Parallel and Perpendicular Lines

Transcription

1 Parallel ad Perpedicular Lies. Pairs of Lies ad gles. Parallel Lies ad Trasversals. Proofs with Parallel Lies. Proofs with Perpedicular Lies.5 Equatios of Parallel ad Perpedicular Lies ike Path (p. ) rosswalk (p. 5) Kiteboardig (p. ) SEE the ig Idea Gastics (p. 0) Tree House (p. 0)

2 Maitaiig Matheatical Proficiec Fidig the Slope of a Lie Exaple Fid the slope of the lie show. Let ( x, ) = (, ) ad ( x, ) = (, 0). slope = x x Write forula for slope. (, 0) = 0 ( ) ( ) = Substitute. Siplif. (, ) x Fid the slope of the lie.. (, ). (, ). (, ) x (, ) x x (, ) (, ) Writig Equatios of Lies Exaple Write a equatio of the lie that passes through the poit (, 5) ad has a slope of. = x + b Write the slope-itercept for. 5 = ( ) + b Substitute for, for x, ad 5 for. 5 = + b Siplif. 8 = b Solve for b. So, a equatio is = x + 8. Write a equatio of the lie that passes through the give poit ad has the give slope.. (, ); = 5. (, 8); =. (, 5); = 7. (, ); = 8. ( 8, 5); = 9. (0, 9); = 0. STRT RESONING Wh does a horizotal lie have a slope of 0, but a vertical lie has a udefied slope? aic Solutios available at igideasmath.co

3 Matheatical Practices Matheaticall profi ciet studets use techological tools to explore cocepts. haracteristics of Lies i a oordiate Plae ore ocept Lies i a oordiate Plae. I a coordiate plae, two lies are parallel if ad ol if the are both vertical lies or the both have the sae slope.. I a coordiate plae, two lies are perpedicular if ad ol if oe is vertical ad the other is horizotal or the slopes of the lies are egative reciprocals of each other.. I a coordiate plae, two lies are coicidet if ad ol if their equatios are equivalet. lassifig Pairs of Lies Here are soe exaples of pairs of lies i a coordiate plae. a. x + = These lies are ot parallel b. x + = These lies are coicidet x = or perpedicular. The x + = because their equatios itersect at (, ). are equivalet. c. x + = These lies are parallel. d. x + = These lies are perpedicular. x + = Each lie has a slope x = The have slopes of = of =. ad =. Moitorig Progress Use a graphig calculator to graph the pair of lies. Use a square viewig widow. lassif the lies as parallel, perpedicular, coicidet, or operpedicular itersectig lies. Justif our aswer.. x + =. x + =. x + =. x + = x = x + = x + = x = hapter Parallel ad Perpedicular Lies

4 . Pairs of Lies ad gles Essetial Questio What does it ea whe two lies are parallel, itersectig, coicidet, or skew? Poits of Itersectio Work with a parter. Write the uber of poits of itersectio of each pair of coplaar lies. a. parallel lies b. itersectig lies c. coicidet lies lassifig Pairs of Lies Work with a parter. The figure shows a right rectagular pris. ll its agles are right agles. lassif each of the followig pairs of lies as parallel, itersectig, coicidet, or skew. Justif our aswers. (Two lies are skew lies whe the do ot itersect ad are ot coplaar.) E F I H G ONSTRUTING VILE RGUMENTS To be proficiet i ath, ou eed to uderstad ad use stated assuptios, defiitios, ad previousl established results. Pair of Lies lassificatio Reaso a. ad b. ad c. EI ad IH d. F ad EH e. EF ad G f. ad GH Idetifig Pairs of gles Work with a parter. I the figure, two parallel lies are itersected b a third lie called a trasversal. a. Idetif all the pairs of vertical agles. Explai our reasoig. b. Idetif all the liear pairs of agles. Explai our reasoig ouicate Your swer. What does it ea whe two lies are parallel, itersectig, coicidet, or skew? 5. I Exploratio, fid three ore pairs of lies that are differet fro those give. lassif the pairs of lies as parallel, itersectig, coicidet, or skew. Justif our aswers. Sectio. Pairs of Lies ad gles 5

