Building Polygons. Lolo\~C~ }Jlcd"" ~ You can use polystrips and fasteners like these:

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1 )~~j"~llimlll~1 ~1~il~11~11! 11I1II11111I Lolo\~C~ }Jlcd"" ~ S~ es ~ "Des~V1s Building Polygons In the last two investigations, you explored the relationship between the number of sides of a polygon and the measure of its interior angles. Now you will turn your attention to the sides of a polygon. How do the side lengths ofa polygon affect its shape? You can use polystrips and fasteners like these: to build polygons with given side lengths and study their properties. 70 Shapes and Designs

$_ TEKS! lks 6{l1jC Select all appropriate problemosoiving strategy inducting looking fora pattern. 6(13} Make conjectures from patterns.$

2 _ TEKS! lks 6{l1jC Select all appropriate problemosoiving strategy inducting looking fora pattern. 6(13} Make conjectures from patterns. Bridges, towers, and other structures contain many triangles in their design. Why are triangles used so frequently in construction? Building Triangles Make a triangle using the steps below. Sketch and label your results. Step 1 Roll three number cubes and record the sum. Do this two more times, so that you have three sums. Step 2 Using polystrips, try to make a triangle with the three sums as side lengths. If you can build one triangle, try to build a different triangle with the same side lengths. Repeat Steps 1 and 2 to make several triangles.. 1. List each set of side lengths that did make a triangle. 2. List each set of side lengths that did not make a triangle. 3. What pattern do you see in each set that explains why some sets of numbers make a triangle and some do not? 4. Use your pattern to find two new sets of side lengths that will make a triangle. Then find two new sets of side lengths that will not make a triangle. B. Can you make two different triangles from the same three side lengths? c. Why do you think triangles are so useful in construction? Homework starts on page 76. Investigation 4 Building Polygons 71

3 _ TEKS! TKS 6(11)( Select an appropriate problem-solving strategy induding looking for a pattern. G(U) Make conjectures from patterns.. You need four side lengths to make a. Will any four side lengths work? Can you make more than one from four side lengths? Building Quadrilaterals. 1. Use polystrips to build s with each of the following sets of numbers as side lengths. Try to build two or more different s using the same set of side lengths. For: Linkage Strips ctivity Visit: PHSchool.com 6,10,15,15 3,5,10,20 Web Code: amd ,8,10,10 12,20,6,9 Sketch and label your results to share with your classmates. Record any observations you make. 2. Choose your own sets of four numbers and try to build s with those numbers as side lengths. B. Use your observations from Question. 1. Is it possible to make a using any four side lengths? If not, how can you tell whether you can make a from four side lengths? 2. Can you make two or more different s from the same four side lengths? 3. What combinations of side lengths are needed to build rectangles? Squares? Parallelograms? C. 1. Use four polystrips to build a. Press on the sides or corners of your. What happens? 2. Use another polystrip to add a diagonal connecting a pair of opposite vertices. Now, press on the sides or corners of the. What happens? Explain. D. 1. Describe the similarities and differences between what you learned about building triangles in Problem 4.1 and building s in this problem. 2. Explain why triangles are used in building structures more often than s. Homework starts on page Shapes and Designs

4 ',;; i"~'."'."......::";"'-" Mechanical engineers use the fact that s are not rigid to design linkages. Below is an example of a linkage. Frame One of the sides is fixed. It is the frame. The two sides attached to the frame are the cranks. One of the cranks is the driver and the other the follower. The fourth side is called the coupler. Quadrilateral linkages are used in windshield wipers, automobile jacks, reclining lawn chairs, and handcars. In 1883, the German mathematician Franz Grashof suggested an interesting principle for linkages: If the sum of the lengths of the shortest and longest sides is less than or equal to the sum of the lengths of the.. remaining two sides, then the shortest side can rotate 360. ~ ~:~ Investigation 4 Building Polygons 73

5 _ TEKS I TKS 6(11)( Select an appropriate problem-solving strategy including looking for a pattern. 6(13) Make conjectures from patterns. The Quadrilateral Game will help you explore the properties of s. The game is played by two teams. To play, you need two number cubes, a game grid, a geoboard, and a rubber band. 74 Shapes and Designs

Properties of Quadrilaterals. Play the Quadrilateral Game, Keep a record of interesting strategies and difficult situations. Make notes about when you do not receive a point during a turn.

