DG4CS_898_C.qxd 11/14/06 11:21 AM Page 77 CHAPTER 5 Discovering and Proving Polygon Properties Overview polygon is equiangular, each interior angle has 180(n Ϫ 2) measure ᎏᎏ degrees, or 180 Ϫ ᎏ360ᎏ degrees. In this chapter, students discover properties of n n polygons. In Lessons 5.1 and 5.2, they discover Properties of the various quadrilaterals can be the sum of the angle measures in a polygon and seen from their symmetry. A kite is symmetric the sum of the measures of a set of exterior angles about the diagonal through its vertex 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 trapezoid the vertex angles and the other diagonal, that the properties in Lesson 5.3 leads students to seven two diagonals are perpendicular, and that the conjectures about these quadrilaterals. In nonvertex angles are congruent. Lesson 5.4, students discover two properties of An isosceles trapezoid also has reflectional the midsegment of a triangle and two properties symmetry, but over the line 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,rhombuses,rectangles,and squares. congruent, as are the diagonals. Between these lessons, Using Your Algebra Skills 5 reviews linear equations. In Lesson 5.7, In contrast, a parallelogram has 2-fold rotational students use paragraph and flowchart proofs to symmetry about the point 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 triangles to polygons in general. Then it focuses on rectangle, adjacent sides are congruent.” That’s special kinds of quadrilaterals: trapezoids, kites, true in some rectangles. But the statement is false, and parallelograms. 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 quadrilateral,”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, square,”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 Ϫ 2) degrees, and the sum of the measures rectangle is a dart.” of their exterior angles is 360°. Consequently, if a CHAPTER 5 INTERLEAF 257A DISCOVERING GEOMETRY COURSE SAMPLER 77 DG4CS_898_C.qxd 11/14/06 11:21 AM Page 78 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 TEACHER’S 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 78 DISCOVERING GEOMETRY COURSE SAMPLER DG4CS_898_C.qxd 11/14/06 11:21 AM Page 79 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)] CHAPTER 5 Discovering and Proving Polygon Properties 257 DISCOVERING GEOMETRY COURSE SAMPLER 79 DG4CS_898_C.qxd 11/14/06 11:21 AM Page 80 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 pentagons, 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 hexagons, try to think TEACHER’S EDITION TEACHER’S calculators of different ways you could draw a hexagon. 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.