G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th re lled orrespodig sides. 1. The two trigles re ogruet. re the two thik sides orrespodig sides? pli. ) ) ) ogruee Rules for Trigles SSS (side-side-side) SS (side-gle-side) S (gle-side-gle) Two trigles re ogruet if the hve three pirs of equl orrespodig sides two pirs of equl orrespodig sides d pir of equl orrespodig gles etwee these sides two pirs of equl orrespodig gles d pir of equl orrespodig sides etwee these gles 2. Idetif the ogruee rule tht tells ou tht eh pir of trigles is ogruet. Write the ogruee stteet. ) ) S ) N OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION Y T R X Z M H ogruee rule: ogruee rule: ogruee rule: O G Geoetr 8-11 99
3. I the trigles i the piture, = = 20, = = 35, d =. ) Mrk the equl sides d gles o the piture. ) o the give equtios of sides d gles fit ogruee rule? ) The su of the gles i trigle is. = ( + ) = = ( + ) = d) Whih ogruee rule ou ppl ow? Write the equtios for the sides d gles tht the ogruee rule requires. =, =, = 4. Ni thiks tht she ppl the se ethod s i Questio 3 to the two trigles show. ) re the trigles ogruet? ) N = = M 45 3 N K 45 J 3 20 J = = 20 L ) pli wh the S ogruee rule does ot work for JKL d NOM. O 5. ) Mesure the gles d the sides of these trigles. Write the esureets o the digr. ) Whih gles re equl? ) Whih sides re equl? d) re the trigles ogruet? pli wh the ogruee rules do ot ppl here. G K H OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 100 Geoetr 8-11
gle-gle-gle () is ot ogruee rule. 6. rw outereple for this stteet: ll trigles tht hve three pirs of orrespodig equl gles re ogruet. 7. ) Wht sides d gles i d re equl? ) ou use the SS ogruee rule? Wh or wh ot? ) o trigles d look ogruet? d) Is side-side-gle ogruee rule? Is gle-side-side ogruee rule? pli. 8. Sietists wt to esure the diste ross rter o Mrs. The sed rover log the pth OO. ) Wht re the equl sides d gles i trigles O d O? = = = ) Wht ogruee rule ou use? ) Wht is the diste ross the rter? pli how ou kow. 2,460 2,691 2,675 2,675 O 2,691 rter 9. rw two trigles tht re ot ogruet. pli wh the trigles re ot ogruet. OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 10. Sketh outereple to show wh this stteet is flse. If two trigles hve two pirs of orrespodig sides tht re equl, the trigles re ogruet. 11. PQR d XYZ hve PQ = XY = 5 d QR = YZ = 7. Sketh the trigles. oes the stteet give esure tht PQR XYZ? If so, ote the ogruee rule. ) P = Y ) Q = Y ) P = X d) PR = XZ 12. d GH hve = G d = H. Sketh the trigles. oes the stteet give esure tht GH? If so, ote the ogruee rule. ) = ) = GH ) = H d) = GH Geoetr 8-11 101
G8-12 ogruee (dved) RMINR: ogruee Rules for Trigles SS (side-gle-side) S (gle-side-gle) If i d, =, =, d =, the. If i d, =, =, d =, the. SSS (side-side-side) If i d, =, =, d =, the. 1. I the digr, M = 7. ill i the lks to fid the legth of M. ) M = d =. ) M = euse the re gles. ) M the ogruee rule. M This es M = =. 2. I the digr, KL = MN = 42, d KN = LM = 50. K = 20 d KNL = 50. ill i the lks to fid M d MLN. ) Whih sides re equl i trigles KLN d LMN? K M ) KLN the ogruee rule. So, M = = d MLN = =. 3. shool wts to istll squre rope liig struture over lrge irulr sdo. otrtor ples pole i the eter O of the o d ples rope hors t the opposite sides of the pit i poits W, X, Y, d Z. ) rw the qudrilterl WXYZ. ) The otrtor lis tht WXYZ is squre. Is he orret? ) Use ogruet trigles to epli our swer i prt ). Z L W O N Y X OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 102 Geoetr 8-12
