-5 What You ll Learn To classify polygons To find the sums of the measures of the interior and exterior angles of polygons... nd Why To find the measure of an angle of a triangle used in packaging, as in Example 5 The Polygon ngle-sum Theorems heck Skills You ll Need GO for Help Lessons -6 and - Find the measure of each angle of quadrilateral. See below.. See below... 65 55 5 0 70 55 87 6 5 ml ml 60; ml ml 0 New Vocabulary polygon convex polygon concave polygon. ml 77; ml 65; ml ; ml 87 equilateral polygon regular polygon equiangular polygon. ml 70; ml 85; ml 5; ml 80-5. Plan Objectives To classify polygons To find the sums of the measures of the interior and exterior angles of polygons Examples Naming Polygons Real-World onnection Finding a Polygon ngle Sum Using the Polygon ngle-sum Theorem 5 Real-World onnection Math ackground lassifying Polygons Real-World onnection Polygons create striking designs on a soccer ball. Quick heck polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no adjacent sides are collinear. E E E polygon Not a polygon; Not a polygon; not a closed figure two sides intersect between endpoints. To name a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction. EXMPLE Special Needs L Have students draw a polygon with an exterior angle at each vertex. Students cut out their exterior angles and tape the vertices together. Students should recognize the sum of the exterior angles is 60. Naming Polygons Name the polygon. Then identify its vertices, sides, and angles. Two names for this polygon are HKMG and MKHG. vertices:, H, K, M, G, sides: H, HK, KM, MG, G, angles: &, &H, &K, &M, &G, & Three polygons are pictured at the right. Name each polygon, its sides, and its angles. See margin, p.59. E Lesson -5 The Polygon ngle-sum Theorems 57 elow Level L Have students draw a polygon on paper and cut out the exterior angles. y placing the angles so that they are adjacent, students can verify Theorem -5. G H M K ecause each interior angle of a regular n-gon measures 80 (n ), one can readily find n the set of all regular n-gons that tesselate a plane. ombinations of regular polygons that tesselate a plane can likewise be found with a bit more work and application of some straightforward number theory. More Math ackground: p. Lesson Planning and Resources See p. E for a list of the resources that support this lesson. ell Ringer Practice heck Skills You ll Need For intervention, direct students to: Using the ngle ddition Postulate Lesson -6: Example Extra Skills, Word Problems, Proof Practice, h. Finding ngle Measures in Triangles Lesson -: Example Extra Skills, Word Problems, Proof Practice, h. learning style: tactile learning style: tactile 57
. Teach Guided Instruction EXMPLE Math Tip Remind students that there are also different ways to name sides and angles in this example. sk: What is another name for HK? KH What is another name for &M? lkmg or lgmk onnection to Science The study of optics teaches that a convex lens causes rays of light to come together and that a concave lens causes rays of light to spread apart. onvex lenses are used in microscopes and telescopes. Eyeglasses may be either convex or concave. Teaching Tip Students may want to know that a seven-sided polygon is called a heptagon. Visual Learners To help students learn the names of polygons, have small groups make charts to be displayed in the classroom with the name of each polygon, its number of sides, and an appropriate figure. Vocabulary Tip diagonal of a polygon is a segment that connects two nonconsecutive vertices. You can classify a polygon by the number of sides it has. The table at the right shows the names of some common polygons. Polygons are classified as convex or concave. R Y convex polygon has no diagonal with points outside T the polygon. iagonal M P concave polygon W K has at least one G Q diagonal with points outside the polygon. S In this textbook, a polygon is convex unless stated otherwise. EXMPLE Real-World onnection Sides 5 6 