Investigation 2 Designing with Angles 33 Two angles are supplementary if the sum of their measures is 180º. 150° 30° If they are adjacent, they form a straight line. 60° 60° 30° 30° 150° 30° Two angles are complementary if the sum of their measures is 90º. 60° 30° If they are adjacent, they form a right angle. 60° 60° 30° 30° 150° 30° INITIAL CHALLENGE In the following figure, lines L1 , L2, and L3 are parallel lines that are intersected (cut) by another line. Look at the angles formed where the single line cuts across each parallel line. m n po q t r s w x z y L1 L2 L3 • What do you notice about the angles? What do you wonder? • Measure the angles. • Describe any relationships you notice in the angle measures. Do they agree with your prediction? • Are any angles vertical, supplementary, adjacent, or complementary? Is so, name some. 2.3 SAMPLE
34 Shapes and Designs WHAT IF . . . ? Situation A. Testing Your Conjectures In the following two figures, two lines are intersected by another line. Are any of the relationships you found in the Initial Challenge true for each set of lines? Explain. Figure 1 Line 1 and line 2 are parallel lines. They are intersected by line 3. L1 L2 L3 a b c g d f e 150° Figure 2 Line 1 and line 2 are not parallel lines. They are intersected by line 3. m n p o q r L t s 2 L1 L3 Situation B. Jamal’s Claim About Parallelograms Is Jamal’s claim correct? Explain. Jamal’s Claim I added a line to Figure 1 in Situation A. I put in L4 parallel to L3 . I was trying to find the angle measures. I think that › the four lines create a parallelogram; › the opposite angles of a parallelogram have the same angle measures; and › the sum of the interior angles of the parallelogram is 360º. L4 L1 L3 L2 m l n o k j h i g d f e a b c 150° Situation C. Hank Again: Angles and the Algebra Connection Hank added cards to his Algebra Connections game to show what he knows about supplementary and vertical angles. Write an equation that will help you find a value of x. Note: x is a rational number. 2.3 SAMPLE
Investigation 2 Designing with Angles 35 NOW WHAT DO YOU KNOW? Describe what you know about angles formed by intersecting lines. Include parallel lines and the angles formed by a line that intersects the parallel lines in your description. 2.3 Card 4 x is a rational number. What is x? L1 L2 x 5 + x Card 5 x is a rational number. What is x? L1 L2 x 5x Card 6 x is a rational number. What is x? Lines L1 amd L2 are parallel. L1 L2 L3 x 3x “Engineers at Washington State University, supported by a grant from the U.S. National Science Foundation, used honey to make components for computer systems that mimic neurons and synapses of the human brain, known as neuromorphic computers. Honey—a natural antibacterial, preservative, and sweetener—contains environmentally safe ingredients that can be used in neuromorphic computer system components called memristors.” These neuromorphic computer systems are inspired by the structure and function of the human brain. And because honey never spoils and is easily removed, these neuromorphic chips are environmentally friendly. It is amazing how much useful information humans have gained by studying bees. Source: National Science Foundation. “Engineers Use Honey to Make Brain-like Computer Chips.” Updated May 3, 2022. https://new.nsf.gov/news/engineers-use-honey-make-brain-computer-chips. Did You Know? SAMPLE
36 Shapes and Designs MR Mathematical Reflection In this unit, we are investigating some general properties of geometric shapes (figures), including angles and polygons, and using this information to design shapes and solve problems. At the end of this investigation, ask yourself, What do you know about geometric shapes? SAMPLE
Investigation 2 Designing with Angles 37 INVESTIGATION 2 APPLICATIONS 1. An angle whose measure is less than 90° is called an acute angle. An angle whose measure is greater than 90° and less than 180° is called an obtuse angle. Which of these angles are acute, which are obtuse, and which are right? APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) a. d. b. e. c. f. 2. Snowboarders use angle measures to describe their flips and spins. Explain what a snowboarder would mean by each statement. a. I did a 720. b. I did a 540. c. I did a 180. 3. Write an equation, and find the measure of the angle labeled x, without measuring. a. x 30° c. x 27° b. x 125° d. 35° x SAMPLE
38 Shapes and Designs 4. Decide whether each angle is closest to 30°, 60°, 90°, 120°, 150°, 180°, 270°, or 360° without measuring. Explain your reasoning. a. c. b. d. e. g. f. h. i. Classify each angle in ACE 4a–h as right, acute, or obtuse. 5. A full turn is 360°. Find the fraction of a turn or number of turns for the given measurement. a. 90° b. 270° c. 720° d. How many degrees is ___ 25 360 of a full turn? ACE SAMPLE
Investigation 2 Designing with Angles 39 6. You can think of a right angle as one-quarter of a complete rotation. a. How many degrees is one-third of a quarter-rotation? b. How many degrees is two times a quarter-rotation? c. How many degrees is two and one-third times a quarter-rotation? 7. A test question asked students to choose the larger angle. In one class, most students chose angle 2. Do you agree? Why or why not? angle 1 angle 2 8. Estimate the measure of each angle. Then check your answers with an angle ruler or a protractor. a. c. b. d. e. 9. Draw an angle for each measure. Include an arc indicating the turn. a. 45° b. 25° c. 180° d. 200° ACE SAMPLE
