22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions n erive properties Why lern this? ngles n e rui lly importnt in some extreme spor ts interior ngle Interior n exterior ngles up to 180 Level 5 exterior ngle The interior ngles in tringle sum to 180 Level 5 The interior ngles in qurilterl sum to 360 The exterior ngle of tringle is equl to the sum of the two interior opposite ngles For exmple, ⴙ ⴝ e e Di you know? onvention is n epte mthemtil wy to show some informtion Level 7 efinition is preise esription For exmple, the efinition of squre is: shpe with extly four equl sies n four equl ngles Level 7 The wor ngle omes from the Ltin wor erive property is informtion tht n e worke out from efinition For exmple, eh ngle of squre is 90 euse they sum to 360 n re ll equl Level 7 ngulus, whih mens orner Level 5 Work out the size of ngle q q is n interior ngle q n 52 lie on stright line, so they sum to 180 q = 180 52 = 128 80 I n use interior n exterior ngles to lulte ngles p q 52 Work out the size of ngle p opy n omplete these sentenes to ientify the interior n exterior ngles E D U Z T Tip S 117 105 E is X 98 84 Y W V W X n interior ngle i YZW is ii SWX is iii XTU is The mrke ngle is or lulte the missing ngles mrke on the igrms lulte the size of the lettere ngles, stting ny ngle fts tht you use 94 107 p 70 68 22 Get in line x y s t 110 58 u onvention q 105 efinition 64 98 e erive property I n lulte interior n exterior ngles of tringles n qurilterls exterior ngle
Sketh this igrm x Then opy n omplete these sentenes ngle x is equl to ngle euse they re ngle y is equl to ngle xⴙⴙyⴝ euse they re euse they lie on Sine x ⴝ n y ⴝ, ⴙⴙⴝ ⴙⴙ ngles y ngles ngles in tringle sum to 180 ngles in qurilterl sum to 360 Sketh this igrm The qurilterl hs een split into two tringles ⴙ ⴙ ⴝ 180 ontinue the proof to show tht ngles in qurilterl sum to 360 e Then opy n omplete this proof tringle sum to ⴙxⴝ x euse they lie on So ⴙ ⴙ ⴝ ⴙ So ⴙ ⴝ x I n follow proof tht the sum of ngles in qurilterl is 360 f Sketh this igrm ⴝ 180 euse ngles in I n follow proof tht the sum of ngles in tringle is 180 Lern this This proves tht ngles in tringle sum to ⴙⴙ I n follow proof tht the exterior ngle of tringle is equl to the sum of the two interior opposite ngles Deie whether eh sttement is efinition, onvention or erive property ngles on stright line sum to 180 n interior ngle is n ngle insie shpe Derive property Definition The exterior ngle of tringle is equl to the sum of the two interior opposite ngles Level 7 I n reognise the ifferene etween onventions, efinitions n erive properties The shes on opposite sies of retngle show tht the sies re the sme length tringle hs three sies n three interior ngles Prllel lines re mrke with rrows pointing in the sme iretion ngle prolems Work with prtner Eh rw tringle with the interior n exterior ngles mrke Tell your prtner two of the interior ngles from your tringle hllenge them to work out the other interior ngle n the exterior ngles hek their nswers to see if they re orret Tringle properties Use ynmi geometry progrm to onstrut tringle with line going through one vertex tht is prllel to the opposite sie Drg ny of the verties to explore wht hppens interior ngle proof qurilterl tringle 22 ngles n proof 23
23 onstruting tringles Drw n ngle urtely using protrtor onstrut tringle using protrtor n ruler onstrut tringle using ompsses n ruler Drw right-ngle tringle using ompsses n ruler Why lern this? Tringles re stron g shpe use in the onstruti on of mny riges You n onstrut tringle using ruler n protrtor if you know either two sies n the inlue ngle (SS) or two ngles n the inlue sie (S) Level 5 You n onstrut tringle using ruler n ompsses if you know the length of ll three sies (SSS) hypotenuse The hypotenuse of right-ngle tringle is the longest sie n is opposite the right ngle Di you know? The wor tringle is me up of tri, whih mens three n ngle tringle hs three ngles Lines tht meet t right ngles re perpeniulr Perpeniulr lines n e onstrute using ompsses You n onstrut