NAME DATE PERIOD Lesson Reading Guide Get Ready for the Lesson Read the introduction at the top of page 316 in your textbook. Write your answers below. 1. Predict the number of triangles and the sum of the angle measures in a polygon with 8 sides. 2. Write an algebraic expression that could represent the number of triangles in an n-sided polygon. Then write an expression to represent the sum of the angle measures in an n-sided polygon. Read the Lesson 3. How many triangles would be in a 12-sided polygon? 4. Why do you think that you need to subtract 2 from the number of sides? 5. What do you call the angles that lie inside a polygon? Remember What You Learned 6. The outside walls of a sports stadium create a giant regular 60-sided figure. Write an equation to find the number of triangles inside the figure. Then write and solve an equation to find the sum of the interior angles of the figure. Lesson 6 3 Chapter 6 19 Course 3
NAME DATE PERIOD Study Guide and Intervention An interior angle is an angle with sides that are adjacent sides of the polygon. A regular polygon is a polygon whose sides and angles are congruent. Example 1 Find the sum of the measures of the interior angles of a tricontagon, which is a 30-sided polygon. S (n 2)180º Write an equation. S (30 2)180º Replace n with 30. Subtract. S (28)180º Multiply. S 5,040º The sum of the measures of the interior angles of a tricontagon is 5,040º. Example 2 The defense department of the United States has its headquarters in a building called the Pentagon because it is shaped like a regular pentagon. What is the measure of an interior angle of a regular pentagon? S (n 2)180º Write an equation. S (5 2)180º Replace n with 5. Subtract. S (3)180º Multiply. S 540º 540º 5 108º Divide by the number of interior angles to find the measure of one angle. The measure of one interior angle of a regular pentagon is 108º. Exercises For Exercises 1 6, find the sum of the measures of the interior angles of the given polygon. 1. nonagon (9-sided) 2. 14-gon 3. 16-gon 4. hendecagon (11-sided) 5. 25-gon 6. 42-gon For Exercises 7 12, find the measure of one interior angle of the given regular polygon. Round to the nearest hundredth if necessary. 7. hexagon 8. 15-gon 9. 22-gon 10. icosagon (20-sided) 11. 38-gon 12. pentacontagon (50-sided) Chapter 6 20 Course 3
NAME DATE PERIOD 6-3 Skills Practice Find the sum of the measures of the interior angles of each polygon. 1. 13-gon 2. 17-gon 3. 18-gon 4. 24-gon 5. 32-gon 6. 35-gon 7. 21-gon 8. 29-gon 9. 54-gon 10. 64-gon 11. 81-gon 12. 150-gon Find the measure of one interior angle of the given regular polygon. Round to the nearest hundredth if necessary. 13. heptagon (7-sided) 14. 26-gon 15. decagon (10-sided) 16. 23-gon 17. 37-gon 18. 51-gon 19. 48-gon 20. 85-gon Lesson 6-3 21. 72-gon 22. 49-gon 23. 66-gon 24. 500-gon Chapter 6 21 Course 3
NAME Practice DATE PERIOD Find the sum of the measures of the interior angles of each polygon. 1. 13-gon 2. 16-gon 3. 17-gon 4. 18-gon 5. 20-gon 6. 25-gon Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. 7. pentagon 8. hexagon 9. 24-gon ALGEBRA For Exercises 10 and 11, determine the angle measures in each polygon. 10. 5 11. 5 1.5 1.5 12. FLOORING A floor is tiled with a pattern consisting of regular octagons and squares as shown. Find the measure of each angle at the circled vertex. Then find the sum of the angles. 13. ART Jose is laying out a pattern for a stained glass window. So far he has placed the 13 regular polygons shown. Find the measure of each angle at the circled vertex. Then find the sum of the angles. 14. REASONING Vanessa s mother made a quilt using a pattern of repeating regular hexagons as shown. Will Vanessa be able to make a similar quilt with a pattern of repeating regular pentagons? Explain your reasoning. Chapter 6 22 Course 3
NAME DATE PERIOD Word Problem Practice For Exercises 1 6, use the formula S (n 2)180 to solve. 1. FLOORING Martha s kitchen floor is made from a tessellation of rows of regular octagons. The space between them is filled with square tiles as shown below. Find the measure of one interior angle in both the octagon and the square tiles. 2. CIRCLES As the number of sides of a regular polygon increase, the polygon gets closer and closer to a true circle. The interior angles of any regular polygon can never actually reach 180º. How many sides would a polygon have whose interior angles are exactly 179º? 3. GEOMETRY A trapezoid has angles that measure 3, 3,, and. What is the measure of x? 4. GEOMETRY An irregular heptagon has angles that measure,, 2, 2, 3, 3, and 4. What is the measure of x? 3 3 5. TILES A bathroom tile consists of regular hexagons surrounded by regular triangles as shown below. Find the measure of one interior angle in both the hexagon and the triangle tiles. x 2x 4x 6. CHALLENGE How many sides does a regular polygon have if the measure of an interior angle is 171º? 3x Lesson 6 3 Chapter 6 23 Course 3
NAME DATE PERIOD Enrichment M.C. Escher Maurits Cornelis Escher (1898-1972) was a Dutch graphic and mathematical artist. Some of his most famous pieces used tessellations, or repeated tiling of one or more shapes. His designs range from artfully simple to extremely intricate. A regular polygon will tessellate a plane if the measure of one of its interior angles is a factor of 360. Other combinations of polygons tessellate if the sum of the measures of the adjoining angles equals 360. The tessellation at the right is made of regular octagons and squares. At any vertex the sum of the measures of the angles is 90 135 135 or 360. 1. Make a list of all regular polygons that will tessellate. 2. Explain why you know there are no other regular polygons that will tessellate. For Steps A E, you will create your own tessellation on a separate piece of paper. Step A Start with a polygon that will tessellate. Trace it, and cut it out. Grid paper or isometric dot paper may help you accurately draw your shape. Step B Cut a portion of the figure on one side, slide it to the opposite side, and tape it on. (This was done three times in the example at right.) Step C Use the modified shape as a tracing template. Trace the template on another sheet of paper. Step D Slide, reflect, and/or rotate the shape so that it fits with your first tracing. Trace the template where it fits with the previous tracing. Repeat the process to cover the page. Step A Step B Step C Step E Color each polygon in the tessellation. Escher often decorated the shapes so that they resembled objects or animals. Chapter 6 24 Course 3