40 Shapes and Designs 10. Without measuring, decide whether the angles in each pair have the same measure. If they do not, tell which angle has the greater measure. Then, find the measure of the angles with an angle ruler or protractor to check your work. a. 1 2 b. 1 2 c. 1 2 11. For each pair of angles, estimate the measure of each angle. Then, check your estimates by measuring with an angle ruler or a protractor. a. A V B c. Q P R b. K J L d. X Y Z ACE SAMPLE
Investigation 2 Designing with Angles 41 12. Ms. Cosgrove asked her students to estimate the measure of the angle shown. Deshawn thought 150° would be a good estimate. Sofia said it should be 210°. Who is closer to the exact measurement? Explain. 13. Use the diagram of the protractor below. Angle 1 and angle 2 are called adjacent angles because they have a common vertex and a common side. Find the angle measures. 90 1 2 3 4 180 0 a. ∠1 b. ∠1 + ∠2 c. ∠1 + ∠2 + ∠3 d. ∠2 e. ∠2 + ∠3 f. ∠3 g. the complement of ∠1 h. the supplement of ∠1 i. the complement of ∠3 j. the supplement of ∠1 combined with ∠2 ACE SAMPLE
42 Shapes and Designs 14. Write an equation using the angle measures shown. Then, find the measures of ∠A and ∠B. 2x = B A = 3x 15. Use what you know about supplemental and vertical angles to find the measure of each angle. Show the equation you could use to find the value of y. 3y y 16. Lines 1 and 2 are perpendicular; they meet at 90° angles. w x y z t v L1 L3 L2 a. Name all the pairs of complementary angles. b. If the measure of ∠t is twice the measure of ∠v, what are the measures of all of the angles? Explain. ACE SAMPLE
Investigation 2 Designing with Angles 43 17. In the diagram, lines L1 and L2 are parallel. L1 L2 T 35° g d f e a b c a. Find the degree measures of angles labeled a–g. b. Name the pairs of opposite or vertical angles in the figure. c. Name three pairs of supplementary angles in the figure. 18. L1 and L2 are not parallel lines. They are intersected by L3 . Explain what you know about the measure of each angle. 105° L2 L1 L3 g e f h a b d 19. In the diagram, L1 and L2 are parallel lines. Without measuring, find the measure of each angle, and explain how you know. L1 L2 L3 50° b a c e f d g ACE SAMPLE
44 Shapes and Designs CONNECTIONS 20. The number 360 has many factors. This may be why it was chosen for the number of degrees in a full turn. a. List all of the factors of 360. b. Find the prime factorization of 360. 21. At the start of each hour, the minute hand of a clock points straight up at 12. In ACE 21a–f determine the angle between the minute hand at the start of an hour and the minute hand after the given amount of time passes. For each situation, sketch the angle, and indicate the rotation of the minute hand. a. 15 minutes b. 30 minutes c. 20 minutes d. one hour e. 5 minutes f. one and one-half hours 22. Write each fraction as a decimal. a. _ 2 5 b. __3 4 c. __3 10 d. __1 4 e. __7 10 f. __7 20 g. __ 4 5 h. __7 8 i. __15 20 j. _ 3 5 ACE SAMPLE
Investigation 2 Designing with Angles 45 23. Find the value of n. a. _1 2 = ___n 360 b. __1 10 = __ 36 n c. __1 n = ___ 40 360 d. __n 3 = ___ 120 360 24. The members of the skateboard club were practicing. They wanted to work on increasing the angle rotation each member did when standing and turning their skateboards. They recorded their rotations: 0° 10° 20° 30° 40° 50° 80° 100° a. What is the median of the turns that the club members can make on their boards? b. What is the mean of the turns that the club members can make on their boards? Three weeks later, they each tried a stand and turn again. They recorded their rotations again: 0° 10° 20° 30° 40° 50° 80° 100° c. How does their new data compare? 25. Find a value of n that makes the equation true compare to the previous data. a. 3 × n = 24 b. 5 + n = 60 c. 144 = 12 × n d. 160 ÷ 8 = n e. 2 + n = 50 ACE SAMPLE
46 Shapes and Designs 26. Multiple Choice Which of the following shaded regions is not a representation of __4 12 ? ACE A. B. 0180 20 10 0 170 10 160 20 150 30 140 40 50 130 60 120 70 110 80 100 90 100 80 110 70 120 60 130 50 140 40 150 30 160 170 180 350 340 330 320 310 300 290 280 260 250 240 230 220 210 200 190 190 200 210 220 230 240 250 260 280 270 290 300 310 320 330 340 350 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 27. For each equation or inequality, find the value of x. a. 3x = 180 b. 3x > 180 c. 3x = 90 d. 3x > 90 e. x + x = 180 f. x + x = 90 g. 72 + x = 180 h. x + 15 = 90 C. D. SAMPLE