To prove tht two polgos re ogruet, igie plig the oe o top of the other, so tht the th etl. If the polgos hve equl orrespodig sides d equl orrespodig gles, the re ogruet. ple: = G, =, =, = H = G, =, = H, = HG So GH. G H 4. The equl sides re rked o the qudrilterls JKLM d NOPQ. lso, J = N d L = P. ) Mrk the equl gles o the digr. ) rw lie segets KM d OQ. J K ) JKM the ogruee rule. This es JKM =. Lel oth these gles w. M L This lso es JMK =. Lel oth these gles. d) KLM the ogruee rule. This es LKM =. Lel oth these gles. N O This lso es LMK =. Lel oth these gles z. e) press the esure of JKL d the gle equl to it i NOPQ usig w,,, z. JKL = = f) press the esure of JML d the gle equl to it i NOPQ usig w,,, z. JML = = g) Write the ogruee stteet for the qudrilterls. Q P OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 5. re the qudrilterls ogruet? Use ruler d protrtor to hek. If es, write the pirs of orrespodig equl sides d orrespodig equl gles. The write the ogruee stteet. ) Y ) ous X W Z W I prllelogrs d PQRS, we hve = QR, = PQ, d = Q. Jke thiks the prllelogrs re ogruet. Is tht orret? pli. H G I O Geoetr 8-12 103
G8-13 terior gles of Trigle eterior gle is fored etedig side of trigle. gle is eterior gle. 1. ) Mesure the gles with protrtor d fill i the tle. i) ii) iii) ) press the esure of usig the esures of d. eterior gle of trigle is equl to the su of the two opposite gles i the trigle ( = + ). ples: = 88 + 45 = 133 88 + 90 = 150 45 = 150 90 = 60 150 2. id the issig gle. Show our work. ) ) ) 110 150 45 48 = = = d) e) f) 53 25 142 53 105 53 z OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 104 Geoetr 8-13
RMINR: gles i trigle dd to 180. Two gles tht oie to ke stright lie dd to 180. + = 180. 3. id the issig gles. Show our work. ) ) ) 95 30 50 110 =, = =, = =, = 4. id the issig gles. ) z ) ) z 35 45 110 138 60 = = = = = = z = z = d) z e) f) 76 130 88 47 48 z z 135 30 30 = = = OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION = = = z = z = z = 5. Prove tht = + usig the su of the gles i trigle d suppleetr gles (gles tht dd to 180 ). ) Wht do ou kow out the esures of,, d? pli. ) Wht do ou kow out the esures of d? pli. ) Write two epressios for the esure of usig our swers i prts ) d ). d) Use the epressios i prt ) to epli wh = +. Geoetr 8-13 105
G8-14 orrespodig gles d Prllel Lies orrespodig gles rete ptter like i the letter show: These pirs of gles re orrespodig gles for lies d : 4 8 3 7 1 5 2 6 1. List the orrespodig gles. ) ) 1 2 3 4 5 6 7 8 w z t s u v 1 d, 2 d w d, d 3 d, 4 d d, z d 2. Mesure the orrespodig gles d. re the equl or ot equl? ) ) =, = =, = The orrespodig gles re. The orrespodig gles re. ) d) =, = =, = The orrespodig gles re. The orrespodig gles re. e) f) =, = =, = The orrespodig gles re. The orrespodig gles re. OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 106 Geoetr 8-14