right-ngle tringle using ruler n ompsses if you know the lengths of the hypotenuse n one of the shorter sies (RHS) Level 7 Level 5 Drw these ngles urtely using ruler n protrtor Lel eh ngle s reflex or otuse 138 294 onstrut these tringles using ruler n protrtor 197 I n use protrtor to rw otuse n reflex ngles to the nerest egree 176 4 m X I n onstrut tringle given two sies n the inlue ngle (SS) 6 m 72 Mke n urte rwing of these tringles 6 m 4 m Z 45 Y 85 5 m 73 I n onstrut tringle given two ngles n the inlue sie (S) 60 3 m n rhitet is lulting the length of woo require to mke trussels for roof The with of the roof is 5 m n the two ngles to the horizontl re 88 n 65 Using sle of 1 m represents 1 m, rw n urte sle rwing of the roof Mesure the length of eh sloping em to fin how muh woo is neee for one truss 24 Get in line ute 88 65 5m ngle ompsses
These tringles re ll right-ngle tringles Whih letter mrks the hypotenuse of eh tringle? e h g l I n ientify the hypotenuse in right-ngle tringle k i j f I n onstrut tringle given three sies Use ompsses n ruler to onstrut tringle with sies ⴝ 7 m, ⴝ 6 m n ⴝ 5 m onstrut tringle with sies of length 9 m, 7 m n 8 m using ompsses n ruler Using ompsses n ruler, rw the perpeniulr to the line t point I n use ruler n ompsses to onstrut the perpeniulr from point on line segment 4 m 5 m Level 7 motor yle stunt mn is uiling rmp so he n jump over four rs Here is the sie-view of his rmp I n onstrut right-ngle tringle if I know the lengths of the hypotenuse n nother sie (RHS) 10 m 8m Drw n urte sle rwing of the rmp using ruler n ompsses Wht is the height of the top of the rmp? 4 m ler lens ginst wll with its se 1 m from the wll Wth out! 4m Drw n urte sle rwing of the ler ginst the wll Use your rwing to fin how fr the ler rehes up the wll Don t ru out your onstru tion lines s they show tht you hve use the ompsses orretly 1m Mrk wnts to onstrut tringle with sies of length 5 m, 3 m n 9 m Explin why Mrk s tringle is impossile to onstrut Hint: Try to onstrut the tringle first Drwing tringles 1 Drwing tringles 2 1 Drw tringle n lel the verties, n 1 Drw right-ngle tringle using ompsses n ruler 2 Mesure the sies n 3 Mesure the ngle 2 Mesure the hypotenuse n one of the other sies 4 Desrie the tringle to your prtner y telling them the informtion out the two sies n the ngle Your prtner rws the tringle you hve esrie 3 Desrie the tringle to your prtner y telling them the informtion out the two sies n the ngle Your prtner rws the tringle you hve esrie 5 hek your prtner s tringle with the originl 4 hek your prtner s tringle with the originl onstrution hypotenuse perpeniulr otuse 23 onstruting tringles 25
24 Speil qurilterls Know the properties of qurilterls Solve geometril prolems involving qurilterls n explin the resons Why lern this? Mny uilings re me of retngles n othe r qurilterls How mny ifferent shpes n you see in this photo? Qurilterl properties: retngle squre prllelogrm rhomus isoseles trpezium kite rrowhe When solving prolems using the properties of shpes it is importnt to explin your resoning & Level 7 I n ientify ngle, sie n symmetry properties of simple qurilterls Nthniel si squre is retngle Is this true? Explin your nswer Whih of these sttements re lwys true for retngle? ll its sies re equl It hs four lines of symmetry It hs four right ngles opy this tle omplete it y writing eh shpe nme in the orret position Numer of pirs of prllel sies Tip Numer of lines of symmetry 0 1 2 Some ells my ontin more thn one shpe 4 0 1 2 Di you know? retngle squre prllelogrm rhomus e kite f rrowhe g isoseles trpezium The prefix quri- omes from the Ltin wor for four n you think of ny other wors tht egin with qu? Drw retngle n ut it out I n ientify n egin to use ngle, sie n symmetry properties of qurilterls ut long one of the igonls Rerrnge the piees to mke nother qurilterl Write the nme of the new qurilterl tht you hve me Write one geometril ft out this shpe 26 Get in line rrowhe isoseles trpezium kite prllelogrm
