Patterns in Geometry. Polygons. Investigation 1 UNIT. Explore. Vocabulary. Think & Discuss

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1 UNIT K Patterns in Geometry In this lesson, you will work with two-dimensional geometric figures. You will classify polygons and find angle measures. Explore Inv 1 Polygons 172 How many squares are in this design? (Hint: The answer is more than 16.) Inv 2 Angles 176 Inv 3 Classify Polygons 180 Inv 4 Triangle Sides 184 Investigation 1 Polygons Vocabulary polygon vertex Polygons are flat, two-dimensional geometric figures that have these characteristics. They are made of line segments. Each segment touches exactly two other segments, one at each of its endpoints. These shapes are polygons. These shapes are not polygons. Think & Discuss Look at the shapes above that are not polygons. Explain why each of these shapes does not fit the definition of a polygon. See Unit K Patterns in Geometry

2 Polygons can be classified according to the number of sides they have. You have probably heard many of these names. Name Sides Examples Triangle 3 Real-World Link In Greek, poly means many and gon means angle. Polygon names refer to the number of angles. For example, pentagon means five angles. Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Investigation 1 173

3 Most polygons with more than ten sides have no special name. A polygon with 11 sides is described as an 11-gon, a polygon with 12 sides is a 12-gon, and so on. Each of the polygons below is a 17-gon. Each corner of a polygon, where two sides meet, is called a vertex. The plural of vertex is vertices. Labeling vertices with capital letters makes it easy to refer to a polygon by name. Math Link A diagonal is a segment that connects two vertices of a polygon but is not a side of the polygon. In quadrilateral ABCD, the diagonal is AC. Example This figure can be seen as two triangles and one quadrilateral. To name one of the polygons in the figure, list its vertices in order as you move around it in either direction. One name for the white triangle is ABC. Other names are possible, including BCA and ACB. One name for the green triangle is ADC. The quadrilateral in the figure could be named quadrilateral ABCD, or BCDA, or DCBA, or DABC. All of these names list the vertices in order as you move around the quadrilateral. The name ACBD is not correct. 174 Unit K Patterns in Geometry

4 Real-World Link Polygons, such as triangles and octagons, are used for traffic signs. Develop & Understand: A You will now search for polygons in given figures. Each figure has a total score that is calculated by adding the following. 3 points for each triangle 4 points for each quadrilateral 5 points for each pentagon 6 points for each hexagon As you work, try to discover a systematic way to find and list all the polygons in a figure. Be careful to give only one name for each polygon. Record your work in a table like this one, which has already been started for Exercise 1. Polygon Names Score Triangle ABC, ADC 6 Quadrilateral Pentagon Hexagon Total Score Now create your own figure that is worth at least 30 points. Label the vertices. List each of the triangles, quadrilaterals, pentagons, and hexagons in your figure. Investigation 1 175

5 Share & Summarize 1. Draw two polygons. Also draw two shapes that are not polygons. Explain why the shapes that are not polygons do not fit the definition of a polygon. 2. In Exercises 1 5, you had to find ways to list all the polygons in a figure without repeating any. Describe one strategy you used. Investigation 2 Angles Vocabulary angle Materials paper polygons or pattern blocks You probably already have a good idea about what an angle is. You may think about an angle as a rotation, or a turn, about a point. Examples include an arm bending at the elbow or hinged boards snapping shut at the start of a movie scene. You may also think about an angle as two sides that meet at a point, like the hands of a clock or the vanes of a windmill. Or you may think of an angle as a wedge, like a piece of cheese or a slice of pizza. 176 Unit K Patterns in Geometry