5 . Lesso What You Will Lear ore Vocabular parallel lies, p. skew lies, p. parallel plaes, p. trasversal, p. 8 correspodig agles, p. 8 alterate iterior agles, p. 8 alterate exterior agles, p. 8 cosecutive iterior agles, p. 8 Previous perpedicular lies Idetif lies ad plaes. Idetif parallel ad perpedicular lies. Idetif pairs of agles fored b trasversals. Idetifig Lies ad Plaes ore ocept Parallel Lies, Skew Lies, ad Parallel Plaes Two lies that do ot itersect are either parallel lies or skew lies. Two lies are parallel lies whe the do ot itersect ad are coplaar. Two lies are skew lies whe the do ot itersect ad are ot coplaar. lso, two plaes that do ot itersect are parallel plaes. k Lies ad are parallel lies ( ). Lies ad k are skew lies. T U Plaes T ad U are parallel plaes (T U ). Lies k ad are itersectig lies, ad there is a plae (ot show) cotaiig the. Sall directed arrows, as show i red o lies ad above, are used to show that lies are parallel. The sbol eas is parallel to, as i. Segets ad ras are parallel whe the lie i parallel lies. lie is parallel to a plae whe the lie is i a plae parallel to the give plae. I the diagra above, lie is parallel to plae U. Idetifig Lies ad Plaes REMEMER Recall that if two lies itersect to for a right agle, the the are perpedicular lies. Thik of each seget i the figure as part of a lie. Which lie(s) or plae(s) appear to fit the descriptio? a. lie(s) parallel to ad cotaiig poit b. lie(s) skew to ad cotaiig poit c. lie(s) perpedicular to ad cotaiig poit E F G H d. plae(s) parallel to plae EFG ad cotaiig poit SOLUTION a., HG, ad EF all appear parallel to, but ol cotais poit. b. oth G ad H appear skew to ad cotai poit. c.,, E, ad F all appear perpedicular to, but ol cotais poit. d. Plae appears parallel to plae EFG ad cotais poit. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. Look at the diagra i Exaple. Nae the lie(s) through poit F that appear skew to EH. hapter Parallel ad Perpedicular Lies

6 Nash Rd Idetifig Parallel ad Perpedicular Lies Two distict lies i the sae plae either are parallel, like lie ad lie, or itersect i a poit, like lie j ad lie. Through a poit ot o a lie, there are ifiitel a lies. Exactl oe of these lies is parallel to the give lie, ad exactl oe of the is perpedicular to the give lie. For exaple, lie k is the lie through poit P perpedicular to lie, ad lie is the lie through poit P parallel to lie. j k P Postulates Postulate. Parallel Postulate If there is a lie ad a poit ot o the lie, the there is exactl oe lie through the poit parallel to the give lie. There is exactl oe lie through P parallel to. Postulate. Perpedicular Postulate If there is a lie ad a poit ot o the lie, the there is exactl oe lie through the poit perpedicular to the give lie. There is exactl oe lie through P perpedicular to. P P Idetifig Parallel ad Perpedicular Lies The give lie arkigs show how the roads i a tow are related to oe aother. a. Nae a pair of parallel lies. b. Nae a pair of perpedicular lies. c. Is FE? Explai. SOLUTION a. M FE b. M F c. FE is ot parallel to, because M is parallel to FE, ad b the Parallel Postulate, there is exactl oe lie parallel to FE through M. 5 8 Seaw a a Tr Oliver Street Oliver Street T ail Pa ae v e Pae v WalckRd 9thv e E M e Pae v F Wheatfield St e Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. I Exaple, ca ou use the Perpedicular Postulate to show that is ot perpedicular to F? Explai wh or wh ot. Sectio. Pairs of Lies ad gles 7