8 9 0 n Name triangle quadrilateral pentagon hexagon octagon nonagon decagon dodecagon n-gon Tilework The tilework in the photo is a combination of different polygons that form a pleasing pattern. lassify the polygon outlined in red by using the table above. Then classify the polygon as convex or concave. dditional Examples Name the polygon. Then identify its vertices, sides, and angles. Quick heck The polygon outlined in red has 6 sides. Therefore, it is a hexagon. No diagonal of the hexagon contains points outside the hexagon. The hexagon is convex. lassify each polygon by its sides. Identify each as convex or concave. a. b. hexagon; convex octagon; concave E E; vertices:,,,, E; sides:,,, E, E; angles: l, l, l, l, le lassify the polygon below by its sides. Identify it as convex or concave. dodecagon; concave 58 hapter Parallel and Perpendicular Lines dvanced Learners L Have students do the ctivity, discuss whether Theorem - applies to concave n-gons, and justify their reasoning. c. the -pointed star at the center of the tilework pictured above -gon; concave English Language Learners ELL Some students may not know the meaning of adjacent sides in the definition of a polygon. Show how adjacent sides share a vertex just as adjacent angles share a side 58 learning style: verbal learning style: visual
Polygon ngle Sums ctivity: The Sum of Polygon ngle Measures You can use triangles and the Triangle ngle-sum Theorem to find the sum of the measures of the angles of a polygon. Record your data in a table like the one begun below. Guided Instruction ctivity Have students work in pairs to draw polygons and display the sum of their angles using geometry software. Have students manipulate the polygons to see that the sum remains constant. Vocabulary Tip n n-gon is a polygon with n sides, where n can be,, 5, 6, c Polygon Number of Sides Number of Triangles Formed Sketch polygons with, 5, 6, 7, and 8 sides. Sum of the Interior ngle Measures 80 ivide each polygon into triangles by drawing all diagonals that are possible from one vertex. Multiply the number of triangles by 80 to find the sum of the measures of the angles of each polygon. See back of book.. Look for patterns in the table. escribe any that you find.. Inductive Reasoning Write a rule for the sum of the measures of the angles of an n-gon. The sum of the measures of the angles of an n-gon is (n )? 80. y dividing a polygon with n sides into n - triangles, you can show that the sum of the measures of the angles of any polygon is a multiple of 80. lternative Method Students can use inductive reasoning to write the sum of the measures of the angles of an n-gon as 80n - 60. Have them draw a hexagon and segments from an interior point to each vertex. sk: How many triangles are there? 6 What is the sum of the angle measures of all the triangles? 80? 6, or 080 What is the sum of the angle measures of the triangles drawn from an interior point to the vertices of a polygon with n sides? 80n Point out that the sum of the angle measures around the interior point is 60. iscuss why 60 must then be subtracted from 80n. Key oncepts Theorem - Polygon ngle-sum Theorem nline Visit: PHSchool.com Web ode: aue-0775 Quick heck The sum of the measures of the angles of an n-gon is (n - )80. EXMPLE Finding a Polygon ngle Sum Find the sum of the measures of the angles of a 5-gon. For a 5-gon, n = 5. Sum = (n - )80 Polygon ngle-sum Theorem = (5 - )80 Substitute. =? 80 Simplify. = 0 The sum of the measures of the angles of a 5-gon is 0. a. Find the sum of the measures of the angles of a -gon. 980 b. ritical Thinking The sum of the measures of the angles of a given polygon is 70. How can you use Sum = (n - )80 to find the number of sides in the polygon? You can solve the equation (n )80 70. You will sometimes use algebra with the Polygon ngle-sum Theorem to find measures of polygon angles. EXMPLE Error Prevention Some students may think the answer should be 5? 80. Review the activity to correct them. dditional Examples Find the sum of the measures of the angles of a decagon. 0 Lesson -5 The Polygon ngle-sum Theorems 59 Quick heck. E; sides:, E, E; angles: l, le, le E; sides:,, E, E; ': le, l, l, le E; sides:,,, E, E; ': l, l, l, l, le 59
