Angles 2.1. Exercise Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example


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1 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel Alternte ngles on prllel lines re equl. In this igrm, the lternte ngles re on lternte sies of the re line. Corresponing ngles on prllel lines re equl. In this igrm, the orresponing ngles re on the sme sie of the re line. Rememer, vertilly opposite ngles re lso equl. mple Clulte the size of the lettere ngles in this igrm. Give resons for your nswers euse euse it is the orresponing ngle to 50 euse it is the lternte ngle to 50 erise Fin the size of the lettere ngles. Give resons for your nswers. ) ) ) Angles on stright line up to ) e) f) 34 e f 14 Mths Connet 2
2 Clulte the size of the lettere ngles. Give resons for your nswers. ) ) ) g f e ) s e) f) q n p r l g) h) i) 47 v k j 82 m t u i h w Clulte the size of the lettere ngles. Give resons for your nswers n write own eh step of your working. ) ) ) ) i f e 78 j k 150 It my e helpful to sketh the igrm n work out some of the other ngles first. Investigtion ) Copy this igrm. ) Mrk ll the ngles tht re equl to. ) Mrk ll the ngles tht re equl to. ) Wht o you notie? e) Wht o you notie out the pttern of ngles in prllelogrm? Angles 15
3 2.2 Clulting ngles Know n use the ft tht the sum of the ngles in tringle is 180 Know n use the ft tht the sum of the ngles in qurilterl is 360 Unerstn tht the eterior ngle of tringle is equl to the sum of the two interior opposite ngles Key wors proof interior eterior p q 180 s r p q r s 360 The sum of the ngles in ny tringle The sum of the ngles in ny is 180. qurilterl is 360. An interior ngle is insie the shpe. An eterior ngle is outsie the shpe. It is me y etening one of the lines. e e 180 (ngles on stright line) 180 (ngles in tringle) so e The eterior ngle of tringle is equl to the sum of the two interior opposite ngles. mple Clulte the size of the lettere ngles, giving resons for your nswers. ) ) ) ) The sum of ngles in tringle is 180. The eterior ngle of tringle is equl to the sum of the two interior opposite ngles. 16 Mths Connet 2
4 erise Clulte the size of the lettere ngles, giving resons for your nswers. ) ) ) 122 f ) g e) f) g) h) i j h k Clulte the size of the lettere ngles, giving resons for your nswers. ) ) e Look t the lue tringle first n fin. Look for the two lrge tringles. 34 f 65 Work out the vlue of in eh of the following: ) ) ) Investigtion rw two stright lines A n C. A C A C Mrk point O etween them. O O rw line through O tht rosses the two lines. Mrk the two ngles, unerneth the line, s n y. y Mesure the size of n y with protrtor. rw some more lines through O. h time mesure the ngles unerneth the line n reor your results in tle. Wht o you notie? Cn you eplin why this hppens? ten A n C so they meet. Clulting ngles 17
5 2.3 Qurilterls Reognise n lssify qurilterls y their geometri properties Key wors prllelogrm rhomus isoseles trpezium kite rrowhe or elt A qurilterl is foursie shpe. There re mny ifferent qurilterls. Some hve speil nmes euse they hve prtiulr geometri properties. Retngle Opposite sies re equl n prllel igonls iset eh other Rottion symmetry orer 2 Squre A retngle with ll four sies the sme length igonls iset eh other t right ngles igonls re lines of symmetry rllelogrm Opposite sies re equl n prllel igonls iset eh other Rottion symmetry orer 2 Rhomus A prllelogrm with ll four sies the sme length igonls iset eh other t right ngles oth igonls re lines of symmetry Isoseles trpezium A trpezium with two opposite nonprllel sies One line of symmetry oth igonls re the sme length Kite Two pirs of jent sies tht re the sme length No interior ngle is lrger thn 180 One line of symmetry igonls ross t right ngles Arrowhe or elt Two pirs of jent sies tht re the sme length One interior ngle is lrger thn 180 One line of symmetry igonls ross t right ngles outsie the shpe mple Julie hs to rw sketh of n isoseles trpezium. She writes the mesurements on the igrm. plin why this sketh nnot show n isoseles trpezium. 4 m m It is not n isoseles trpezium euse it oes not hve ny equl ngles. This mens there is no line of symmetry. 18 Mths Connet 2