7 Idetifig Pairs of gles trasversal is a lie that itersects two or ore coplaar lies at differet poits. ore ocept gles Fored b Trasversals t 5 t Two agles are correspodig agles whe the have correspodig positios. For exaple, ad are above the lies ad to the right of the trasversal t. Two agles are alterate iterior agles whe the lie betwee the two lies ad o opposite sides of the trasversal t. 8 t 5 t Two agles are alterate exterior agles whe the lie outside the two lies ad o opposite sides of the trasversal t. Two agles are cosecutive iterior agles whe the lie betwee the two lies ad o the sae side of the trasversal t. Idetifig Pairs of gles Idetif all pairs of agles of the give tpe. a. correspodig b. alterate iterior c. alterate exterior d. cosecutive iterior SOLUTION a. l ad 5 b. ad 7 c. l ad 8 d. ad 5 ad ad 5 ad ad 7 ad 7 ad 8 Moitorig Progress lassif the pair of ubered agles. Help i Eglish ad Spaish at igideasmath.co hapter Parallel ad Perpedicular Lies

8 . Exercises aic Solutios available at igideasmath.co Vocabular ad ore ocept heck. OMPLETE THE SENTENE Two lies that do ot itersect ad are also ot parallel are lies.. WHIH ONE OESN T ELONG? Which agle pair does ot belog with the other three? Explai our reasoig. ad ad 5 ad 8 ad Moitorig Progress ad Modelig with Matheatics I Exercises, thik of each seget i the diagra as part of a lie. ll the agles are right agles. Which lie(s) or plae(s) cotai poit ad appear to fit the descriptio? (See Exaple.)? Explai. 9. Is PN KM? Explai. 0. Is PR NP I Exercises, idetif all pairs of agles of the give tpe. (See Exaple.) G F H E. correspodig. lie(s) parallel to. alterate iterior. lie(s) perpedicular to. alterate exterior 5. lie(s) skew to. cosecutive iterior. plae(s) parallel to plae H USING STRUTURE I Exercises 5 8, classif the I Exercises 7 0, use the diagra. (See Exaple.) agle pair as correspodig, alterate iterior, alterate exterior, or cosecutive iterior agles. N M L K Q S P R 5 7. Nae a pair of parallel lies ad. ad 8. Nae a pair of perpedicular lies. 7. ad 8. ad Sectio. hs_geo_pe_00.idd 9 Pairs of Lies ad gles 9 /9/5 9: M

9 ERROR NLYSIS I Exercises 9 ad 0, describe ad correct the error i the coditioal stateet about lies If two lies do ot itersect, the the are parallel. If there is a lie ad a poit ot o the lie, the there is exactl oe lie through the poit that itersects the give lie.. HOW O YOU SEE IT? Thik of each seget i the figure as part of a lie. a. Which lies are parallel to NQ? b. Which lies itersect NQ? c. Which lies are skew to NQ? d. Should ou have aed all the lies o the cube i parts (a) (c) except NQ? Explai. N Q K R M P L S. MOELING WITH MTHEMTIS Use the photo to decide whether the stateet is true or false. Explai our reasoig. I Exercises 5 8, cop ad coplete the stateet. List all possible correct aswers. E G F J H 5. G ad are correspodig agles.. G ad are cosecutive iterior agles. a. The plae cotaiig the floor of the tree house is parallel to the groud. b. The lies cotaiig the railigs of the staircase, such as, are skew to all lies i the plae cotaiig the groud. c. ll the lies cotaiig the balusters, such as, are perpedicular to the plae cotaiig the floor of the tree house. 7. FJ ad are alterate iterior agles. 8. F ad are alterate exterior agles. 9. MKING N RGUMENT Your fried clais the ueve parallel bars i gastics are ot reall parallel. She sas oe is higher tha the other, so the caot be i the sae plae. Is she correct? Explai.. THOUGHT PROVOKING If two lies are itersected b a third lie, is the third lie ecessaril a trasversal? Justif our aswer with a diagra.. MTHEMTIL ONNETIONS Two lies are cut b a trasversal. Is it possible for all eight agles fored to have the sae easure? Explai our reasoig. Maitaiig Matheatical Proficiec Use the diagra to fid the easures of all the agles. (Sectio.) 0. = 7. = 59 Reviewig what ou leared i previous grades ad lessos 0 hapter Parallel ad Perpedicular Lies