5 EXMPLE Point out that the angles of a polygon can be called interior angles. lso point out that the exterior angles of a regular polygon are congruent. dditional Examples Find m&x in quadrilateral XYZW. Y X 00 W 85 5 Explain how you know that all the angles labeled & in Example 5 have equal measures. Sample: ecause the hexagon is regular, all its angles are congruent. n exterior angle is the supplement of a polygon s angles, and supplements of congruent angles all have the same measure. Z Quick heck EXMPLE Using the Polygon ngle-sum Theorem lgebra Find m&y in pentagon TVYMR at the right. Use the Polygon ngle-sum Theorem for n = 5. m&t + m&v + m&y + m&m + m/r = (5 - )80 90 + 90 + m&y + 90 + 5 = 50 Substitute. m&y + 05 = 50 m&y = 5 R 5 Simplify. Y Subtract 05 from each side. Pentagon E has 5 congruent angles. Find the measure of each angle. 08 You can draw exterior angles at any vertex of a polygon. The figures below show that the sum of the measures of the exterior angles, one at each vertex, is 60. This can be proved as a theorem in a way suggested in Exercise 6. 80 75 59 5 86 50 98 99 0 7 76 80+50+0=60 5+75+99+7=60 86+ 59+98++76=60 Key oncepts Theorem -5 Polygon Exterior ngle-sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 60. For the pentagon, m& + m& + m& + m& + m&5 = 60. M 5 T V Resources aily Notetaking Guide -5 L aily Notetaking Guide -5 dapted Instruction L 5 n equilateral polygon has all sides congruent. n equiangular polygon has all angles congruent. regular polygon is both equilateral and equiangular. EXMPLE Real-World onnection losure If the sum of the interior angles of a polygon equals the sum of the exterior angles, what is the name of the polygon? quadrilateral If each exterior angle of a regular polygon measures 0, how many sides does the polygon have? For: Regular Polygon ctivity Use: Interactive Textbook, -5 Quick heck Packaging The game board at the right has the shape of a regular hexagon. It is packaged in a rectangular box outlined beneath it. The box uses four right triangles made of foam in its four corners. Find m& in each foam triangle. Find the measure of an angle of the hexagon first. regular hexagon has 6 sides and 6 congruent angles. The sum of the measures of the interior angles = (6 - )80, or 70. 70 The measure of one interior angle is 6, or 0. The measure of its adjacent exterior angle, &, is 80-0, or 60. 5 a. Find m& by using the Polygon Exterior ngle-sum Theorem. 60 b. Find m&. Is & an exterior angle? Explain. 0; no, it is not formed by extending one side of the polygon. 60 hapter Parallel and Perpendicular Lines 60
EXERISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving GO Practice by Example for Help Example (page 57) 5. MWFX; sides: MW, W, F, FX, XM; ': lm, lw, l, lf, lx 7. HEPTGN; sides: HE, EP, PT, T, G, GN, NH; ': lh, le, lp,lt, l, lg, ln Example (page 58) Is the figure a polygon? If not, tell why..... yes No; it has no sides. No; it is not a plane figure. Name each polygon by its vertices. Then identify its sides and angles. 5. M 6. K 7. X F W Find a polygon in each photograph. lassify the polygon by its number of sides. Tell whether the polygon is convex or concave. 8. 9. 0. P KLP; sides: K, L, LP, PK; ': lk, l, ll, lp L. No; two sides intersect between endpoints. H E N P T G. Practice ssignment Guide -0, 50-5 -9, 5-56 hallenge 57-6 Test Prep 6-70 Mixed Review 7-86 Homework Quick heck To check students understanding of key skills and concepts, go over Exercises 8,,, 7, 55. Exercises 8 0 Point out that it is usually easier to count vertices than sides of a polygon. ecause the number of vertices and sides are equal, either method is acceptable. Example (page 59) pentagon; convex decagon; concave pentagon; concave Find the sum of the measures of the angles of each polygon.. 