Topic 6 Polygons and Quadrilaterals TOPIC OVERVIEW VOCABULARY 6-1 The Polygon Angle-Sum Theorems English/Spanish Vocabulary Audio Online: 6-2 Properties of Parallelograms English Spanish 6-3 Proving That a Quadrilateral equiangular polygon, p. 249 polígono equiángulo Is a Parallelogram equilateral polygon, p. 249 polígono equilátero 6-4 Properties of Rhombuses, isosceles trapezoid, p. 281 trapecio isósceles Rectangles, and Squares kite, p. 282 cometa 6-5 Conditions for Rhombuses, midsegment of a trapezoid, p. 281 segmento medio de un trapecio Rectangles, and Squares parallelogram, p. 255 paralelogramo rectangle, p. 269 rectángulo 6-6 Trapezoids and Kites regular polygon, p. 249 polígono regular rhombus, p. 269 rombo square, p. 269 cuadrado trapezoid, p. 281 trapecio DIGITAL APPS PRINT and eBook Access Your Homework . ONLINE HOMEWORK You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. HOMEWORK TUTOR APP YOUR DIGITAL Do your homework anywhere! You can access the Practice and RESOURCES Application Exercises, as well as Virtual Nerd tutorials, with this PearsonTEXAS.com Homework Tutor app, available on any mobile device. STUDENT TEXT AND HOMEWORK HELPER Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 246 Topic 6 Polygons and Quadrilaterals The Mystery Sides Have you every looked closely at honeycombs? What shape are they? How do you know? Math Most often the cells in the honeycombs look like hexagons, but they might also look like Act Math Act Act - - circles. Scientists now believe 3 3 that the bees make circular cells that become hexagonal due to the bees’ body heat and natural physical forces. What are some strategies you use to identify shapes? Think about this as you watch the 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . VOCABULARY ONLINE LEARNING ANIMATIONS INTERACTIVE MATH TOOLS You’ll find definitions of math You can also access all of the These interactive math tools terms in both English and stepped-out learning animations give you opportunities to Spanish. All of the terms have that you studied in class. explore in greater depth audio support. key concepts to help build understanding. INTERACTIVE EXPLORATION STUDENT COMPANION VIRTUAL NERD You’ll have access to a robust Refer to your notes and Not sure how to do some of assortment of interactive solutions in your Student the practice exercises? Check explorations, including Companion. Remember that out the Virtual Nerd videos interactive concept explorations, your Student Companion is also for stepped-out, multi-level dynamic activitites, and topic- available as an ACTIVebook instructional support. level exploration activities. accessible on any digital device. PearsonTEXAS.com 247 Technology Lab Exterior Angles of Polygons USE WITH LESSON 6-1 TEKS (5)(A), (1)(E) Use geometry software. Construct a polygon similar to the one at the right. Extend each side as shown. Mark a point on each ray so that you can measure the exterior angles. Use your figure to explore properties of a polygon. • Measure each exterior angle. • Calculate the sum of the measures of the exterior angles. • Manipulate the polygon. Observe the sum of the measures of the exterior angles of the new polygon. hsm11gmse_0601a_t06178 Exercises 1. Write a conjecture about the sum of the measures of the exterior angles (one at each vertex) of a convex polygon. Test your conjecture with another polygon. 2. The figures below show a polygon that is decreasing in size until it finally becomes a point. Describe how you could use this to justify your conjecture in Exercise 1. 1 5 5 1 4 4 5 2 1 4 3 2 3 2 3 3. The figure at the right shows a square that has been copied several times. Notice that you can use the square to completely cover, or tile, a plane, without Ӈ gapshsm11gmse_0601a_t06179 or overlaps. hsm11gmse_0601a_t06180hsm11gmse_0601a_t06181 a. Using geometry software, make several copies of other regular polygons with 3, 5, 6, and 8 sides. Regular polygons have sides of equal length and angles of equal measure. Ӈ b. Which of the polygons you made can tile a plane? c. Measure one exterior angle of each polygon (including the square). d. Write a conjecture about the relationship between the measure of an exterior angle and your ability to tile a plane with the polygon. Test your conjecture with another regular polygon. hsm11gmse_0601a_t06182 248 Technology Lab Exterior Angles of Polygons 6-1 The Polygon Angle-Sum Theorems TEKS FOCUS VOCABULARY TEKS (5)(A) Investigate patterns to make conjectures about • Equiangular polygon – An equiangular geometric relationships, including angles formed by parallel