10 . Parallel Lies ad Trasversals Essetial Questio Whe two parallel lies are cut b a trasversal, which of the resultig pairs of agles are cogruet? Explorig Parallel Lies Work with a parter. Use daic geoetr software to draw two parallel lies. raw a third lie that itersects both parallel lies. Fid the easures of the eight agles that are fored. What ca ou coclude? E F TTENING TO PREISION To be proficiet i ath, ou eed to couicate precisel with others. 0 0 Writig ojectures Work with a parter. Use the results of Exploratio to write cojectures about the followig pairs of agles fored b two parallel lies ad a trasversal. 5 a. correspodig agles b. alterate iterior agles c. alterate exterior agles d. cosecutive iterior agles ouicate Your swer. Whe two parallel lies are cut b a trasversal, which of the resultig pairs of agles are cogruet?. I Exploratio, = 80. Fid the other agle easures. Sectio. Parallel Lies ad Trasversals

11 . Lesso What You Will Lear ore Vocabular Previous correspodig agles parallel lies suppleetar agles vertical agles t p q Use properties of parallel lies. Prove theores about parallel lies. Solve real-life probles. Usig Properties of Parallel Lies Theores Theore. orrespodig gles Theore If two parallel lies are cut b a trasversal, the the pairs of correspodig agles are cogruet. Exaples I the diagra at the left, ad 7. Proof Ex., p. 80 Theore. lterate Iterior gles Theore If two parallel lies are cut b a trasversal, the the pairs of alterate iterior agles are cogruet. Exaples I the diagra at the left, ad 5. Proof Exaple, p. Theore. lterate Exterior gles Theore If two parallel lies are cut b a trasversal, the the pairs of alterate exterior agles are cogruet. Exaples I the diagra at the left, 8 ad 7. Proof Ex. 5, p. NOTHER WY There are a was to solve Exaple. other wa is to use the orrespodig gles Theore to fid 5 ad the use the Vertical gles ogruece Theore (Theore.) to fid ad 8. Theore. osecutive Iterior gles Theore If two parallel lies are cut b a trasversal, the the pairs of cosecutive iterior agles are suppleetar. Exaples I the diagra at the left, ad 5 are suppleetar, ad ad are suppleetar. Proof Ex., p. Idetifig gles The easures of three of the ubered agles are 0. Idetif the agles. Explai our reasoig. SOLUTION 0º the lterate Exterior gles Theore, 8 = 0. 5 ad 8 are vertical agles. Usig the Vertical gles ogruece Theore (Theore.), 5 = 0. 5 ad are alterate iterior agles. the lterate Iterior gles Theore, = 0. hapter Parallel ad Perpedicular Lies So, the three agles that each have a easure of 0 are, 5, ad 8.

12 Usig Properties of Parallel Lies Fid the value of x. 5 (x + 5) a b heck 5 + (x + 5) = (0 + 5) =? = 80 SOLUTION the Vertical gles ogruece Theore (Theore.), = 5. Lies a ad b are parallel, so ou ca use the theores about parallel lies. + (x + 5) = 80 osecutive Iterior gles Theore 5 + (x + 5) = 80 Substitute 5 for. x + 0 = 80 x = 0 So, the value of x is 0. obie like ters. Subtract 0 fro each side. Usig Properties of Parallel Lies Fid the value of x. c (7x + 9) d heck = (7x + 9) =? 7(5) + 9 = SOLUTION the Liear Pair Postulate (Postulate.8), = 80 =. Lies c ad d are parallel, so ou ca use the theores about parallel lies. = (7x + 9) lterate Exterior gles Theore = (7x + 9) Substitute for. 5 = 7x Subtract 9 fro each side. 5 = x ivide each side b 7. So, the value of x is 5. Moitorig Progress Use the diagra. Help i Eglish ad Spaish at igideasmath.co. Give = 05, fid, 5, ad 8. Tell which theore ou use i each case.. Give = 8 ad 8 = (x + ), what is the value of x? Show our steps Sectio. Parallel Lies ad Trasversals