6 erise Nme eh shpe, using its geometri properties to eplin your nswers. You will nee to mesure lines n ngles. ) ) ) The igonls re shown s she lines. Jules rws some skethes of qurilterls, mrks the size of their sies n ngles n nmes them. Some of the nmes re inorret. Fin whih skethes hve inorret nmes n eplin why they re inorret. ) 3 m ) 3 m 2 m ) 3 m 3 m 6 m ISOSCLS TRAZIUM ARALLLOGRAM ) e) 4 m 50 4 m 50 3 m RHOMUS 3 m m 50 4 m KIT LTA ) rw long oth eges of ruler. Tke the ruler wy n then ple it ross your lines t n ngle. rw long oth eges of the ruler gin. She the shpe enlose y the lines. Wht is the nme of the shpe? plin how you know. ) It is possile to mke nother qurilterl using the sme metho. Wht is the nme of the qurilterl? plin how you know. Investigtion You nee 3 y 3 pin or, or Resoure sheet 61, n ruer n. Mke s mny ifferent qurilterls s you n. rw n lel eh shpe. Cut out your shpes n rrnge them into groups tht hve similr properties. Stik the groups into your workook n eplin the properties tht link the shpes in eh group. Qurilterls 19
7 2.4 Solving geometril prolems Solve geometril prolems using sie n ngle properties of speil tringles n qurilterls Key wors lternte orresponing vertilly opposite Angles in tringle up to 180. Angles on stright line up to 180. Alternte ngles re equl. Corresponing ngles re equl. Vertilly opposite ngles re equl. We n use this knowlege to fin the size of missing ngles. Sine we re lulting the sizes of the ngles y following hin of resoning n not mesuring them, our igrms o not hve to e urte. mple 1 Clulte the size of ngles,,, n e, giving resons for your nswers. 53 e Angle is 53 euse tringle AC is isoseles n the se ngles of n isoseles tringle re equl. Angle is 53 euse it is n lternte ngle to ngle. Angle is 74 euse ngles on stright line up to 180 n Angle is 53 euse it is orresponing ngle to ngle. Angle e is 53 euse it is n lternte ngle to ngle. A 53 C We n see from the igrm tht AC is n isoseles tringle n tht n C re prllel. C n re lso prllel. mple 2 Clulte the size of the missing ngles giving resons for your nswers. 3 y 2 z euse ngles in tringle up to so 3 54 n 2 36 y euse ngles in stright line up to 180 so y 126 z 36 euse it is vertilly opposite to the ngle Mths Connet 2
8 erise Clulte the size of the lettere ngles, giving resons for your nswers. ) 76 ) A C F ) 58 A C Clulte the size of the lettere ngles, giving resons for your nswers. ) A ) ) A C y C 33 Clulte the size of eh ngle, giving resons for your nswers. ) ) Investigtion The ngle t the verte of regulr pentgon is 108. Two igonls re rwn to the sme verte to mke three tringles. ) Clulte the sizes of the ngles in eh tringle. ) The mile tringle n one of the other tringles re ple together like this: 108 plin why the tringles fit together to mke new tringle. Wht re its ngles? ) Investigte the other shpes you n mke y putting the three tringles together. Solving geometril prolems 21
9 2.5 Constrution Construt isetor of n ngle, using ruler n ompsses Construt the mipoint n perpeniulr isetor of line segment, using ruler n ompsses Key wors isetor ompsses equiistnt perpeniulr isetor mipoint equiistnt The isetor of n ngle is line tht ivies the ngle into two equl prts. You n onstrut the isetor of n ngle using ompsses. In this igrm, is the isetor of the ngle AC. very point on the line is equiistnt from the lines A n C. A C The perpeniulr isetor of line segment ivies the line segment into two equl prts t right ngles. In this igrm, is the perpeniulr isetor of AC. It rossses AC t the mipoint (M) of the line. very point on the line is equiistnt from oth A n C. If you join AC, rhomus is forme. A M C mple 1 F is 100. Construt the isetor of F. G An r is prt of irle. You n rw rs with ompsses. F F F 1) Open the ompsses n put the point on. rw n r tht intersets with n F. mple 2 1) 2.5 m 22 Mths Connet 2 2) o not hnge the opening of the ompsses. ut the point first on the intersetion of the r with n then with F. rw new rs to interset t G. Join G. The line segment Q is 2.5 m long. Construt the perpeniulr isetor of Q. Q 2) Q 3) Q 1) rw line Q of 2.5 m. Open up the ompsses to over hlf the length of Q. 2) le the point t n rw n r. Keep the opening of the ompsses the sme n repet t Q. Join the points where the rs interset.