6 In mathematics, an angle is defined as two rays with the same endpoint. A ray is straight, like a line. It has an endpoint where it starts, and it goes forever in the other direction. Ray 2 Ray 1 Angles can be measured in degrees. Below are some angles with which you may be familiar. The angle at the vertex of a square measures 90. You can think of a 90 angle as a rotation 1 _ 4 of the way around a circle. Ray 2 90 Ray 1 Two rays pointing in opposite directions form a 180 angle. A 180 angle is a rotation 1 _ 2 of the way around a circle. 180 Ray 2 Ray 1 Real-World Link In snowboarding, skateboarding, and other sports, the term 360 is used to mean a full turn. The term 180 is used to refer to a half turn. A 360 angle is a rotation around a complete circle. In a 360 angle, the rays point in the same direction. 360 Ray 1, Ray 2 You can use 90, 180, and 360 angles to help estimate the measures of other angles. For example, the angle below is about a third of a 90 angle, so it has a measure of about 30. Investigation 2 177

7 Think & Discuss Copies of the polygon at right can be arranged to form a star. What is the measure of the angle that is marked in the star? How do you know? Develop & Understand: A 1. You will be given several copies of each polygon below. Your job is to determine the angle measures for each polygon. To find the measures of the angles, you can use 90, 180, and 360 angles as guides, and you can compare the angles of the polygons with one another. Your answers should be a record of each vertex, A Y, and the measure of the angle at that vertex. For many of the polygons, two or more of the angles are identical. So, you only have to find the measure of one of them. You will now use the angles you found in Exercise 1 to help estimate the measures of other angles. 178 Unit K Patterns in Geometry

8 Develop & Understand: B Estimate the measure of each angle. To help make your estimates, you can compare the angles to 90, 180, and 360 angles and to the angles of the polygons in Exercise 1. For each angle, explain how you made your estimate Real-World Link Think about the corners of index cards when estimating angle measures. They form approximate 90 angles Share & Summarize 1. Describe how you can estimate the measure of an angle. 2. Moria said the angles below have the same measure. Stella said Angle 2 is larger than Angle 1. Who is correct? Explain. Angle 1 Angle 2 3. Explain the difference between a 1 _ 4 -rotation and a 1 _ 2 -rotation. Investigation 2 179

9 Investigation 3 Classify Polygons Vocabulary concave polygon line symmetry regular polygon Polygons can be divided into groups according to certain properties. Concave polygons look like they are collapsed or have a dent on one or more sides. Any polygon with an angle measuring more than 180 is concave. The polygons below are concave. Materials set of polygons and category labels large Venn diagram The polygons below are not concave. Such polygons are sometimes called convex polygons. Regular polygons have sides that are all the same length and angles that are all the same size. The polygons below are regular. The polygons below are not regular. Such polygons are sometimes referred to as irregular. A polygon has line symmetry, or reflection symmetry, when you can fold it in half along a line and the two halves match exactly. The folding line is called the line of symmetry. 180 Unit K Patterns in Geometry

10 The polygons below have line symmetry. The lines of symmetry are shown as dashed lines. Notice that three of the polygons have more than one line of symmetry. Real-World Link The United Nations building located in New York City is an example of line symmetry in modern-day architecture. These polygons do not have line symmetry. Think & Discuss Consider the polygons below. This diagram shows how these four polygons can be grouped into the categories concave and not concave. Concave Not Concave Now make a diagram to show how the four polygons can be grouped into the categories line symmetry and not concave. Use a circle to represent each category. Investigation 3 181

11 Develop & Understand: A You will now play a polygon-classification game with your group. Your group will need a set of polygons, category labels, and a large Venn diagram. Here are the polygons used for the game. Math Link A Venn diagram uses circles to represent relationships among sets of objects. Here are the category labels. Regular Concave Triangle Not Regular Not Concave Not Triangle Quadrilateral Pentagon Hexagon Not Quadrilateral Not Pentagon Not Hexagon Line Symmetry No Line Symmetry 1. As a warm-up for the game, put one of the labels Regular, Concave, and Triangle next to each of the circles on the diagram. Work with your group to place each of the polygons in the correct region of the diagram. Record your work. Sketch the three-circle diagram, label each circle, and record the polygons you placed in each region of the diagram. Record just the letters. You do not need to draw the polygons. 182 Unit K Patterns in Geometry