13 Provig Theores about Parallel Lies Provig the lterate Iterior gles Theore Prove that if two parallel lies are cut b a trasversal, the the pairs of alterate iterior agles are cogruet. STUY TIP efore ou write a proof, idetif the Give ad Prove stateets for the situatio described or for a diagra ou draw. SOLUTION raw a diagra. Label a pair of alterate iterior agles as ad. You are lookig for a agle that is related to both ad. Notice that oe agle is a vertical agle with ad a correspodig agle with. Label it. Give p q t p q Prove STTEMENTS RESONS. p q. Give.. orrespodig gles Theore.. Vertical gles ogruece Theore (Theore.).. Trasitive Propert of ogruece (Theore.) Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. I the proof i Exaple, if ou use the third stateet before the secod stateet, could ou still prove the theore? Explai. Solvig Real-Life Probles Solvig a Real-life Proble Whe sulight eters a drop of rai, differet colors of light leave the drop at differet agles. This process is what akes a raibow. For violet light, = 0. What is? How do ou kow? SOLUTION ecause the Su s ras are parallel, ad are alterate iterior agles. the lterate Iterior gles Theore,. So, b the defiitio of cogruet agles, = = 0. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. WHT IF? I Exaple 5, ellow light leaves a drop at a agle of =. What is? How do ou kow? hapter Parallel ad Perpedicular Lies

14 . Exercises aic Solutios available at igideasmath.co Vocabular ad ore ocept heck. WRITING How are the lterate Iterior gles Theore (Theore.) ad the lterate Exterior gles Theore (Theore.) alike? How are the differet?. WHIH ONE OESN T ELONG? Which pair of agle easures does ot belog with the other three? Explai. ad ad ad ad 5 5 Moitorig Progress ad Modelig with Matheatics I Exercises, fid ad. Tell which theore ou use i each case. (See Exaple.) (8x + ) I Exercises ad, fid,, ad. Explai our reasoig.. 80 I Exercises 7 0, fid the value of x. Show our steps. (See Exaples ad.) 7. 8 x 8. 7 (7x + ). 9.. ERROR NLYSIS escribe ad correct the error i the studet s reasoig. 5º 5 (x 7)º b the orrespodig gles Theore (Theore.). Sectio. Parallel Lies ad Trasversals 5

15 . HOW O YOU SEE IT? a. b. 9. RITIL THINKING Is it possible for cosecutive Use the diagra. iterior agles to be cogruet? Explai. ad Nae two pairs of cogruet agles whe are parallel. Explai our reasoig. Nae two pairs of suppleetar agles whe are parallel. Explai our reasoig. ad PROVING THEOREM I Exercises 5 ad, prove the theore. (See Exaple.) 5. lterate Exterior gles Theore (Th..) 0. THOUGHT PROVOKING The postulates ad theores i this book represet Euclidea geoetr. I spherical geoetr, all poits are poits o the surface of a sphere. lie is a circle o the sphere whose diaeter is equal to the diaeter of the sphere. I spherical geoetr, is it possible that a trasversal itersects two parallel lies? Explai our reasoig. MTHEMTIL ONNETIONS I Exercises ad, write ad solve a sste of liear equatios to fid the values of x ad.. (x 0).. osecutive Iterior gles Theore (Th..) 7. PROLEM SOLVING group of capers tie up their food betwee two parallel trees, as show. The rope is pulled taut, forig a straight lie. Fid. Explai our reasoig. (See Exaple 5.) x (x + ) ( + ) 5x. MKING N RGUMENT urig a gae of pool, our fried clais to be able to ake the shot show i the diagra b hittig the cue ball so that = 5. Is our fried correct? Explai our reasoig RWING ONLUSIONS You are desigig a box like the oe show. 5. RESONING I the diagra, 5 ad SE bisects RSF. Fid. Explai our reasoig. E a. The easure of is 70. Fid ad. F b. Explai wh is a straight agle. c. If is 0, will still be a straight agle? Will the opeig of the box be ore steep or less steep? Explai. Maitaiig Matheatical Proficiec T 5 S R Reviewig what ou leared i previous grades ad lessos Write the coverse of the coditioal stateet. ecide whether it is true or false. (Sectio.) 5. If two agles are vertical agles, the the are cogruet.. If ou go to the zoo, the ou will see a tiger. 7. If two agles for a liear pair, the the are suppleetar. 8. If it is war outside, the we will go to the park. hapter hs_geo_pe_00.idd Parallel ad Perpedicular Lies /9/5 9: M