080. dodecagon 800. decagon 0. 0-gon 0 5. 00-gon 80,000 onnection to Language rts Exercises, Point out that the two polygons differ only in the prefix do-, which means two. y adding it and the prefix deca-, which means ten, you get + 0 =. Example (page 60) Example 5 (page 60) x lgebra Find the missing angle measures. 6. y 0 7. 7 00 8. 6 9 0 05 0 x 0 85 5 5 0 5 y 5 5 7 9. 0. h. a h 6, h h 9 8 60, 60, 0, 0 (n 6) n Find the measures of an interior angle and an exterior angle of each regular polygon.. pentagon. dodecagon. 8-gon 5. 00-gon 08; 7 50; 0 60; 0 76.;.6 Lesson -5 The Polygon ngle-sum Theorems 6 5 0 5 6 GPS Guided Problem Solving L Enrichment Reteaching dapted Practice L L L Pearson Education, Inc. ll rights reserved. Practice Name lass ate Practice -5 Lines in the oordinate Plane Write an equation of the line with the given slope that contains the given point.. F(, -6), slope. Q(5, ), slope -. (, ), slope 7. (-, -), slope - 5 5. L(-, -), slope 6. R(5, 0), slope 7. (, -9), slope 8. W(0, 6), slope - 6 Graph each line using slope-intercept form. 9. y = 8x - 0. y = x - 0. x + 9y = 8. -x + y =-. y + 7 = x. x - y = 6 5. 5 - y = x 6. x = y - Graph each line. 7. y = 5x + 8. y = x - 9. x =- 0. y =-x. y =-5. y = x. y =- x +. x =.5 Write an equation of the line containing the given points. 5. (, 7), (, ) 6. P(-, ), Q(0, ) 7. S(0, ), T(, ) 8. (7, -), E(-5, ) 9. G(-, 0), H(, 0) 0. (, 5), (-6, ). X(-, -), Y(, ). M(8, -), N(7, ) Write equations for (a) the horizontal line and (b) the vertical line that contain the given point.. Z(, -). (0, ) 5. R(-, -) 6. F(-, 8) Graph each line using intercepts. 7. x - y = 8. x + y =- 9. x + y = 0. x - y = 6. x - y = 8. x + y =. -6x +.5y = 8. 0.x + 0.y =.8 5. Hourly Wages The equation P = $.90 + $0.0x represents the hourly pay (P) a worker receives for loading x number of boxes onto a truck. a. What is the slope of the line represented by the given equation? b. What does the slope represent in this situation? c. What is the y-intercept of the line? d. What does the y-intercept represent in this situation? 6. Inclines The lackberrys driveway is difficult to get up in the winter ice and snow because of its slope. What is the equation of the line that represents the lackberrys driveway? 8 yd 0 yd L 6
Error Prevention! Exercise 6 sk: How many sides does this polygon have? 5 What is the sum of the interior angles for this figure? 50 Exercise 7 Have students investigate the shapes of sports stadiums and report how polygons are used in designing them. Exercise 8 If necessary, review how to solve equations with a variable in the denominator. Remind students that this is a regular polygon. Technology Tip Exercise 57 This would be a nice classroom exercise using the table feature of a graphing calculator with overhead display. onnection to lgebra Exercise 58 Part b introduces the idea of the asymptote of a function, which students will encounter in their next algebra course. 8. 9. 0. 7. 90 Real-World pply Your Skills onnection t a theater-in-the-round, seats are arranged so the audience surrounds the stage. GO nline Homework Help Visit: PHSchool.com Web ode: aue-005 x Packaging The nut container at the right has the shape of a regular octagon. It fits in a square box. cheese wedge fills each corner of the box. 6. Find the measure of each angle of a cheese wedge. 5, 5, 90 7. ritical Thinking Show how to rearrange the four pieces of cheese to make a regular polygon. What is the measure of each angle of the polygon? See left. Use a protractor. Sketch each type of regular polygon. Sample: dodecagon Use the protractor to equally space points around a circle. (608 = 08, so mark a point every 08.) onnect these points to form a regular dodecagon. 8. See margin. 8. triangle 9. quadrilateral 0. hexagon. octagon The sum of the measures of the angles of a polygon with n sides is given. Find n.. 80. 080 8. 980 5. 880 8 6. Multiple hoice In the figure at right, / > / and m/ 5 0. What is m /? 