Investigation 2 Designing with Angles 47 ACE EXTENSIONS 28. Design a new polar coordinate grid for Four in a Row in Problem 1.2. Play your game with a friend or family member. What ideas did you use to design your new grid? Explain. How does playing on your grid compare to playing on the original grids? 29. List all polygons in the Shapes Set that have the following characteristics: a. only right-angle corners b. only obtuse-angle corners c. only acute-angle corners d. at least one angle of each type—acute, right, and obtuse 30. A compass is a tool used in wilderness navigation. On a compass, north is assigned the direction label 0°, east is 90°, south is 180°, and west is 270°. Directions that are between those labels are assigned degree labels such as NE (northeast) at 45°, for example. NW (315°) NE (45°) W (270°) E (90°) SW (225°) SE (135°) N (0°) S (180°) a. What degree measures would you expect for the direction southsouthwest? For north-northwest? b. A ship at sea is on a heading of 300°. Approximately what direction is it traveling? 31. Major airports label runways with the numbers by the compass heading. For example, a plane on runway 15 is on a compass heading of 150°. A plane on runway 9 is on a compass heading of 90°. (Refer to information in Exercise 30 on the design of a compass.) a. What is the runway number of a plane that is taking off on a heading due west? On a heading due east? b. What is the compass heading of a plane landing on runway 6? On runway 12? W E N S SAMPLE
48 Shapes and Designs c. Each actual runway has two direction labels. The label depends on the direction in which a landing or taking-off plane is headed. How are those labels related to each other? 32. Soledad extended the sides of the green parallelogram below. She looked at the vertical angles and supplementary angles. Soledad claims that the opposite angles in a parallelogram will always have the same measures. Do you agree? Explain. L3 L4 L1 L2 33. In the parallelogram, find the measure of each numbered angle. 4 1 117° 3 2 5 34. In the triangle ABC, a line has been drawn through vertex A, parallel to side BC. 4 5 1 2 3 B C A 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 ACE 34a and 34b to show that the angle sum of a triangle is 180°? ACE SAMPLE
49 In Investigation 1, we explored the role of side lengths on shape. We have seen that polygons with the same number of sides can have different shapes. There is also an important relationship between the number of sides and the angle sum of any polygon. Honeybees live in colonies. They build nests called hives. A typical hive might be home for as many as 60,000 bees. Bees are small insects, but packing a hive with that many bees and the honey they make is tricky. Why do bees create their honey storage tubes in the shape of hexagonal prisms? Why not some other shape? In this investigation, we explore the role of angles in determining the unique shape of a polygon. We will use the information about shapes to explore how these shapes occur in many natural objects like the combs made by bees to store their honey. Designing Polygons: The Angle Connection INVESTIGATION 3 SAMPLE
50 Shapes and Designs PROBLEM 3.1 Back to the Bees: Tiling a Plane Experiment The diagram here shows a pattern that uses regular hexagons to cover a flat surface without any gaps or overlaps. Notice that three angles fit together exactly around any point in the beehive pattern. These patterns are called tilings or tessellations of the surface. INITIAL CHALLENGE To answer the question of why the bees choose a hexagon, Ms. Bennet’s class explored other shapes to see if they would tile a plane. They started with the following shapes and recorded their information in a chart. T 1 3 2 B 4 3 1 2 K 3 2 4 1 S 3 2 4 1 D 2 4 3 5 6 1 E 1 2 3 45 6 7 SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 51 Make a Prediction • Which shapes will tile a plane? Conduct the Experiment Equipment › several copies of the Shape Set Directions (individually or with a partner or small group) › Use the copies of each shape to see if it will tile. › Record the following information in a table. Tiling a Plane Experiment Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Triangle T 1 3 2 Quadrilateral B 4 3 1 2 Quadrilateral S 3 2 4 1 Quadrilateral K 3 2 4 1 3.1 SAMPLE
52 Shapes and Designs Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Hexagon D 2 4 3 5 6 1 Heptagon E 1 2 3 45 6 7 Analyze the Data • Which polygons can be used to cover a flat surface without gaps or overlap? Explain why they fit tightly with no gaps around the points where they meet. WHAT IF . . . ? Situation A. Angle Measure and Tiling Maisie makes a claim. Is she correct? Explain why or why not. Maisie’s Claim I was thinking about the bees using regular hexagons. I think that the size of an angle has something to do with regular polygons that tile. I measured the angles of a regular hexagon. Each angle is 120º. So three regular hexagons tile around a point since the sum of the vertex angles is 360º. I looked at other regular polygons. I know that the vertex angles in regular polygons are equal. 3.1 SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 53 Situation B. Tiling with More Than One Polygon Luke wondered if he could use more than one polygon to tile a flat surface. Is this possible? Explain why or why not. Regular Polygon Number of Sides Measure of One Angle in Degrees Number of Shapes Needed to Tile Triangle 3 60 6 (60 ) = 360 Square 4 90 4 (90 ) = 360 Pentagon 5 Does not tile Hexagon 6 120 3 (120 ) = 360 I don’t need to try regular polygons with more than six sides. If I put more than two together, the sum of the vertex angles is greater than 360º, which means they will not tile. So the hexagon is the last regular polygon that will tile. NOW WHAT DO YOU KNOW? Which polygons can be used to tile a surface? Explain why they tile. Give some examples of how tiling polygons can be useful. 3.1 SAMPLE
