Proving Lines Parallel
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1 rovig Lies arallel ecogize agle coditios that occur with parallel lies. rove that two lies are parallel based o give agle relatioships. do you kow that the sides of a parkig space are parallel? Have you ever bee i a tall buildig ad looked dow at a parkig lot? The parkig lot is full of lie segets that appear to be parallel. The workers who pait these lies ust be certai that they are parallel. IDETIFY LLEL LIE Whe each stripe of a parkig space itersects the ceter lie, the agles fored are correspodig agles. If the lies are parallel, we kow that the correspodig agles are cogruet. Coversely, if the correspodig agles are cogruet, the the lies ust be parallel. ostulate. If two lies i a plae are cut by a trasversal so that correspodig agles are cogruet, the the lies are parallel. bbreviatio: If corr. are, the lies are. Eaples: If 5, 6,, or 8, the ostulate. justifies the costructio of parallel lies. arallel Lie Through a oit ot o Lie tudy Tip Look Back To review copyig agles, see Lesso -. Use a straightedge to draw a lie. Label two poits o the lie as ad. Draw a poit that is ot o. Draw. Copy so that is the verte of the ew agle. Label the itersectio poits ad. Draw. Because by costructio ad they are correspodig agles,. Lesso -5 rovig Lies arallel 5 David ailors/cobi
2 The costructio establishes that there is at least oe lie through that is parallel to. I 95, cottish physicist ad atheaticia Joh layfair provided the oder versio of Euclid s arallel ostulate, which states there is eactly oe lie parallel to a lie through a give poit ot o the lie. ostulate.5 arallel ostulate If give a lie ad a poit ot o the lie, the there eists eactly oe lie through the poit that is parallel to the give lie. arallel lies with a trasversal create ay pairs of cogruet agles. Coversely, those pairs of cogruet agles ca deterie whether a pair of lies is parallel. Theores Eaples.5 If two lies i a plae are cut by a trasversal so that If 8 or if a pair of alterate eterior agles is cogruet, the, the. the two lies are parallel. bbreviatio: If alt. et. are, the lies are..6 If two lies i a plae are cut by a trasversal so that If 5 80 a pair of cosecutive iterior agles is suppleetary, or if 6 the the lies are parallel. 80, the. bbreviatio: If cos. it. are suppl., the lies are.. If two lies i a plae are cut by a trasversal so that a If 6 or if pair of alterate iterior agles is cogruet, the the 5, the. lies are parallel. bbreviatio: If alt. it. are, the lies are..8 I a plae, if two lies are perpedicular to the sae lie, If ad, the they are parallel. the. bbreviatio: If lies are to the sae lie, the lies are. rovig Lies arallel Eaple Idetify arallel Lies I the figure, BG bisects BH. Deterie which lies, if ay, are parallel. The su of the agle easures i a triagle ust be 80, so BDF 80 (5 65) or 0. ice BDF ad BGH have the sae easure, they are cogruet. Cogruet correspodig agles idicate parallel lies. o, DF GH. BD DBF, because BG bisects BH. o, BD 5. BD ad BDF are alterate iterior agles, but they have differet easures so they are ot cogruet. Thus, B is ot parallel to DF or GH. G D 0 B 5 65 F H 5 Chapter arallel ad erpedicular Lies
3 gle relatioships ca be used to solve probles ivolvig ukow values. Eaple olve robles with arallel Lies LGEB Fid ad U so that. Eplore Fro the figure, you kow that U 8 ad TV 9. You also kow that U ad TV are correspodig agles. (8 ) U (9 ) T V la For lie to be parallel to lie, the correspodig agles ust be cogruet. o, U TV. ubstitute the give agle easures ito this equatio ad solve for. Oce you kow the value of, use substitutio to fid U. olve U TV Correspodig agles 8 9 ubstitutio ubtract 8 fro each side. 5 dd to each side. ow use the value of to fid U. U 8 Origial equatio 8(5) 5 iplify. Eaie Verify the agle easure by usig the value of to fid TV. That is, 9 9(5) or. ice U TV, U TV ad. tudy Tip rovig Lies arallel Whe provig lies parallel, be sure to check for cogruet correspodig agles, alterate iterior agles, alterate eterior agles, or suppleetary cosecutive iterior agles. OVE LIE LLEL The agle pair relatioships fored by a trasversal ca be used to prove that two lies are parallel. Eaple Give: rove: roof: r s 5 6 rove Lies arallel 5 6 tateets easos. r s, 5 6. Give. ad 5 are suppleetary.. Cosecutive Iterior gle Theore Defiitio of suppleetary agles Defiitio of cogruet agles ubstitutio roperty () 6. ad 6 are suppleetary. 6. Defiitio of suppleetary agles.. If cos. it. are suppl., the lies are. r s Lesso -5 rovig Lies arallel 5