5 50 5 60 E 7. Stage esign The diagram at the right shows platforms GPS constructed for a theater-in-the-round stage. escribe the largest platform by the type of regular polygon it suggests. Find the measure of each numbered angle. octagon; ml 5; ml 5 8. Error nalysis Miles said that he measured an angle of a regular polygon to be 08. Explain why this result (n )80 is impossible. If you solve n 5 0, you get n 7.. This number is not an integer. 9. ritical Thinking triangle has two congruent angles and an exterior angle with measure 00. Find two possible sets of measures for the angles of the triangle. 0-80-80; 50-50-80 The measure of an exterior angle of a regular polygon is given. Find the measure of an interior angle, and find the number of sides. 80 x; 60 x 0. 7 08; 5. 6 ; 0. 8 6; 0. 0 50;. x 5. Probability Find the probability that the measure of an angle of a regular n-gon is a positive integer if n is an integer and # n #. 5 6. lgebra polygon has n sides. n interior angle of the polygon and an adjacent exterior angle form a straight angle. a. What is the sum of the measures of the n straight angles? n 80 b. What is the sum of the measures of the n interior angles? (n )80 c. Using your answers above, what is the sum of the measures of the n exterior angles? 80n 80(n ) 60 d. What theorem do the steps above lead to? Polygon Ext. l-sum Thm. 80 50 0 0 0 0 0 0 0 0 0 0 50 50 60 60 60 70 70 80 80 90 90 90 80 80 90 70 70 60 60 60 50 50 0 0 0 0 0 0 0 0 0 0 0 50 6 hapter Parallel and Perpendicular Lines. 57. a. (0, 6), (0, 7), (60, 7), (80, 75.5), (00, 76.), (0, 77), (0, 77.), (60, 77.8), (80, 78), (00, 78.) c. It is very close to 80. d. No, two sides cannot be collinear. 58. b. s n gets larger, the size of the angles get closer to 80. The more sides it has, the closer the polygon is to a circle. 6
8. w 7, x 59, y 9, z ; k Problem Solving Hint In Exercises 5-5, sketch a figure to meet the first condition. Then adjust it to meet the second condition. hallenge x 5. Yes; the sum of the measures of ' at the int. point is 60. The sum of the measures of all the > is 80n. 80n 60 (n )80 To graph the ordered pairs, use and STT PLOT on your graphing calculator. PHSchool.com For: Graphing calculator procedures Web ode: aue-0 lesson quiz, PHSchool.com, Web ode: aua-005 lgebra Find each missing angle measure. Then name the polygon. 7. y 00 8. 9. z x x x 87 0 z (z ) x w y x y 0; z 70; quad. (z 0) x 6, x 7, x 08, x ; quad. Open-Ended Sketch each figure described in Exercises 50 5. 50. a quadrilateral that is not equiangular 50 5. See back of book. 5. an equiangular quadrilateral that is not regular 5. an equilateral polygon that is not equiangular 5. an equiangular polygon that is not equilateral 5. ritical Thinking Ellen says she has another way to find the sum of the measures of the angles of a polygon. She picks a point inside the polygon, draws a segment to each vertex, counts the number of triangles, multiplies by 80, and then subtracts 60. oes her method work? Explain. See left. 55. Writing Tell what you know about the figure at the right. See above right. 55. nswers may vary. Sample: The figure is a convex equilateral quadrilateral. The sum of its angles is 80 or 60. 56. The measure of an interior angle of a regular polygon is three times the measure of an exterior angle of the same polygon. What is the name of the polygon? octagon 57. a. Graphing alculator Find the measure of an angle of a regular n-gon for n = 0, 0, 60, 80, c, 00. Record your results to the nearest tenth as ordered pairs in the form (n, measure of each angle). See margin p. 6. b. Plot the ordered pairs using a window Xmin = 0 Ymin = 60 like the one shown at the right. See back of Xmax = 00 Ymax = 8 c. ata nalysis ased on the graph book. Xscl = 0 Yscl = from part (b), make a statement about the measure of an angle of a regular 000-gon. c d. See margin p. 6. d. Is there a regular n-gon with an angle of 808? Explain. 58. a. Explain why the measure of an angle of a regular n-gon is given by the 80(n ) 60 formulas and 80 -. 