54 Shapes and Designs PROBLEM 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment Mr. Pulaski’s class thought that knowing the angle measure of polygons would be useful information to answer questions about shape. Is there a way to predict the angle measure of regular polygons? How can we find the sum of the angles of a polygon? Why is this important? How does the size of an angle affect the shape of the polygon? Devon thought there might be a pattern relating the angle sum to the number of sides that would work for any polygon. INITIAL CHALLENGE Devon used the shapes from Problem 3.1. He suggested starting with a triangle. Devon’s Strategy The Sum of the Angles of a Triangle I began by drawing irregular triangles. I tore the corners off the triangle and then rearranged them. The angles form a straight line or a straight angle. The sum of the angles of a triangle is 180º. 1 1 2 3 3 2 Make a Prediction • Does Devon’s strategy make sense? Will it work on other triangles? Other polygons? SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 55 Conduct the Experiment 3.2 Equipment › a copy of the Shape Set from Problem 3.1 Directions (individually or with a partner or small group) › Tear off each angle, and rearrange them around a point. › Record the following information in a table. Relating Angle Measures to Number of Sides of Polygons Experiment Polygon Number of Sides (n) Angle Sum (S) Triangle Square Pentagon Hexagon Heptagon Octagon Nonagon Decagon n-gon Analyze the Data • What patterns do you observe in the table? Do your observations agree with your predictions? • What is the sum of the angles of an octagon? Nonagon? Decagon? N-gon? • What is the angle measure of regular polygons? Octagon? Nonagon? Decagon? N-gon? SAMPLE
56 Shapes and Designs 3.2 WHAT IF . . . ? Situation A. Trevor’s, Casey’s, and Maria’s Strategies Below are students’ strategies for finding the sum of the angles of a polygon. Are they correct? Explain why. Trevor’s Strategy The Sum of the Angles of a Polygon I used Devon’s results for a triangle. I divided polygons into smaller triangles by drawing diagonals from one vertex. Casey’s Strategy I used Devon’s results for triangles. I divided polygons into triangles by drawing line segments from a point within the polygon. Maria’s Strategy I found a different way to find the sum of the angles of a triangle. I used parallel lines L1 and L2, which are intersected by two line segments, AB and AC. L1 L2 2 1 4 B C A 5 3 I used Devon’s thinking and angle relationships to find the sum of the angles of other polygons. Situation B: Zane Checks His Conjecture Zane wondered about his group’s conjecture about the sum of the angles of a polygon in the Initial Challenge. What is the sum of the angles in these shapes? Do the measures match what you found about the sum of angles in the Initial Challenge? SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 57 Shape 1 Shape 2 Shape 3 Shape 3 3.2 Situation C. Right Triangles and Algebra Triangle ABC is a right triangle. 40° x A B C Neveah’s and Amy’s Strategies Neveah and Amy use different equations to find the measure of the angle whose measure is x°. Amy wrote: 90 + 40 + x = 180. Neveah wrote 40 + x = 90. 1. Are these equations equivalent? How is each student thinking about this problem? 2. Neveah claims that her equation shows that the two acute angles in a right triangle are always complementary. Is she correct? Why? NOW WHAT DO YOU KNOW? What is the relationship between the angle sum S of a polygon with n sides and the number of sides? How can you find the measure of an angle in a regular polygon with n sides? SAMPLE
58 Shapes and Designs 3.2 A golf ball manufacturer developed a hexagon pattern for the cover of golf balls. The manufacturer claims it is the first design to cover 100% of the surface area of a ball. This pattern of mostly hexagons almost eliminates flat spots that interfere with performance. The new design produces a longer, better flight for the golf ball. Tessellations can also be formed with irregular polygons and other shapes. All triangles and quadrilaterals tile a plane. It was proved in 1963 that there are exactly three types of convex hexagons that tile a plane. And no convex heptagon, octagon, or any other n-gon tiles a plane. But full classification of the pentagons is still an open area of research. As of 1985, there were only 15 pentagons that tile. (See picture on the left.) A new pentagon tiling was found in 2015. (See picture on the right.) In 2023, mathematicians finally discovered an elusive “einstein” tile, a single shape that forms a special tiling of a plane. A 13-sided shape called “the hat” forms a pattern that never repeats. This is the only example of an “einstein.” David Smith, a nonprofessional mathematician, headed a team of mathematicians who made this discovery. Source: Conover, Emily, “Mathematicians Have Finally Discovered an Elusive ‘Einstein’ Tile,” ScienceNews, March 24, 2023. Accessed online. Did You Know? SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 59 3.3 PROBLEM The Ins and Outs of Polygons: Using Supplementary Angles In previous investigations, we used estimation as well as tools to measure interior angles of polygons. By extending a side of a convex polygon, we can make an exterior angle that lies outside the polygon. Exterior angle Interior angle Notice that the measures of the exterior angle and its adjacent interior angle add up to 180°. Arturo claims that the exterior and interior angles are supplementary. Is this true? Why? The figures that follow show two ways to form exterior angles. We can extend the sides as we move in either direction around the polygon. Exterior angles as you move counterclockwise Exterior angles as you move clockwise Measuring exterior angles provides useful information about the interior angles of a polygon. SAMPLE