Whe the lies re prllel, orrespodig gles re equl. 3. List the orrespodig equl gles. ) ) ) e 1 2 3 4 d 5 6 7 8 f 1 2 3 4 5 6 7 8 14 10 11 15 13 9 12 16 1 = 5, 2 = =, = 3 =, 4 = 3 =, 4 = e = 5 =, 10 = 4. Give the esure of the orrespodig gles. ) ) ) 78.86 115 81 = = = d) e) f) 79 64 116 105 101 85 OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION Geoetr 8-14 =, = =, = =, = Whe the orrespodig gles re equl, the lies re prllel. 5. id usig suppleetr gles. irle the gle orrespodig to. The drw rrows to show if the lies re prllel. ) ) ) 95 85 81 101 85 85 = = = 107
RMINR: Vertil gles re equl. 6. or eh pir of lies tht do ot iterset i the piture rk pir of orrespodig gles d fid their esures. rk prllel lies. ) ) ) 115 115 59 115 58 115 d) e) f) 54.31 89 98 82 125.79 89 89 You use sll letters to lel lies, too. 7. Lies d re prllel. Use wht ou kow out orrespodig d vertil gles to epli wh + = 180. o-iterior gles rete ptter like i the letter : gles 3 d 5, d gles 4 d 6, re o-iterior gles. These gles re soeties lled se-side iterior gles. If the lies d re prllel, o-iterior gles dd to 180. If the o-iterior gles dd to 180, the lies re prllel. 8. ll gles i the qudrilterl re right gles. ) Idetif pir of o-iterior gles for lies d. Idetif pir of o-iterior gles for lies d. ) Use the gle esures of o-iterior gles to epli tht these pirs of sides re prllel. ) I hve just prove tht retgle is. 1 2 3 4 5 6 7 8 OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 108 Geoetr 8-14
G8-15 lterte gles d Prllel Lies lterte gles or lterte iterior gles ke ptter like the letter Z: 3 d 6 re lterte gles for prllel lies d. 4 d 5 re lterte gles for lies p d r. k 3 6 k 4 5 p r 1. List the pirs of lterte gles. ) ) ) 1 2 3 4 8 7 6 5 8 5 6 7 4 1 2 3 1 2 3 4 12 11 10 9 2 d 3 d 2 d 3 d 4 d 11 d 2. Mesure the lterte gles d. re the equl or ot equl? ) ) =, = =, = The lterte gles re. The lterte gles re. ) d) OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION =, = =, = The lterte gles re. The lterte gles re. 3. ill i the lks to prove tht lterte gles for prllel lies re equl. 3 d re lterte gles. k 6 d 7 re gles, so 6 =. 7 8 Lies d re. 1 2 3 4 3 d 7 re gles, so 3 = 7. 5 6 This es 3 =. Geoetr 8-15 109
4. id the issig lterte gles. ) ) ) 48 109.8 = = = d) e) f) 97 89 72 92 68 88 =, = =, = =, = 5. id the issig lterte, orrespodig, suppleetr, or vertil gles. ) ) ) 88 67 113 =, = =, = =, = d) e) 88.1 f) 97 72 102 =, = =, = =, = If the lterte gles re equl, the lies re prllel. 6. id usig suppleetr or vertil gles. irle the gle tht is lterte to. The deide if the lies or rs re prllel. ) ) ) 43 127 55 55 = = = prllel ot prllel prllel ot prllel prllel ot prllel 105 OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION 110 Geoetr 8-15
7. ill i the issig gles. Use rrows to rk prllel lies if the re ot rked lred. ) ) ) 80.3 65 106 125 74 99.7 81.3 d g d k =, = =, = =, = p 8. ill i ll the issig gles. eide whih lies or rs re prllel. ) ) g ) p 65 115 65 f 129 d 117.2 63.8 k 51 e 9. Shro wts to prove (usig logi) tht the su of the gles i trigle is 180. ) Shro drws trigle d leled the gle esures,, d. Write equtio usig,, tht she eeds to prove. ) Shro drws lie prllel to through. Lel the lterte gles o the digr with their esures. OPYRIGHT 2015 JUMP MTH: NOT TO OPI. ITION ) Whih gles o the digr ke stright gle? Wht do their esures dd to? Write equtio. d) id ou prove wht ou eeded to prove? ous ) id the esure of N. pli how ou kow. ) If ou eted the lies M d N, will the iterset? pli. M N Geoetr 8-15 111