12 2. Now you are ready to play the game. Choose one member of your group to be the leader. Use the following rules. The leader selects three category cards and looks at them without showing them to the other group members. The leader uses the cards to label the regions, placing one card face down next to each circle. The other group members take turns selecting a polygon. The leader places the polygon in the correct region of the diagram. After a player s shape has been placed in the diagram, he or she may guess what the labels are. The first player to guess all three labels correctly wins. At the end of each game, work with your group to place the remaining shapes. Then copy the final diagram. Take turns being the leader until each member of the group has had a chance. 3. Work with your group to create a diagram in which no polygons are placed in an overlapping region, that is, no polygon belongs to more than one category. 4. Work with your group to create a diagram in which all of the polygons are placed either in the overlapping regions or outside the circles, that is, no polygon belongs to just one category. Target Market Share & Summarize 1. Determine what the labels on this diagram must be. Use the category labels on page 182. Label 2 Real-World Link Venn diagrams are named after John Venn ( ) of England, who made them popular. Venn diagrams are used in business to create visual models. Label 3 Label 1 2. Explain why there are no polygons in the overlap of the Label 1 circle and the Label 2 circle. 3. Explain why there are no polygons in the Label 3 circle that are not also in one of the other circles. Investigation 3 183

13 Inquiry Inquiry Investigation 4 Triangle Sides Materials linkage strips and fasteners In many ways, triangles are the simplest polygons. They are the polygons with the fewest sides. Any polygon can be split into triangles. For this reason, learning about triangles can help you understand other polygons as well. In this investigation, you will build triangles from linkage strips. The triangles will look like the one below. The sides of this triangle are 2, 3, and 4 units long. Notice that a unit is the space between two holes. Do you think any three segments can be joined to make a triangle? You will investigate this question. Build the Triangles 1. Copy the table on the next page. Do the following steps for each row. Try to build a triangle with the given side lengths. In the Triangle? column, enter yes if you could make a triangle and no if you could not. If you could make a triangle, try to make a different triangle from the same side lengths. (Hint: For two triangles to be different, they must have different shapes.) In the Different Triangle? column, enter yes if you could make another triangle and no if you could not. 184 Unit K Patterns in Geometry

14 Inquiry Side 1 Side 2 Side 3 Triangle? Different Triangle? 4 units 4 units 4 units 5 units 4 units 3 units 4 units 4 units 2 units 4 units 4 units 1 unit 4 units 3 units 1 unit 4 units 2 units 2 units 3 units 5 units 6 units 3 units 3 units 1 unit 3 units 2 units 2 units 3 units 2 units 1 unit 3 units 1 unit 1 unit Analyze the Results 2. Do you think you could make a triangle with segments 4, 4, and 10 units long? Explain your answer. 3. Do you think you could make a triangle with segments 10, 15, and 16 units long? Explain your answer. What Did You Learn? 4. Describe a rule you can use to determine whether three given segments will make a triangle. Test your rule on a few cases different from those in the table until you are convinced it is correct. 5. Do you think you can make more than one triangle with the same set of side lengths? Explain.

15 On Your Own Exercises Unit K Practice & Apply 1. How many triangles are in this figure? Do not count just the smallest triangles. 2. Look at the figure in Exercise 1. a. Copy the figure. Label each vertex with a capital letter. b. In your figure, find at least one of each of the following polygons. quadrilateral pentagon hexagon Use your vertex labels to name each shape. c. Find the polygon with the maximum number of sides in your figure. Use the vertex labels to name the shape. 3. List all the polygons in the figure below. Compute the figure s score using the following point values. 3 points for each triangle 4 points for each quadrilateral 5 points for each pentagon 6 points for each hexagon Record your work in a table like the one below. Polygon Names Score Triangle Quadrilateral Pentagon Hexagon Total Score 186 Unit K Patterns in Geometry