60 Shapes and Designs 3.3 INITIAL CHALLENGE Members of the Columbia Triathlon Club train by bicycling around the following polygonal path shown. GPS Route Menu E A D C Bike Path B Pentagonal Path They start at vertex A and go on to vertices B, C, D, and E. Then they return to A and start another lap. At each vertex, the cyclists have to make a left turn through an exterior angle of the polygon. • What is the sum of the left-turn exterior angles that the cyclists make on one full lap? Explain how you can determine an answer without measuring. Then, measure the exterior angles to check your thinking. More Polygonal Paths • Draw several other polygons, including a triangle, quadrilateral, and hexagon. • Is the angle you turn at each point as you walk around these polygons the same as that for a pentagon? Explain. • Find the sum of the turn angles if you cycle around each figure and return to your starting point. SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 61 WHAT IF . . . ? Situation A. More Angle Patterns Students were discussing some patterns they noticed in the pentagonal training track. 1. How does their thinking compare to the strategies you and your classmates used in the Initial Challenge? 2. Will the claims work for other polygons? Explain. 180° 180° 180° 180° 180° D C A B E Amy’s Claim › There are five supplementary or straight angles in the pentagonal diagram. › The total angle measure for the supplementary angles T is T = 5 • 180°. › If we subtract the interior angle sum (540°) of a pentagon, we get 360°. › This is the sum of the five exterior angles of the pentagon. 5 • 180° − 540°= 360° D C A B E Neveah’s Claim › The total angle measure T of the interior and exterior angles of any polygon is T = n • 180, where n is the number of angles. › So the sum S of the interior angles of any polygon is that total minus the exterior angles. S = (n • 180°) − 360° A B C 180° 180° 180° Nic’s Claim I think exterior angles are like “walking around” a polygon. I use this idea to show that the sum of the interior angles of a triangle is 180°. › The sum of the supplementary angles is 3 • 180° = 540°. › I subtract the sum of the exterior angles, which is 360°. › This leaves 180° for the sum of the interior angles of the triangle. 3.3 SAMPLE
62 Shapes and Designs Situation B. Quadrilaterals and Algebra The following quadrilateral is formed by the intersection of four lines. The measures of some angles are given. Note that none of the lines are parallel to each other. 60° 70° 75° x y z w 1. Find the measures of the angles represented by w, x, y, and z. Write equations to help show your reasoning. 2. Are any of the labeled angles vertical? Supplementary? Complementary? Explain why. 3.3 NOW WHAT DO YOU KNOW? What do you know about the exterior angles of a polygon? How might this knowledge be useful? These insects’ brains are tiny, but they know numbers and communication. Here are five smart things honeybees can do. Use Numbers Scientists, headed by Dr. Adrian Dyer trained 14 bees to perform 100 training exercises. The bees were correct about 70% of the time. The bees were good students. Solve Problems Researchers at Queen Mary University of London showed that bees could perform a task to gain a reward and then the bees trained other bees to do the same task and gain a reward. The task was to move a small ball to a specified location. The bees were also able to shorten the path. Did You Know? SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 63 3.3 Give Directions The decoding of the bees’ “waggle dance” was done by Karl von Frisch who won a Nobel Prize for it. A bee indicates the location of food sources to other bees by performing a special dance using the location of food in relation to the sun. Understand Zero Researchers at RMIT University in Melbourne showed that bees were the first insect species to conceive of zero as a number. How they used this skill is unclear. Make Decisions Bees collaborate in making decisions. For example, 300 older bees from a colony of 1,000 bees search for a suitable new home. They use the “waggle dance” to communicate possible locations. The repetitions of the dance provide information on the quality of the site. Share Decisions Bees have a sense of democracy. When looking for a new home for, say 10,000 bees, 300 older bees form a “senate” and fly off looking for options. Using the waggle dance, they communicate possible locations. The number of dance repetitions tells the quality of the site. Source: Tucker, Ian. “Five Smart Things Honeybees Can Do.” The Guardian, February 10, 2019. SAMPLE