lines polygon is a polygon with all angles cut by a transversal, criteria required for triangle congruence, special congruent. segments of triangles, diagonals of quadrilaterals, interior and • Equilateral polygon – An equilateral polygon exterior angles of polygons, and special segments and angles of circles is a polygon with all sides congruent. choosing from a variety of tools. • Regular polygon – A regular polygon is TEKS (1)(C) Select tools, including real objects, manipulatives, paper and a polygon that is both equilateral and pencil, and technology as appropriate, and techniques, including mental equiangular. math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(E), (1)(F) • Number sense – the understanding of what numbers mean and how they are related ESSENTIAL UNDERSTANDING The sum of the interior angle measures of a polygon depends on the number of sides the polygon has. Key Concept Classifying Polygons Based on Sides and Angles An equilateral polygon An equiangular polygon A regular polygon is is a polygon with all is a polygon with all a polygon that is both sides congruent. angles congruent. equilateral and equiangular. Theorem 6-1 Polygonhsm11gmse_0601_t06300 Angle-Sum Theorem hsm11gmse_0601_t06299 hsm11gmse_0601_t06301 The sum of the measures of the interior angles of an n-gon is (n - 2)180. For a proof of Theorem 6-1, see the Reference section on page 683. Corollary to the Polygon Angle-Sum Theorem (n - 2)180 The measure of each interior angle of a regular n-gon is n . You will prove the Corollary to the Polygon Angle-Sum Theorem in Exercise 16. PearsonTEXAS.com 249 Theorem 6-2 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at 3 2 each vertex, is 360. For the pentagon, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360. 4 1 5 You will prove Theorem 6-2 in Exercise 9. Problem 1 TEKShsm11gmse_0601_t06313.ai Process Standard (1)(C) Investigating Interior Angles of Polygons A Choose from among a variety of tools (such as a ruler, a compass, or geometry software) to investigate the sums of the measures of the interior angles of different polygons. Explain your choice. Geometry software is a good way to identify the measures of the interior angles of polygons. You can quickly make many different polygons and use the software to find the measures of their angles. B Use geometry software to make several triangles, quadrilaterals, pentagons, and hexagons. Then complete the table. Sum of Interior Angle Sum of Interior Angle Polygon Measures Polygon Measures Triangle 1 180 Pentagon 1 540 Triangle 2 180 Pentagon 2 540 Triangle 3 180 Pentagon 3 540 Quadrilateral 1 360 Hexagon 1 720 Quadrilateral 2 360 Hexagon 2 720 Quadrilateral 3 360 Hexagon 3 720 How can recording data in a table C Use the data in the table in part B to make a conjecture about the sum of the help you make a measures of the interior angles of a polygon. conjecture? Recording data in a table Notice that the numbers in the table are all multiples of 180. Look at the patterns: is an organized way to present and analyze Triangle 1 # 180 = 180 Pentagon 3 # 180 = 540 information. You can look Quadrilateral 2 180 360 Hexagon 4 180 720 for patterns in the data # = # = and make a conjecture. Conjecture: If you subtract 2 from the number of sides and multiply by 180, you will get the sum of the measures of the interior angles of any polygon. 250 Lesson 6-1 The Polygon Angle-Sum Theorems Problem 2 Finding a Polygon Angle Sum How many sides does What is the sum of the interior angle measures of a heptagon? a heptagon have? A heptagon has 7 sides. Sum = (n - 2)180 Polygon Angle-Sum Theorem = (7 - 2)180 Substitute 7 for n. = 5 # 180 Simplify. = 900 The sum of the interior angle measures of a heptagon is 900. Problem 3 Using the Polygon Angle-Sum Theorem STEM Biology The common housefly, Musca domestica, has eyes How does the word that consist of approximately 4000 facets. Each facet is regular help you a regular hexagon. What is the measure of each answer the question? interior angle in one hexagonal facet? The word regular tells you that each angle has the same measure. (n - 2)180 Measure of an angle = n Corollary to the Polygon Angle-Sum Theorem (6 - 2)180 = 6 Substitute 6 for n. 4 # 180 = 6 Simplify. = 120 The measure of each interior angle in one hexagonal facet is 120. PearsonTEXAS.com 251 Problem 4 Using the Polygon Angle-Sum Theorem What is mjY in pentagon TODAY? How does the T diagram help you? Use the Polygon Angle-Sum Theorem for n = 5. 110Њ You know the number of O sides and four of the five m∠T + m∠O + m∠D + m∠A + m∠Y = (5 - 2)180 Y angle measures.