2.4 Angle Properties in Polygons YOU WILL NEED dynmic geometry softwre OR protrctor nd ruler EXPLORE A pentgon hs three right ngles nd four sides of equl length, s shown. Wht is the sum of the mesures of the ngles in the pentgon? GOAL Determine properties of ngles in polygons, nd use these properties to solve problems. INVESTIGATE the Mth In Lesson 2.3, you proved properties involving the interior nd exterior ngles of tringles. You cn use these properties to develop generl reltionships involving the interior nd exterior ngles of polygons.? How is the number of sides in polygon relted to the sum of its interior ngles nd the sum of its exterior ngles? Prt 1 Interior Angles A. Giuseppe sys tht he cn determine the sum of the mesures of the interior ngles of this qudrilterl by including the digonls in the digrm. Is he correct? Explin. B. Determine the sum of the mesures of the interior ngles of ny qudrilterl. C. Drw the polygons listed in the tble below. Crete tringles to help you determine the sum of the mesures of their interior ngles. Record your results in tble like the one below. Polygon Number of Sides Number of Tringles Sum of Angle Mesures tringle 3 1 180 qudrilterl 4 pentgon 5 hexgon 6 heptgon 7 octgon 8 D. Mke conjecture bout the reltionship between the sum of the mesures of the interior ngles of polygon, S, nd the number of sides of the polygon, n. E. Use your conjecture to predict the sum of the mesures of the interior ngles of dodecgon (12 sides). Verify your prediction using tringles. 94 Chpter 2 Properties of Angles nd Tringles NEL
Prt 2 Exterior Angles F. Drw rectngle. Extend ech side of the rectngle so tht the rectngle hs one exterior ngle for ech interior ngle. Determine the sum of the mesures of the exterior ngles. Tip Communiction When side of polygon is extended, two ngles re creted. The ngle tht is considered to be the exterior ngle is djcent to the interior ngle t the vertex. djcent interior ngle exterior ngle G. Wht do you notice bout the sum of the mesures of ech exterior ngle of your rectngle nd its djcent interior ngle? Would this reltionship lso hold for the exterior nd interior ngles of the irregulr qudrilterl shown? Explin. w z b x d c y H. Mke conjecture bout the sum of the mesures of the exterior ngles of ny qudrilterl. Test your conjecture. I. Drw pentgon. Extend ech side of the pentgon so tht the pentgon hs one exterior ngle for ech interior ngle. Bsed on your digrm, revise your conjecture to include pentgons. Test your revised conjecture. J. Do you think your revised conjecture will hold for polygons tht hve more thn five sides? Explin nd verify by testing. Reflecting K. Compre your results for the sums of the mesures of the interior ngles of polygons with your clssmtes results. Do you think your conjecture from prt D will be true for ny polygon? Explin. L. Compre your results for the sums of the mesures of the exterior ngles of polygons with your clssmtes results. Do you think your conjecture from prt I will pply to ny polygon? Explin. NEL 2.4 Angle Properties in Polygons 95
APPLY the Mth exmple 1 Resoning bout the sum of the interior ngles of polygon convex polygon A polygon in which ech interior ngle mesures less thn 180. Prove tht the sum of the mesures of the interior ngles of ny n-sided convex polygon cn be expressed s 180 1n 2 22. Viktor s Solution convex non-convex (concve) C 2 C 1 C 3 C 4 A C 5 C 6 C 7 I drew n n-sided polygon. I represented the nth side using broken line. I selected point in the interior of the polygon nd then drew line segments from this point to ech vertex of the polygon. The polygon is now seprted into n tringles. The sum of the mesures of the ngles in n tringles is n(180 ). C 2 C 1 C 3 C n C n The sum of the mesures of the interior ngles of the polygon, S(n), where n is the number of sides of the polygon, cn be expressed s: S1n2 5 180 n 2 360 S1n2 5 180 1n 2 22 The sum of the mesures of the interior ngles of convex polygon cn be expressed s 180 1n 2 22. Your Turn C 4 A C 8 C 8 C 5 C 6 C 7 The sum of the mesures of the ngles in ech tringle is 180. Two ngles in ech tringle combine with ngles in the djcent tringles to form two interior ngles of the polygon. Ech tringle lso hs n ngle t vertex A. The sum of the mesures of the ngles t A is 360 becuse these ngles mke up complete rottion. These ngles do not contribute to the sum of the interior ngles of the polygon. Explin why Viktor s solution cnnot be used to show whether the expression 180 1n 2 22 pplies to non-convex polygons. 96 Chpter 2 Properties of Angles nd Tringles NEL