6 Properties of Quadrilaterals. Play the Quadrilateral Game, Keep a record of interesting strategies and difficult situations. Make notes about when you do not receive a point during a turn. Why did you not need to move any corners on those turns? B. Write two new descriptions of s that you could include in the game grid. For: Quadrilateral Game ctivity Visit: PHSchool.com Web Code: amd-3403 _ Homework starts on page 76. Quadrilateral Game Grid ~ ".'~._. ~.,,, >:~,, - dd 1 point to d '\ t I: "q"ua-drii"ateral 1. quadnlateral I d. rectangle that. qu?thr~ era.. with exactly one with one pair of. your score an I ' ' that IS a square f k' t IS not a square b~1 0 pair of parallel ~. opposite side s Ip your urn i o use ang 1es I,:.;> y,> S bt act 2 i I ~ / f I with two pairs q,uadrilateral f i pom s rom h' f ' with all four. 'th f r our score and t at IS not a I 0 consecutive angles the., WI our mes that is a ~. same size. " equa " ~kip your turn I rectangle. angles th~t are, 1 of symmetry rectangle ~~= ~~~ I with exactly one ", ", with no, I. f t quadnlaterall: quadnlateral reflection or Iwith four right I Skip a turn pairof: with exactly one' with two 45 0 consecu Ive '" I rotation angles.' Sl 'd e I eng th s nght angle angles ~mm~ry... that are equal h'*'~'<r'~d'" '31,. ' ', quadnlateral r. Iquad,,'ate,a' 'th 'f with exactly one t dd 2...nts I. quadnlateral.,l quadnlateral. with no angles WI ~ne palr,~ pair of opposite 0JO~! score: with no sides..,with exactly two equa OppOSIelan s Ip your :,. equal I I angles that are ~ t 1 parallel ',nght angles ang es equa I' urn.,. I quadd'atera' I I -.~"-' -~~~-~: : '.~ T-o/"' K ~ft W" quad"'ate,al Iquad,,'ate,al. h d ' d., t '.. quad,,'ate,a' Subtract 1. h b h.. h '. Wit a lagona qua n a era. h 1800 Wit ot pairs. Wit two pairs Ith t d' 'd 't th' Wit point from of adjacent side' of equal, ~ t Iv~es I h at I~ a.', rotation your score and, In 0 0 r om us ': lengths equal opposite. angles 'd f I h symmetry skip your turn I en lea s ap~s Iquad,lIate,a' dd 3 points quadnlateral, 'th ctl parallelogram with no side. WI ~xa ytone: that is not a to your score with two pairs ang e grea er and skip your of opposite side lengths equal ' than 1800 rectangle l: turn lengths equal Investigation 4 Building Polygons 75

7 pplications Follow these directions for Exercises 1-4. If possible, build a triangle with the given set of side lengths. Sketch your triangle. Tell whether your triangle is the only one that is possible. Explain. If a triangle is not possible, explain why. 1. Side lengths of 5, 5, and 3 2. Side lengths of 8, 8, and 8 3. Side lengths of 7,8, and Side lengths of 5,6, and Which set(s) of side lengths from Exercises 1-4 can make each of the following shapes? a. an equilateral triangle b. an isosceles triangle c. a scalene triangle d. a triangle with at least two angles of the same measure For Exercises 6 and 7, draw the polygons described to help you answer the questions. 6. What must be true of the side lengths in order to build a triangle with three angles measuring 60? What kind of triangle is this? 7. What must be true of the side lengths in order to build a triangle with only two angles the same size? What kind of triangle is this? 8. Giraldo and Maria are building a tent. They have two 3-foot poles. In addition, they have a 5-foot pole, a 6-foot pole, and a 7-foot pole. They want to make a triangular-shaped doorframe for the tent using both 3-foot poles and one of the other poles. Which of the other poles could be used to form the base of the door? 76 Shapes and Designs

8 .. Follow these directions for Exercises If possible, build a with the given set of side lengths. Sketch your. Tell whether your is the only one that is possible. Explain. If a is not possible, explain why. 9. Side lengths of 5, 5, 8, and Side lengths of 5, 5, 6, and Side lengths of 8, 8, 8, and Side lengths of 4, 3, 5, and Which set(s) of side lengths from Exercises 9-12 can make each of the following shapes? a. a square b. a with all angles the same size c. a parallelogram d. a that is not a parallelogram... ~.... Which..., ".-_,<,---:.,,-: 14. with four equal sides is called a fthgmlj~,.x -_"_X >' ;"":":;:':i~'" set(s) of side lengths from Exercises 9-12 can make a rhombus? 15. with at least one pair of parallel sides is called a lllilil'" Which set(s) of side lengths from Exercises 9-12 can make a trapezoid? For Exercises 16 and 17, draw the polygons described to help you answer the questions. 16. What must be true of the side lengths of a polygon to build a square? 17. What must be true of the side lengths of a polygon to build a rectangle that is not a square? 18. Li Mei builds a with sides that are each five inches long. To help stabilize the, she wants to insert a ten-inch diagonal. Is this possible? Explain. 19. You are playing the Quadrilateral Game. The shape currently on the geoboard is a square. Your team rolls the number cubes and gets the description " parallelogram that is not a rectangle." What is the minimum number of vertices your team needs to move to form a shape meeting this description? line...~ PlISchool.com For: Help with Exercise 18 Web Code: ame-3418 Investigation 4 Building Polygons

9 20. You are playing the Quadrilateral Game. The shape currently on the geoboard is a non-rectangular parallelogram. Your team rolls the number cubes and gets the description " with two obtuse angles." What is the minimum number of vertices your team needs to move to create a shape meeting this description? Connections 21. Multiple Choice Which one of the following shaded regions is not a representation of ~?. 80m c. D. 78 Shapes and Designs

22. a. How are all three s below alike? b. How does each differ from the other two? 1 2 3 23. In this parallelogram, find the measure of each numbered angle. 2 (Got::!