16 On Your Own Exercises In Exercises 4 7, several identical angles have the same vertex. Find the measure of the marked angle. Explain how you found it A 180 angle is sometimes called a straight angle. Explain why that name makes sense. Real-World Link During a Ferris wheel ride, the wheel makes several complete rotations. 9. You know that a 360 rotation is one complete rotation around a circle. Find the degree measures for each of these rotations. a. half a rotation b. two complete rotations c. 1 1_ 2 rotations d. three complete rotations e. 2 1_ 4 rotations f. five and one-half rotations 10. Draw two angles that each measure more than 90. Explain how you know they measure more than Draw two angles that each measure less than 90. Explain how you know they measure less than 90. On Your Own Exercises 187

17 On Your Own Exercises 12. The diagram shows the result of one round of the game of polygon classification described on page 182. Label 2 Label 3 Label 1 a. Figure out what the labels must be. Use the category labels from the polygon-classification game. b. Where would you place each of these shapes? In Exercises 13 15, draw a polygon that fits the given description, if possible. If it is not possible, say so. 13. a regular polygon with four sides 14. a concave polygon with a line of symmetry 15. a triangle with just one line of symmetry Connect & Extend 16. A diagonal of a polygon is a segment that connects two vertices but is not a side of the polygon. In each polygon below, the dashed segment is one of the diagonals. The number of diagonals you can draw from a vertex of a polygon depends on the number of vertices the polygon has. 188 Unit K Patterns in Geometry

18 On Your Own Exercises Math Link Triangles are the only polygons that are rigid. If you use linkage strips to build a polygon with more than three sides, you can push on the sides or vertices to create an infinite number of different shapes. a. Copy each of these regular polygons. On each polygon, choose a vertex. Draw every possible diagonal from that vertex. b. Copy and complete the table. Polygon Vertices 3 Quadrilateral 5 Hexagon Heptagon 7 Octagon Diagonals from a Vertex c. Describe a rule that connects a polygon s number of vertices to the number of diagonals that can be drawn from each vertex. d. Explain how you know your rule will work for polygons with any number of vertices. e. Challenge Describe a rule for predicting the total number of diagonals you can draw if you know the number of vertices in a polygon. Explain how you found your rule. Add a column to your table to help you organize your thinking. Total Diagonals Look for polygons in your home or school. Describe at least three different polygons that you find. Tell where you found them. On Your Own Exercises 189

19 On Your Own Exercises 18. Find three angles in your home or school with measures equal to 90, three with measures less than 90, and three with measures greater than 90. Describe where you found each angle. 19. Order the angles below from smallest to largest. b a c d 20. Statistics In a survey for the school yearbook, students were asked to name their favorite class. Conor made a circle graph to display the results. He forgot to label the wedges. a. Of the students surveyed, _ 1 liked math best. 3 Which color wedge represents these students? What is the angle measure of that wedge? b. Conor remembers that he used light blue to represent students who like science best. What fraction of the students surveyed chose science as their favorite subject? c. Drama and English tied with _ 1 of the students choosing each. 8 Which wedges represent drama and English? What is the angle measure of each wedge? In Exercises 21 23, describe a rule for creating each shape based on the preceding shape. Then draw the next two shapes Unit K Patterns in Geometry

20 On Your Own Exercises 24. Circle diagrams, like those you used to classify polygons, are sometimes used to solve logic puzzles like this one. Camp Maple Leaf offers two sports, soccer and swimming. Of 30 campers, 24 play soccer, 20 swim, and 4 play no sport at all. How many campers both swim and play soccer? The diagram below includes a circle for each sport. The 4 outside the circles represents the four campers that do not play either sport. Use the diagram to help you solve the logic puzzle. Play Soccer Swim In Your Own Words Explain what each of the following words means. Give at least two facts related to each word. polygon angle triangle On Your Own Exercises 191