles and for each triangle the sum of angle measures is 180°, so the total is 180°(n 2). B The proof, with adaptations in 48 cm particular cases, holds for concave polygons. 2. 27. Developing Proof Prove the conjecture: The diagonals of a rhombus bisect the angles. It’s not possible for a “rect- angle” on a sphere to have four Given: Rhombus DENI, with diagonal DN I N right angles, because the sum of Show: Diagonal DN bisects D and N 4 3 the angles of any quadrilateral on Flowchart Proof a sphere is greater than 360°. The DI 2 proof from activity 1 extends to DE ? 2 1 the sphere, on which the sum of ? 1 DENI is Definition of rhombus D E angle measures of each triangle is a rhombus NI DEN DIN 2, 4 more than 180°. Sample answer: 3 5 6 7 ? Given NE ? ? ? 1 ? DN bisects On a sphere, the sum of the angle 3 ? IDE and INE Definition of ? ? SSS measures of an n-sided polygon rhombus ? ? Definition of is greater than 180°(n 2). 4 CPCTC angle bisector DN ? DN ? Same segment

28. Developing Proof Use a labeled diagram to prove the Parallelogram Opposite Sides Conjecture: The opposite sides of a parallelogram are congruent. TAKE ANOTHER LOOK ᮣ 1. Developing Proof Draw several polygons that have four or more sides. In each, draw all the

diagonals from one vertex. Explain how the EDITION TEACHER’S Polygon Sum Conjecture follows logically from the Triangle Sum Conjecture. Does the Polygon Sum Conjecture apply to concave polygons?

2. A triangle on a sphere can have three right angles. Can you find a “rectangle” with four right angles on a sphere? Investigate the Polygon Sum Conjecture on a sphere. Explain how it is related to the Triangle Sum Conjecture on a sphere. Be sure to test your The small, precise polygons in the painting, Boy With Birds (1953, oil on canvas), by American artist David C. Driskell conjecture on polygons with the smallest and (b 1931), give it a look of stained glass. largest possible angle measures.

28. possible answer: 2 AD CB 4 Є1 Є3 8 AB CD A B 4 Definition of AIA CPCTC 3 parallelogram

1 ABCD is a parallelogram 5 BD BD 7 ᭝ABD ᭝CDB 1 2 Given Same segment ASA D C 3 6 Є2 Є4 9 Given: Parallelogram ABCD AB CD AD CB Definition of AIA CPCTC Show: AB CD and parallelogram AD CB

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3. Arranging the angles about a EW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPT point is one way of finding their sum. Because this arrangement forms a complete circle about the 3. Developing Proof Draw a polygon and one set of its exterior angles. Label the exterior point, the sum of the angle angles. Cut out the exterior angles and arrange them all about a point. Explain how measures is 360°. this activity demonstrates the Exterior Angle Sum Conjecture. 4. In Lesson 5.2, students may 4. Is the Exterior Angle Sum Conjecture also true for concave polygons? Are the kite have seen the generality of the conjectures also true for darts (concave kites)? Choose your tools and investigate. Exterior Angle Sum Conjecture 5. Investigate exterior angle sums for polygons on a sphere. Be sure to test polygons by imagining walking around the with the smallest and largest angle measures. figure. In effect, the measure of an exterior angle on a concave side is 180° minus the measure of the interior angle and thus is Assessing What You’ve Learned negative. Other definitions of GIVING A PRESENTATION exterior angle are possible, so results may vary. Giving a presentation is a powerful way to demonstrate your understanding of a topic. Presentation skills are also among the most useful skills you can develop All the kite conjectures also hold in preparation for almost any career. The more practice you can get in school, for darts (if the diagonal joining the better. the vertices is extended to inter- Choose a topic to present to your class. There are a number of things you can do to

TEACHER’S EDITION sect the diagonal lying outside make your presentation go smoothly. the figure). ᮣ Work with a group. Make sure your group presentation involves all group 5. Because the sum of the angle members so that it’s clear everyone contributed equally. measures at a vertex is still 360°, ᮣ Choose a topic that will be interesting to your audience. ᮣ the same reasoning that applies Prepare thoroughly. Make an outline of important points you plan to cover. Prepare visual aids—like posters, models, handouts, and overhead in a plane shows that the exterior transparencies—ahead of time. Rehearse your presentation. angle sum is less than 360°. ᮣ Communicate clearly. Speak up loud and clear, and show your enthusiasm about your topic.

ASSESSING ORGANIZE YOUR NOTEBOOK Your conjecture list should be growing fast! Review your notebook to be sure it’s complete and well organized. Write a one-page chapter summary. Use either form of the Chapter Test, or substitute one or two WRITE IN YOUR JOURNAL Write an imaginary dialogue between your teacher and Constructive Assessment items a parent or guardian about your performance and progress in geometry. for part of the test. If your students are struggling with UPDATE YOUR PORTFOLIO Choose a piece that represents your best work from this proof, give them flowchart proofs chapter to add to your portfolio. Explain what it is and why you chose it. to complete. If your students are PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a comfortable with proof, ask them teacher observes, carry out one of the investigations from this chapter. Explain what to do an entire proof. you’re doing at each step, including how you arrived at the conjecture.

FACILITATING SELF-ASSESSMENT Discuss norms for group presen- tation and for equal participation by team members. Give students pointers on public speaking: • Look at everyone in your audience. Don’t hide Choose topics, or give students a behind your poster or projector. • Don’t try to memorize everything. Prepare an choice. Investigations, projects, outline and visual aids. explorations, Take Another Look • Most important, speak clearly and show your enthusiasm about your topic! activities, and even some • Rehearse your presentation in front of your group Improving Your Skills puzzles or family or with a tape recorder or video can all be good topics. camera.

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