10 22. a. How are all three s below alike? b. How does each differ from the other two? In this parallelogram, find the measure of each numbered angle. 2 (Got::!c~~com For: Multiple-Choice Skills Practice Web Code: ame Think about your polystrip experiments with triangles and s. What explanations can you now give for the common use of triangular shapes in structures like bridges and antenna towers for radio and television? Investigation 4 Building Polygons

11 Multiple Choice For Questions 25-28, choose the symmetry or symmetries of each shape. 25. rhombus (four equal sides) F. rotational G. reflectional H. both F and G J. none 26. regular pentagon. rotational B. reflectional c. both and B D. none 27. square F. rotational G. reflectional H. both F and G J. none 28. a parallelogram that is not a rhombus or a rectangle. rotational B. reflectional C. both and B D. none Extensions 29. In the triangle, a line has been drawn through vertex, parallel to line segment Be of the triangle. 8.c Ie a. What is the sum of the measures of angles 1,2, and 3? b. Explain why angle 1 has the same measure as angle 4 and why angle 3 has the same measure as angle 5. c. How can you use the results of parts (a) and (b) to show that the angle sum of a triangle is 180? 80 Shapes and Designs

12 30. In parts (a)-(c), explore pentagons by using polystrips or by making sketches. a. If you choose five numbers as side lengths, can you always build a pentagon? Explain. b. Can you make two or more different pentagons from the same side lengths? 31. Refer to the Did You Know? after Problem 4.2. a. Make a model that illustrates Grashof's principle using polystrips or paper fasteners and cardboard strips. Describe the motion of your model. b. How can your model be used to make a stirring mechanism? windshield wiper? 32. Build the figure below from polystrips. Note that the vertical sides are all the same length, the distance from B to C equals the distance from E to D, and the distance from B to C is twice the distance from tob. a. Experiment with holding various strips fixed and moving the other strips. In each case, tell which strips you held fixed, and describe the motion of the other strips. b. Fix a strip between points F and B and then try to move strip CD. What happens? Explain why this occurs. Investigation 4 Building Polygons

13 In this investigation, you experimented with building polygons by choosing lengths for the sides and then connecting those sides to make a polygon. These questions will help you summarize what you have learned. Think about your answers to these questions. Discuss your ideas with other students and your teacher. Then write a summary of your findings in your notebook. 1. a. How can you tell whether three line segments will form a triangle? b. If it is possible to build one triangle, is it also possible to build a different triangle with the same three segments? Explain. 2. a. How can you tell whether four line segments will form a? b. If it is possible to build one, is it also possible to build a different with the same four segments? Explain. 3. Explain why triangles are useful in building structures. What's Next? What information about shapes can you add to your Shapes and Designs project? 82 Shapes and Designs

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17 Name Date Class _ Labsheet Shapes and Designs The Quadrilateral Game that is a square dd 1 point to your score and skip your turn rectangle that is not a square with two obtuse angles with exactly one pair of parallel sides with one pair of opposite side lengths equal Subtract 2 points from your score and skip your turn that is not a rectangle with two pairs of consecutive angles that are equal with all four angles the same size with four lines of symmetry that is a rectangle with no reflection or rotation symmetry with four right angles Skip a turn with exactly one pair of consecutive side lengths that are equal with exactly one right angle with two 45 0 angles -d Q) c: ~ ~.t!2 -= 0> 00=; «""E =.~ c: ~ 0... c: 0 l!! ~ 0> '" c:.:e ~ ::::I c. -= J c: 0 ""8 ::::I -c... c: 0 l!! Q) 0... with no angles equal with one pair of equal opposite angles with exactly one pair of opposite angles that are equal with both with two with a pairs of pairs of diagonal that adjacent equal divides it into side lengths opposite two identical equal angles shapes with one with exactly diagonal that with no one angle is a line of side lengths greater than symmetry equal dd 2 points to your score and skip your turn that is a rhombus parallelogram that is not a rectangle with no sides parallel with rotation symmetry dd 3 points to your score and skip your turn with exactly two right angles Subtract 1 point from your score and skip your turn with two pairs of opposite side lengths equal 111

For Exercises 6 and 7, draw the polygons described to help you answer the questions.

For Exercises 6 and 7, draw the polygons described to help you answer the questions. Applications Follow these directions for Exercises 1 4. If possible, build a triangle with the given set of side lengths. Sketch your triangle. Tell whether your triangle is the only one that is possible.