80n 60 60 n n [80(n )] n n 580 n. b. Use the second formula to explain what happens to the measures in the angles of regular n-gons as n becomes a large number. Explain also what happens to the polygons. See margin p. 6. 59. Two rays bisect two consecutive angles of a regular decagon and intersect in the decagon s interior. Find the measure of the acute angles formed by the intersecting rays. 6 raw, if possible, the concave quadrilateral described. If not possible, explain. 60 6. See margin. 60. with two pairs of congruent 6. with two pairs of congruent adjacent sides opposite sides 6. with three congruent sides 6. with four congruent sides Lesson -5 The Polygon ngle-sum Theorems 6. ssess & Reteach Lesson Quiz For Exercises and, if the figure is a polygon, name it by its vertices and identify its sides. If the figure is not a polygon, explain why not.. quadrilateral ;,,,. E not a polygon because two sides intersect at a point other than endpoints. Find the sum of the measures of the angles in an octagon. 080. pentagon has two right angles, a 00 angle and a 0 angle. What is the measure of its fifth angle? 0 EFGHIJ is a regular decagon. 5. Find m&. 6. &X is an exterior angle at vertex. Find m&x. 6 lternative ssessment Have students draw a polygon with 5 sides and a polygon with 8 sides. For each polygon, students should show the exterior angles and find the sum of the interior angles and the sum of the exterior angles. Then have them write a paragraph explaining how the sums illustrate the Polygon ngle-sum Theorem and the Exterior ngle-sum Theorem. 60 6. nswers may vary. Samples are given. 60. 6. Not possible; opp. sides would overlap. 6. 6. Not possible; opp. and adj. sides would overlap. 6
Test Prep sheet of blank grids is available in the Test-Taking Strategies with Transparencies booklet. Give this sheet to students for practice with filling in the grids. Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 98 Test-Taking Strategies, p. 88 Test-Taking Strategies with Transparencies Test Prep Gridded Response Lesson - Mixed Review For Exercises 6 70, you may need the formula (n - )80 for the sum of the angle measures in a polygon with n sides. 6. What is the sum of the measures of the angles of a 5-gon? 0 65. company is manufacturing a gear that has the shape of a regular polygon. The measure of each angle of the gear is 6. How many sides does the gear have? 0 66. The car at each vertex of a Ferris wheel holds a maximum of 5 people. The sum of the measures of the angles of the Ferris wheel is 770. What is the maximum number of people that the Ferris wheel can hold? 5 67. What is the sum of the measures of the exterior angles, one at each vertex, of an octagon? 60 68. Exactly three angles of a pentagon are congruent. The other two angles are complementary. What is the measure of one of the three congruent angles? 50 69. The sum of the measures of the angles of a regular polygon is 500. How many sides does the polygon have? 7 70. What is the measure of an exterior angle of a regular polygon with 6 sides? 0 GO for Help Lesson - x Find each missing angle measure. 7. 5 (y 5) 0, 5 y 7. 7. 50, 0 0, 76, 5, 69 (y ) (x ) (x ) x (x 8) y Lesson - Lessons -, -6 8. nswers may vary. Sample: R and TK 8. nswers may vary. Sample: lrm Name the property that justifies each statement. istr. Prop. 7. (a - ) = 8a - Subst. Prop. 75. If b + c = 7 and b =, then + c = 7. 76. RS > RS 77. If & > &, then & > &. Reflexive Prop. of O Symm. Prop. of O 78. If r = 8, then r = 9. 79. If = and =, then =. iv. Prop. Trans. Prop. Identify the following in the diagram. 80. a pair of opposite rays RT, RK M 8. two right angles lrt, lrk 8. two segments T R K 8. an acute angle 8. an obtuse angle ltrm 85. a straight angle ltrk 86. a midpoint R 6 hapter Parallel and Perpendicular Lines 6 page 65 lgebra Review 0. The line is not as steep as line <, but has the same tilt. y O x. The line is steeper than line r, but has the same tilt. y x O