exmple 2 Resoning bout ngles in regulr polygon Outdoor furniture nd structures like gzebos sometimes use regulr hexgon in their building pln. Determine the mesure of ech interior ngle of regulr hexgon. Nzr s Solution Let S(n) represent the sum of the mesures of the interior ngles of the polygon, where n is the number of sides of the polygon. S1n2 5 180 1n 2 22 S162 5 180 3162 2 24 S162 5 720 720 5 120 6 The mesure of ech interior ngle of regulr hexgon is 120. A hexgon hs six sides, so n 5 6. Since the mesures of the ngles in regulr hexgon re equl, ech ngle must mesure 1 6 of the sum of the ngles. Your Turn Determine the mesure of ech interior ngle of regulr 15-sided polygon ( pentdecgon). NEL 2.4 Angle Properties in Polygons 97
exmple 3 Visulizing tesselltions A floor tiler designs custom floors using tiles in the shpe of regulr polygons. Cn the tiler use congruent regulr octgons nd congruent squres to tile floor, if they hve the sme side length? Vness s Solution S1n2 5 180 1n 2 22 S182 5 180 3182 2 24 S182 5 1080 1080 5 135 8 The mesure of ech interior ngle in regulr octgon is 135. The mesure of ech internl ngle in squre is 90. Since n octgon hs eight sides, n 5 8. First, I determined the sum of the mesures of the interior ngles of n octgon. Then I determined the mesure of ech interior ngle in regulr octgon. Two octgons fit together, forming n ngle tht mesures: 21135 2 5 270. I knew tht three octgons This leves gp of 90. would not fit together, s the 21135 2 1 90 5 360 sum of the ngles would be A squre cn fit in this gp if the sides greter thn 360. of the squre re the sme length s the sides of the octgon. 2(135 ) 90 360 I drew wht I hd visulized using dynmic geometry softwre. The tiler cn tile floor using regulr octgons nd squres when the polygons hve the sme side length. Your Turn Cn tiling pttern be creted using regulr hexgons nd equilterl tringles tht hve the sme side length? Explin. 98 Chpter 2 Properties of Angles nd Tringles NEL
In Summry Key Ide You cn prove properties of ngles in polygons using other ngle properties tht hve lredy been proved. Need to Know The sum of the mesures of the interior ngles of convex polygon with n sides cn be expressed s 180 (n 2 2). 180 1n 2 22 The mesure of ech interior ngle of regulr polygon is. n The sum of the mesures of the exterior ngles of ny convex polygon is 360. CHECK Your Understnding 1. ) Determine the sum of the mesures of the interior ngles of regulr dodecgon. b) Determine the mesure of ech interior ngle of regulr dodecgon. L M N O P Q W R 2. Determine the sum of the mesures of the ngles in 20-sided convex polygon. 3. The sum of the mesures of the interior ngles of n unknown polygon is 3060. Determine the number of sides tht the polygon hs. PRACTISING 4. Honeybees mke honeycombs to store their honey. The bse of ech honeycomb is roughly regulr hexgon. Explin why regulr hexgon cn be used to tile surfce. V U T S NEL 2.4 Angle Properties in Polygons 99