4 I Lesso -, you leared that parallel lies have the sae slope. You ca use the slopes of lies to prove that lies are parallel. Eaple Deterie whether g f. lope ad arallel Lies slope of f: 0 or 6 0 slope of g: or 0 () ice the slopes are the sae, g f. g y (0, ) (6, ) (, 0) (, 0) O f Cocept Check. uarize five differet ethods to prove that two lies are parallel.. Fid a coutereaple for the followig stateet. If lies ad are cut by trasversal t so that cosecutive iterior agles are cogruet, the lies ad are parallel ad t is perpedicular to both lies.. OE EDED Describe two situatios i your ow life i which you ecouter parallel lies. How could you verify that the lies are parallel? Guided ractice Give the followig iforatio, deterie which lies, if ay, are parallel. tate the postulate or theore that justifies your aswer Fid so that (5 90) (9 5) 6 5 ( ) p q ( 9) 0. OOF Write a two-colu proof of Theore.5.. Deterie whether p q. C(, ) 6 8 O 6 B(0, ) (, ) 6 y D(6, ) p q pplicatio. HYIC The Hubble Telescope gathers parallel light rays ad directs the to a cetral focal poit. Use a protractor to easure several of the agles show i the diagra. re the lies parallel? Eplai how you kow. 5 Chapter arallel ad erpedicular Lies
5 ractice ad pply For Eercises 6 5, 8 9 ee Eaples Etra ractice ee page 60. Give the followig iforatio, deterie which lies, if ay, are parallel. tate the postulate or theore that justifies your aswer.. 8 a b EF BFG 8. EB DBC 9. EFB CBF 0. GFD CBD E 6 5 B C D F G. HLK JL. L L. L L 80. H, JT K H L J T Latitude lies are parallel, ad logitude lies appear parallel i certai locatios o Earth. Visit co/webquest to cotiue work o your Webuest project. 5. OOF Copy ad coplete the proof of Theore.8. Give: t t rove: roof: tateets easos. t, t.?. ad are right agles..?..?..? t Fid so that (9 ) 0 (8 ) (9 ) ( ) ( 5) ( 00) (5 90) ( 9). OOF Write a two-colu proof of Theore.6.. OOF Write a paragraph proof of Theore.. (8 ) ( 8) Lesso -5 rovig Lies arallel 55
6 OOF Write a two-colu proof for each of the followig.. Give: 5. Give: D CD rove: T UV rove: BC CD V C D W B T U 6. Give: J K. Give: ad are suppleetary. rove: K L rove: J K L Deterie whether each pair of lies is parallel. Eplai why or why ot. 8. y 9. B(, ) (, ) D(, ) y D(0,.5) C(.5,.8) O C(0, 0) (, 0.5) O B(,.5) Joh layfair I 95, Joh layfair published his versio of Euclid s Eleets. I his editio, layfair stadardized the otatio used for poits ad figures ad itroduced algebraic otatio for use i proofs. ource: athworld.wolfra.co 0. HOE IOVEET To build a fece, Ji positioed the fece posts ad the placed a board at a agle betwee the fece posts. s he placed each picket, he easured the agle that the picket ade with the. Why does this esure that the pickets will be parallel?. FOOTBLL Whe stripig the practice football field, r. Hawkiso first paited the sidelies. et he arked off 0-yard icreets o oe sidelie. He the costructed lies perpedicular to the sidelies at each 0-yard ark. Why does this guaratee that the 0-yard lies will be parallel? pickets. CITICL THIKIG Whe deel was workig o a art project, he drew a four-sided figure with two pairs of opposite parallel sides. He oticed soe patters relatig to the agles i the figure. List as ay patters as you ca about a -sided figure with two pairs of opposite parallel sides. fece posts board 56 Chapter arallel ad erpedicular Lies Brow Brothers. EECH Use the Iteret or other resource to fid atheaticias like Joh layfair who discovered ew cocepts ad proved ew theores related to parallel lies. Briefly describe their discoveries.
7 tadardized Test ractice. WITIG I TH swer the questio that was posed at the begiig of the lesso. How do you kow that the sides of a parkig space are parallel? Iclude the followig i your aswer: a copariso of the agles at which the lies forig the edges of a parkig space strike the ceterlie, ad a descriptio of the type of parkig spaces that for cogruet cosecutive iterior agles. 5. I the figure, lie is parallel to lie. Lie itersects both ad. Which of the followig lists icludes all of the agles that are suppleetary to? C agles,, ad B agles,, 6, ad agles, 5, ad 8 D agles,,, ad LGEB Kedra has at least oe quarter, oe die, oe ickel, ad oe pey. If she has three ties as ay peies as ickels, the sae uber of ickels as dies, ad twice as ay dies as quarters, the what is the least aout of oey she could have? $0. B $0.8 C $0.58 D $0.6 aitai Your kills ied eview Write a equatio i slope-itercept for for the lie that satisfies the give coditios. (Lesso -). 0., y-itercept is 6 8., cotais (, 5) 9. cotais (5, ) ad (, ) 50. perpedicular to y, cotais (, ) Fid the slope of each lie. (Lesso -) 5. BD 5. CD 5. B 5. EO 55. ay lie parallel to DE 56. ay lie perpedicular to BD y E(, ) B(0, ) O (, ) C(, ) D(, ) Gettig eady for the et Lesso Costruct a truth table for each copoud stateet. (Lesso -) 5. p ad q 58. p or q 59. p q 60. p q 6. CETY carpeter ust cut two pieces of wood at agles so that they fit together to for the corer of a picture frae. What type of agles ust he use to ake sure that a corer results? (Lesso -5) EEUIITE KILL Use the Distace Forula to fid the distace betwee each pair of poits. (To review the Distace Forula, see Lesso -.) 6. (, ), (, 9) 6. (8, 0), (, ) 6. (6, ), (8, ) Lesso -5 rovig Lies arallel 5
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