Sketh n equilterl tringle in one of its sies Write the nme of the qurilterl tht is forme Whih of these sttements re lwys true for this speil qurilterl? The igonls iset t right ngles The ngles re ll equl It hs two pirs of prllel sies D It hs four lines of symmetry Look t this retngle One of the igonls is rwn Work out the sizes of ngles ngles, n 30 In rhomus, one of the ngles is 40 Work out the sizes of the other ngles x 40 The squre n rhomus re qurilterls with equl length sies T Look t this rrowhe TSV ⴝ 45, STV ⴝ 30 I n solve geometril prolems using properties of tringles n qurilterls 30 TVU SVU 45 V S In this retngle, lulte ngle ED Show your steps for solving this prolem n explin your resoning U 72 E D Work out the sizes of these ngles Explin your resoning E Speil qurilterls gme for two plyers Eh seretly rw speil qurilterl Tke turns to tell eh other one property of your shpe Try to guess eh other s shpe Sore 1 point if you guess orretly from one property, 2 points from two properties, n so on The plyer with the lowest sore wins qurilterl retngle rhomus Level 7 I n use resoning to solve more omplex geometril prolems D F I n use resoning to solve geometril prolems 56 E F I n solve simple geometril prolems using properties of qurilterls Lern this y z lulte TUV I n ientify n egin to use ngle, sie n symmetry properties of qurilterls 70 E 65 D Prllelogrms Drw prllelogrm like this Lel three ngles with their sizes hllenge your prtner to work out the missing ngles n explin their resons Use wht you know out the properties of prllelogrms to hek their nswers squre symmetry 24 Speil qurilterls 27
25 More onstrutions Know the nmes of prts of irle Use stright ege n ompsses to onstrut the perpeniulr isetor of line n n ngle, n the perpeniulr to line Use stright ege n ompsses to investigte the properties of overlpping irles Why lern this? Unerstning perpeniulr lines n help you ppreite their use in Lines tht meet t right ngles re perpeniulr Perpeniulr uilings n on ro s lines n e onstrute using ompsses The ngle isetor uts the ngle in hlf The perpeniulr isetor uts the line in hlf t right ngles oth n e onstrute using ompsses The perpeniulr from point to line segment is the shortest istne to the line When the points of intersetion of two ientil overlpping irles re joine to the entres, rhomus is forme Level 7 right-ngle tringle n e onstrute using ruler n ompsses if you know the length of the hypotenuse n one of the shorter sies Level 7 I n onstrut the mi-point n perpeniulr isetor of line segment Using only ruler n ompsses, rw the perpeniulr isetors of these line segments Mrk the mi-point of eh line segment stright line segment of length 6 m stright line segment D of length 8 m onstrution ompny is uiling two houses, 10 m prt The rhitet s plns look like this 10 m opy the pln, using sle of 1 m to represent 1 m onstrut the perpeniulr isetor of the 10 m line: fene will e uilt on the perpeniulr isetor Wht n you sy out the position of the fene? I n nme the prts of irle opy this irle with rius 4 m these lels rius imeter hor r e tngent f irumferene Use ompsses n ruler to rw the isetor of these ngles Lern this n ute ngle of your hoie iset mens to ut something into two equl prts n ngle of 90 rwn with protrtor n otuse ngle of your hoie 28 Get in line I n onstrut the isetor of n ngle r isetor (iset) hor irle ompsses
I n onstrut the perpeniulr from point to line segment Tip opy the igrm Using ompsses n ruler, rw the perpeniulr t X hek fter you hve rwn perpeniulr line to see if it looks to e t right ngle X 3 m Mke opy of this igrm onstrut the perpeniulr from point to the line 5 m onstrution ompny is uiling rige ross river opy the igrm n rw the perpeniulr from point S ross the river to show where the rige shoul e uilt S Level 7 Using ompsses, rw two irles of rius 4 m tht overlp Join the entres of the irles with stright line n rw the hor tht is ommon to oth irles Join the entres of the irles the points where the irles interset Wht o you notie out the qurilterl tht is forme? I n explin how stnr onstrutions using ruler n ompsses relte to the properties of two interseting irles with equl rii Tringles in irles 1 Drw irle, using ompsses or ynmi geometry softwre 2 Mrk three points on the irumferene of the irle 3 Join up these points to mke tringle 4 onstrut the perpeniulr isetor of eh sie of your tringle 5 Wht o you notie? 