64 Shapes and Designs PROBLEM 3.4 Designing Polygons In this problem, we will use our knowledge of properties of shapes to design triangles that meet a set of conditions. Recall the following: The notation AB¯ means “line segment AB.” B A C The notation ∠A means “angle A.” B A C INITIAL CHALLENGE The drawing shows a triangle with measures of its angles and sides. Suppose you want to text a friend to give directions for making a drawing that is an exact copy of the figure with the shortest message possible. B A C 3 cm 3.6 cm 4 cm 74° 60° 46° • Which of these messages give enough information to draw a triangle that has the same shape and size as triangle ABC? SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 65 WHAT IF . . . ? Situation A. Designing Polygons For each design, do the following: 1. Decide if the criteria can be used to design the polygon. 2. If it is not possible, explain why. 3. If so, make a sketch of the polygon. Label the angle and side measurements. Is the polygon unique? Explain. Design 1 Triangle ∠A = 40º AB¯ = 6 cm AC¯ = 6 cm Design 2 Triangle ∠A = 50º ∠B = 100º ∠C = 30º Design 3 Triangle AB¯ = 5 cm BC¯ = 2 cm AC¯ = 10 cm Design 4 Triangle AB¯ = 12 cm ∠A = 60º ∠B = 40º Design 5 Right Triangle The angle measures are x, 2x, and 3x. Design 6 Right Triangle The angle measures are x, 3x, and 3x. Design 7 Right Triangle The angle measures are x, __1 2x , and __1 2x . Design 8 Quadrilateral ∠A = 90º ∠B = 60º ∠C = 60º ∠D = 100º 3.4 Message 1 BC¯ = 4 cm ∠B = 60° AB¯ = 3 cm Message 2 AB¯ = 3 cm BC¯ = 4 cm ∠C = 46° Message 3 ∠B = 60° BC¯ = 4 cm ∠C = 46° Message 4 ∠B = 60° ∠A = 74° ∠C = 46° Message 5 ∠B = 60° ∠C = 46° AC¯ = 3.6 cm SAMPLE
66 Shapes and Designs Design 9 Hexagon ∠A = 120º ∠B = 120º ∠C = 120º ∠D = 120º ∠E = 120º ∠F = 120º Design 10 Polygon A polygon that requires exactly two diagonals to make it rigid Design 11 Intersecting Lines Two intersecting lines with one pair of vertical angles with a measure of 60º and the other pair with a measure of 100º Design 12 Quadrilateral A quadrilateral with exactly one pair of parallel sides NOW WHAT DO YOU KNOW? What information about a triangle allows you to draw a triangle? Draw a unique triangle? Explain why this true. What information about a quadrilateral allows you to draw a quadrilateral? Explain why this true. Note: Shapes that have the same size and shape are called congruent shapes. We will learn more about congruent shapes and investigate what happens to a shape if is stretched, shrunk, or distorted in the Stretching and Shrinking and Flip, Spin, Slide, and Stretch units. In Looking for Pythagoras, we will explore an important geometric and algebraic property of right triangles and the role right triangles plays in our daily lives. 3.4 SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 67 Mathematical Reflection MR In this unit, we are investigating some general properties of geometric shapes (figures), including angles and polygons, and using this information to design shapes and solve problems. At the end of this investigation, ask yourself: What do you know about geometric shapes? SAMPLE
68 Shapes and Designs INVESTIGATION 3 APPLICATIONS 1. Multiple Choice Which of the following combinations will not tile a flat surface? A. regular heptagons B. equilateral triangles C. regular hexagons D. squares 2. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. regular octagons C. regular pentagons and regular hexagons D. regular hexagons and equilateral triangles 3. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. squares and regular octagons C. regular pentagons and regular hexagons D. regular hexagons and squares 4. A right triangle has one right angle and two acute angles. Without measuring the angles, what is the sum of the measures of the two acute angles? Explain your reasoning. APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 69 5. Without measuring, find the measure of the angle labeled x in each regular polygon. a. x b. x 6. Write an equation, and find the measure of each angle labeled x. a. 90° 30° x c. 93° 135° 70° x b. 45° 45° x d. 60° 60° x x e. 37° 120° x g. This figure is a parallelogram. 70° x f. This figure is a regular hexagon. 120° x h. This figure is a trapezoid. 67° x 7. The following figure is a regular dodecagon. It has 12 sides. a. What is the sum of the measures of the angles of this polygon? b. What is the measure of each angle? c. Can copies of this polygon be used to tile a flat surface? Explain. ACE SAMPLE
70 Shapes and Designs 8. Kele claims that the angle sum of a polygon he has drawn is 1660°. Can he be correct? Explain. 9. Find the measure of each internal angle and the sum of the internal angles for the following concave polygons. a. b. c. 10. A figure is called a regular polygon if all sides are the same length and all angles are equal. List the members of the Shapes Set that are regular polygons. 11. Suppose in-line skaters make one complete lap around a park shaped like the quadrilateral below. GPS Route Menu c b a d 40° 76° 140° 104° Hardware Coffee Bagels Bank What is the sum of the angles through which they turn? 12. Suppose in-line skaters complete one lap around a park that has the shape of a regular pentagon. a. What is the sum of the angles through which they turn? b. How many degrees will the skaters turn if they go once around a regular hexagon? A regular octagon? A regular polygon with n sides? Explain. ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 71 13. A class was asked what convinced them that the sum of exterior angles in any polygon is 360°. Here are three different points of view. "We were convinced when we drew a bunch of different figures and used my angle ruler to measure the exterior angles. They all came out close to 360°." "We were convinced when we thought about walking around the figure and realized that we made one complete turn or 360°." "We used the results about sums of interior angles and the fact that the measure of each interior angle plus its adjacent exterior angle is 180° to deduce the formula using algebra." What are the pros and cons of each argument? 14. Find the measures of the angles represented by a, b, c, and d. Explain your reasoning. 90° 60° 75° a b c d 15. Are any of the labeled angles vertical? Supplementary? Complementary? Explain why. 90° 60° 75° a b c d ACE SAMPLE
72 Shapes and Designs 16. If possible, draw the triangle described. Explain if the triangle is unique or if many triangles can fit the given information. a. A triangle with side = 2 in., side = 1 in., and ∠A = 75°. b. A triangle with ∠A = 75° and ∠C = 75°. c. A triangle with angles 45° and 60° and one side of length 2 in. d. A triangle with side KL = 1 in., side LM = 1.5 in., ∠L = 135°, and side KM = 1 in. e. Triangle with all sides of length 1.5 in and all angles of 60°. 17. Draw the polygons described. a. A trapezoid PQRS. ∠P = 45°. ∠Q = 45°. Side = 1 in. Side = 2 in. b. A parallelogram ABCD with two sides of length 2 in., two sides of length 1 in., and angles of 60° and 120° c. A quadrilateral EFGH. Side = 1 in. Side = 2 in. Side = 3 in. Side = 6 in. 18. Which of these descriptions of a triangle ABC are directions that can be followed to draw exactly one shape? a. AB¯ = 2.5 in., AC¯ = 2 in., ∠B = 40° b. AB¯ = 2.5 in., AC¯ = 1 in., ∠A = 40° c. AB¯ = 2.5 in., ∠B = 60°, ∠A = 40° d. AB¯ = 2.5 in., ∠B = 60°, ∠A = 130° 19. In the parallelogram, find the measure of each numbered angle. 4 1 117° 3 2 5 ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 73 20. Which of the following statements are true? Be able to justify your answers. a. All squares are rectangles. b. No squares are rhombuses. c. All rectangles are parallelograms. d. Some rectangles are squares. e. Some rectangles are trapezoids. f. No trapezoids are parallelograms. g. Every quadrilateral is a parallelogram, a trapezoid, a rectangle, a rhombus, or a square. 21. Suppose you want to build a triangle with three angles measuring 60°. a. What do you think must be true of the side lengths? b. What kind of triangle is this? c. Would this be a unique triangle? CONNECTIONS 22. Arun writes the equation d = 6t to represent the distance in miles, d, that riders could travel in t hours at a speed of 6 miles per hour. Make a table that shows the distance traveled every hour, up to 5 hours, if riders travel at this constant speed. Time and Distance Time (h) Distance (mi) ACE SAMPLE
74 Shapes and Designs 23. Use the equations to fill in the quantities in a copy of the table. a. m = 100 − k k 1 2 5 10 20 m 50 c. d = 3.5t t 1 2 5 10 20 d 140 24. The product of two numbers is 20. Find the value of n. a. n • 2 _1 2 = 20 b. 1 _1 4 • n = 20 c. n • 3 _1 3 = 20 25. The coordinate grid below shows four polygons. 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 11 12 y x 4 3 2 1 a. Give the coordinates of all vertices of each polygon. b. Use the coordinates to find the lengths of as many sides as you can. c. Describe as precisely as possible each type of triangle or quadrilateral shown. 26. Find the area of each of the following figures. a. b. 8 cm 6 cm c. ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 75 27. Find the area and perimeter of each figure. a. 5 cm 5 cm 9 cm 10 cm b. 10 cm 4 cm 2 cm 4 cm 3 cm 6 cm 5 cm c. 9 in. 1 2 9 in. 1 2 8 in. 8 in. 9 in. d. 10 cm 7 cm 81 cm 2 8 cm 7 10 e. 13 cm 13 cm 12 cm 10 cm ACE SAMPLE
76 Shapes and Designs 28. Multiple Choice How many feet are in one yard? A. 12 feet B. 3 feet C. 9 feet D. 27 feet Multiple Choice How many feet are in one inch? A. 12 feet B. 3 feet C. 9 feet D. __1 12 foot Multiple Choice How many feet are in one mile? A. 5,280 ft. B. 3 ft. C. 2,300 ft. D. 100 ft. 29. Find the area of each polygon. a. = 1 square meter b. c. ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 77 30. Multiple Choice Figure QSTV is a rectangle. The lengths QR and QV are equal. What is the measure of angle x? Q R S V T x A. 20° B. 45° C. 90° D. 120° 31. Find the volume and surface area of each rectangular prism. a. 12 cm 12 cm 12 cm b. 12 in. 3 in. 1 2 7 in. 32. Write an equation to represent each question. a. Zuri charges $12 per hour for babysitting in her neighborhood. What equation relates her pay for a job to the number of hours she works? b. A gasoline service station offers 20 cents off the regular price per gallon every Tuesday. What equation relates the discounted price to the regular price on that day? c. What equation shows how the perimeter of a square is related to the length of a side of the square? d. A middle school wants to have its students see a movie at a local theater. The total cost of the theater and movie rental is $1,500. What equation shows how the cost per student depends on the number of students who attend? ACE SAMPLE