5. Is it possible to crete tiling pttern with prllelogrms? Explin. 6. Determine the mesure of ech interior ngle of loonie. 7. Ech interior ngle of regulr convex polygon mesures 140. ) Prove tht the polygon hs nine sides. b) Verify tht the sum of the mesures of the exterior ngles is 360. 8. ) Determine the mesure of ech exterior ngle of regulr octgon. b) Use your nswer for prt ) to determine the mesure of ech interior ngle of regulr octgon. c) Use your nswer for prt b) to determine the sum of the interior ngles of regulr octgon. d) Use the function S1n2 5 180 1n 2 22 to determine the sum of the interior ngles of regulr octgon. Compre your nswer with the sum you determined in prt c). 9. ) Wllce clims tht the opposite sides in ny regulr hexgon re prllel. Do you gree or disgree? Justify your decision. b) Mke conjecture bout prllel sides in regulr polygons. Mth in Action Circulr Homes A building bsed on circulr floor pln hs bout 11% less outdoor wll surfce re thn one bsed on squre floor pln of the sme re. This mens less het is lost through the wlls in winter, lowering utility bills. Most circulr buildings ctully use regulr polygons for their floor plns. Determine the exterior ngle mesures of floor pln tht is regulr polygon with ech of the following number of sides: 12, 18, 24. Explin why building would be closer to circulr s the number of sides increses. List some prcticl limittions on the number of sides building could hve. Bsed on the prcticl limittions, suggest n optiml number of sides for home. Sketch floor pln for home with this number of sides. 100 Chpter 2 Properties of Angles nd Tringles NEL
10. LMNOP is regulr pentgon. ) Determine the mesure of /OLN. b) Wht kind of tringle is ^LON? Explin how you know. P L M 11. Sndy designed this logo for the jerseys worn by her softbll tem. She told the grphic rtist tht ech interior ngle of the regulr decgon should mesure 162, bsed on this clcultion: 180 110 2 12 S1102 5 10 S1102 5 1620 10 S1102 5 162 Identify the error she mde nd determine the correct ngle. 12. Astrid clims tht drwing lines through polygon cn be used s test to determine whether the convex non-convex polygon is convex or non-convex (concve). ) Describe test tht involves drwing single line. b) Describe test tht involves drwing digonls. 13. Mrtin is plnning to build hexgonl picnic tble, s shown. ) Determine the ngles t the ends of ech piece of wood tht Mrtin needs to cut for the sets. b) How would these ngles chnge if Mrtin decided to mke n octgonl tble insted? O N NEL 2.4 Angle Properties in Polygons 101
14. Three exterior ngles of convex pentgon mesure 70, 60, nd 90. The other two exterior ngles re congruent. Determine the mesures of the interior ngles of the pentgon. 15. Determine the sum of the mesures of the indicted ngles. 16. In ech figure, the congruent sides form regulr polygon. Determine the vlues of, b, c, nd d. ) b) c d c b d b 17. Determine the sum of the mesures of the indicted ngles. h g b c d e f 18. Given: ABCDE is regulr pentgon with centre O. ^EOD is isosceles, with EO 5 DO. DO 5 CO Prove: ^EFD is right tringle. E F A O B D C 102 Chpter 2 Properties of Angles nd Tringles NEL
Closing 19. The function representing the sum of the mesures of the interior ngles of polygon with n sides is: S1n2 5 180 1n 2 22 Explin how the expression on the right cn be deduced by considering polygon with n sides. Extending 20. A pentgon tile hs two 90 ngles.the other three ngles re equl. Is it possible to crete tiling pttern using only this tile? Justify your nswer. 21. Ech interior ngle of regulr polygon is five times s lrge s its corresponding exterior ngle. Wht is the common nme of this polygon? History Connection Buckyblls Polygons in 3-D Richrd Buckminster Bucky Fuller (1895 1983) ws n Americn rchitect nd inventor who spent time working in Cnd. He developed the geodesic dome nd built fmous exmple, now clled the Montrél Biosphere, for Expo 1967. A spin-off from Fuller s dome design ws the buckybll, which becme the officil design for the soccer bll used in the 1970 World Cup. In 1985, scientists discovered crbon molecules tht resembled Fuller s geodesic sphere. These molecules were nmed fullerenes, fter Fuller. The Montrél Biosphere nd its rchitect FIFA soccer bll, 1970 Crbon molecule, C 60 A. Identify the polygons tht were used to crete the buckybll. B. Predict the sum of the three interior ngles t ech vertex of the buckybll. Check your prediction. C. Explin why the vlue you found in prt B mkes sense. NEL 2.4 Angle Properties in Polygons 103