10 erise rw ny ute ngle. Construt the ngle isetor using only ruler n ompsses. rw line 10 m in length. Construt the perpeniulr isetor of the line using only ruler n ompsses. Mrk the mipoint (M) of the line. rw ny otuse ngle. Construt the ngle isetor using only ruler n ompsses. Chek it is orret y mesuring the two ngles n mking sure they re the sme. A new wter hnnel is to e uilt t the zoo. It is to e ple etween two niml houses so tht the houses re equiistnt from the hnnel. Copy this igrm, n onstrut re line to show the position of the wter hnnnel. Animl house 1 Animl house 2 A new fene is to e uilt in the prk. The fene is to e ple etween two lrge trees so tht the trees re equiistnt from the fene. The he grener eies to pint in line to mrk the position of the fene. Copy this igrm n onstrut lue line to show the position of the fene. Tree 1 Tree 2 Investigtion ) rw ny tringle with sies longer thn 4 m ut shorter thn 8 m. Construt the ngle isetor for eh of the three ngles. Mke sure the ngle isetors re long enough to ross eh other. Wht o you notie? ) rw nother tringle pproimtely the sme size. Construt the perpeniulr isetor for eh of the three sies. Mke sure the perpeniulr isetors re long enough to ross eh other. Wht o you notie? Constrution 23
11 2.6 erpeniulrs Use stright ege n ompsses to onstrut the perpeniulr from point to line n from point on line Key wors onstrut perpeniulr r We n use ruler n ompsses to: onstrut the perpeniulr from point to line N onstrut the perpeniulr from point on line An r is prt of irle. You n rw rs with your ompsses. mple 1 Mke opy of the igrm. Using only ruler n ompsses rw perpeniulr from the point on the line. 1) Q 2) Q 1) Using ompsses, rw two rs from to mke two intersetions on the line. From the two intersetions, rw two rs tht interset n lel the intersetion Q. 2) le stright ege from to Q n join them with stright line. mple 2 Mke opy of the igrm. Using only ruler n ompsses rw perpeniulr from the point A to the line. A 1) A 2) A 1) Using ompsses, rw two rs from the point A tht interset with the line. From the two rs rw two more rs tht interset n lel the intersetion. 2) le stright ege from A to n join A to the stright line. 24 Mths Connet 2
12 erise Mke opy of the igrm. Using ruler n ompsses, onstrut perpeniulr from the point on the line. 2 m 3.5 m Mke opy of the igrm. Using ruler n ompsses, onstrut perpeniulr from the point to the line. Jmin is wlking her og in the prk when it strts to rin. She wnts to tke the shortest route k to the pth. Copy the igrm n onstrut the shortest route to the pth, using ruler n ompsses. Jmin th ) rw n equilterl tringle, using ruler n protrtor, with sies 5 m in length, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? ) rw n isoseles tringle with se 5 m n se ngles of 50, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? 5 m m 5 m ) rw slene tringle with se 5 m n se ngles of 70 n 30, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? m m erpeniulrs 25
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