78 Shapes and Designs 33. Draw a polygon with the given properties (if possible). Decide if the polygon is unique. If not, design a different second polygon with the same properties. a. a triangle with a height of 5 cm and a base of 10 cm b. a triangle with a base of 6 cm and an area of 48 cm c. a triangle with an area of 12 square centimeters d. a parallelogram with an area of 24 square centimeters e. a parallelogram with a height of 4 cm and a base of 8 cm 34. Draw the rectangle described. If there is more than one (or no) shape that you can draw, explain how you know that. perimeter = 24 cm and side of 8 cm 35. Multiple Choice A triangle has a base of 4 units and an area of 72 square units. Which of the following is true? A. These properties do not make a triangle. B. These properties make a unique triangle. C. There are at least two different triangles with these properties. D. The height of the triangle is 18 units. 36. Multiple Choice Which of the following could not be the dimensions of a parallelogram with an area of 18 square units? A. base = 18 units, height = 1 unit B. base = 9 units, height = 3 units C. base = 6 units, height = 3 units D. base = 2 units, height = 9 units 37. Determine whether the measures in each pair are the same. If not, tell which measure is greater. Explain your reasoning. a. 1 square yard or 1 square foot b. 5 feet or 60 inches c. 12 meters or 120 centimeters d. 12 yards or 120 feet e. 50 centimeters of 500 millimeters f. 1 square meter or 1 square yard ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 79 EXTENSIONS 38. Copy and complete the table. Sort the quadrilaterals from the Shapes Set into groups by name and description. Sides and Angles from the Shapes Set Sides and Angles of Quadrilaterals Name Examples in the Shapes Set All sides are the same length. All sides are the same length, and all angles are right angles. All angles are right angles. Opposite sides are parallel. Only one pair of opposite sides are parallel. 39. Multiple Choice Which equation describes the relationship in the table? n 0 1 2 3 4 5 6 c 10 20 30 40 50 60 70 A. C = 10n B. C = 10 + n C. C = 10 D. C = 10 + 10n ACE SAMPLE
80 Shapes and Designs 40. The table begun here shows a pattern for calculating the measures of interior angles in regular polygons with even numbers of sides. Regular Polygons Number of Sides Measure of Interior Angle 4 _1 2 of 180° 6 _ 2 3 of 180° 8 __3 4 of 180° 10 12 a. What entry would give the angle measures for decagons and dodecagons? Are those entries correct? How do you know? b. Is there a similar pattern for regular polygons with odd numbers of sides? If so, what is the pattern? 41. Below are a quadrilateral and a pentagon with the diagonals drawn from all of the vertices. a. How many diagonals does the quadrilateral have? How many diagonals does the pentagon have? b. Find the total number of diagonals for a hexagon and for a heptagon. c. Copy the table below, and record your results from ACE 41a and 41b. Number of Sides 4 5 6 7 8 9 10 11 12 Number of Diagonals d. Look for a pattern relating the number of sides and the number of diagonals. Complete the table. e. Write a rule for finding the number of diagonals for a polygon with n sides. ACE SAMPLE
Investigation 3 Designing Polygons: The Angle Connection 81 42. The following drawing shows a quadrilateral with measures of all angles and sides. 2.3 cm 3 cm 3.7 cm 4 cm A D B C 110° 75° 60° 115° Suppose you wanted to text a friend giving directions for drawing an exact copy of it. For each of the following short messages, tell whether it gives enough information to draw a quadrilateral that has the same size and shape as ABCD. a. AB¯ = 3 cm, BC¯ = 4 cm, CD¯ = 2.3 cm b. AB¯ = 3 cm, ∠B = 60°, BC¯ = 4 cm, ∠C =115°, ∠A = 110° c. AB¯ = 3 cm, ∠B = 60°, BC¯ = 4 cm, ∠C =115°, CD¯ = 2.3 cm 43. In ACE 43a–d write the shortest possible message that tells how to draw each quadrilateral so that it will have the same size and shape as those below. a. c. b. d. e. What is the minimum information about a quadrilateral that will allow you to draw an exact copy? ACE SAMPLE
82 adjacent angles Two angles in a plane that share a common vertex and common side but do not overlap are adjacent angles. In the parallelogram below, a and b are adjacent angles. Other pairs of adjacent angles are a and c, b and d, and c and d. ángulos adyacentes Dos ángulos en un plano que comparten un lado común y el vértice común, pero no se superponen son ángulos adyacentes. En el paralelogramo a continuación, a y b son ángulos adyacentes. Otros pares de ángulos adyacentes son a y c, b y d, y c y d. a b c d angle The figure formed by two rays or line segments that have a common vertex. Angles are measured in degrees. The sides of an angle are rays that have the vertex as a starting point. Each of the three angles below is formed by the joining of two rays. The angle at point A on the triangle below is identified as angle BAC or ∠BAC. ángulo Figura que forman dos rayos o segmentos que se juntan en un vértice. Los ángulos se miden en grados. Los lados de un ángulo son rayos que tienen el vértice como punto de partida. Cada uno de los tres ángulos está formado por la unión de dos rayos. El ángulo del punto A del triángulo representado a continuación se identifica como el ángulo BAC o ∠BAC. B A C ENGLISH/SPANISH GLOSSARY SAMPLE