6 Wht hppens when the verties of the tringle re move to ifferent points on the irumferene? Polygons in irles 1 Drw irle, using ompsses or ynmi geometry softwre 2 Mrk four points on the irumferene of the irle 3 Join up these points to mke qurilterl 4 onstrut the perpeniulr isetor of eh sie of your qurilterl 5 Wht o you notie? 6 Investigte other polygons insie irle imeter perpeniulr rius right ngle tngent 25 More onstrutions 29
26 ngles in polygons Fin the sum of the interior n exterior ngles of polygons Fin n interior n exterior ngle of regulr polygon Use the interior n exterior ngles of regulr n irregulr Why lern this? polygons to solve prolems Polygons re foun in mny ples in nture Whe n lv ools it n form olum ns in the shpe of polygon s n interior ngle n its orresponing exterior ngle sum to 180 Level 5 The sum of the interior ngles in n n-sie polygon is (n ⴚ 2) ⴛ 180 The sum of the exterior ngles in ny polygon is lwys 360 regulr polygon hs ll sies of equl length n ll ngles equl sum of interior ngles The interior ngle of regulr polygon ⴝ numer of sies You n use interior n exterior ngles in polygons to solve prolems Level 7 Explin how you lulte the sum of the interior ngles in I n explin how to fin the interior ngle sum of polygon qurilterl pentgon I n explin how to lulte the interior n exterior ngles of regulr polygons Explin how you fin the size of n interior ngle in regulr pentgon Explin how you fin the size of n exterior ngle in regulr pentgon I n lulte the sums of the interior n exterior ngles of irregulr polygons Wht is the sum of the interior ngles in i qurilterl ii pentgon iii hexgon? lulte the sum of the interior ngles in 10-sie polygon Look t this qurilterl t eh vertex the sum of the interior n exterior ngles is 180 I E I ⴙ E ⴝ 180 Explin why this is true Drw qurilterl with the exterior ngles mrke, like the one in Q4 Use protrtor to mesure eh exterior ngle Fin the sum of the exterior ngles Repet prts n for pentgon n hexgon Wht o you notie out the sum of the exterior ngles of polygon? 30 Get in line exterior ngle hexgon interior ngle irregulr polygon
lulte the exterior ngle of regulr hexgon I n lulte the interior n exterior ngles of regulr polygons lulte the size of eh interior ngle in regulr hexgon opy n omplete this tle Regulr polygon Numer of sies equilterl tringle 3 squre 4 regulr pentgon 5 Sum of interior Size of eh Sum of exterior Size of eh ngles interior ngle ngles exterior ngle 180 360 regulr hexgon regulr otgon How o you fin the sum of the interior ngles in n n-sie polygon? lulte the size of the interior n exterior ngles in regulr 16-sie shpe The exterior ngle of regulr polygon is 18 ii How mny sies oes the polygon hve? ii lulte the size of eh interior ngle Level 7 Lern this The interior ngle of regulr polygon is 156 How mny sies oes the polygon hve? The exterior ngles of polygon lwys up to 360 It is possile to rw polygon tht hs interior ngles tht sum to 1300? Explin your resoning The igrm shows regulr otgon The line is prllel to the line D lulte the size of D D H D I n solve hrer prolems using properties of ngles, prllel n interseting lines, n tringles n other polygons E 675 G DH I n use the interior n exterior ngles of regulr polygons to solve prolems F Di you know? Polygon poster Mke poster of ll the fts you know out the interior n exterior ngles of polygons Polygons re use to rete omplex-shpe omputer grphis Next time you ply omputer gme, see how mny polygons you n spot Tessellting polygons Investigte whih regulr polygons tessellte Look t the interior ngles How n you tell y looking t the interior ngles whether shpe will tessellte? Why will regulr hexgon n squre tessellte? otgon pentgon qurilterl regulr polygon tringle 26 ngles in polygons 31