16 . Proofs with Parallel Lies Essetial Questio For which of the theores ivolvig parallel lies ad trasversals is the coverse true? Explorig overses ONSTRUTING VILE RGUMENTS To be proficiet i ath, ou eed to ake cojectures ad build a logical progressio of stateets to explore the truth of our cojectures. Work with a parter. Write the coverse of each coditioal stateet. raw a diagra to represet the coverse. eterie whether the coverse is true. Justif our coclusio. a. orrespodig gles Theore (Theore.) If two parallel lies are cut b a trasversal, the the pairs of correspodig agles are cogruet. overse b. lterate Iterior gles Theore (Theore.) If two parallel lies are cut b a trasversal, the the pairs of alterate iterior agles are cogruet. overse c. lterate Exterior gles Theore (Theore.) If two parallel lies are cut b a trasversal, the the pairs of alterate exterior agles are cogruet. overse d. osecutive Iterior gles Theore (Theore.) If two parallel lies are cut b a trasversal, the the pairs of cosecutive iterior agles are suppleetar. overse ouicate Your swer. For which of the theores ivolvig parallel lies ad trasversals is the coverse true?. I Exploratio, explai how ou would prove a of the theores that ou foud to be true. Sectio. Proofs with Parallel Lies 7

17 . Lesso What You Will Lear ore Vocabular Previous coverse parallel lies trasversal correspodig agles cogruet alterate iterior agles alterate exterior agles cosecutive iterior agles Use the orrespodig gles overse. ostruct parallel lies. Prove theores about parallel lies. Use the Trasitive Propert of Parallel Lies. Usig the orrespodig gles overse Theore.5 below is the coverse of the orrespodig gles Theore (Theore.). Siilarl, the other theores about agles fored whe parallel lies are cut b a trasversal have true coverses. Reeber that the coverse of a true coditioal stateet is ot ecessaril true, so ou ust prove each coverse of a theore. Theore Theore.5 orrespodig gles overse If two lies are cut b a trasversal so the correspodig agles are cogruet, the the lies are parallel. Proof Ex., p. 80 j k j k Usig the orrespodig gles overse Fid the value of x that akes. SOLUTION 5 (x + 5) Lies ad are parallel whe the arked correspodig agles are cogruet. (x + 5) = 5 Use the orrespodig gles overse to write a equatio. x = 0 Subtract 5 fro each side. x = 0 ivide each side b. So, lies ad are parallel whe x = 0. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. Is there eough iforatio i the diagra to coclude that? Explai Explai wh the orrespodig gles overse is the coverse of the orrespodig gles Theore (Theore.). 8 hapter Parallel ad Perpedicular Lies

18 ostructig Parallel Lies The orrespodig gles overse justifies the costructio of parallel lies, as show below. ostructig Parallel Lies Use a copass ad straightedge to costruct a lie through poit P that is parallel to lie. SOLUTION Step Step Step Step P Q P P Q P Q P Q raw a poit ad lie Start b drawig poit P ad lie. hoose a poit Q awhere o lie ad draw QP. raw arcs raw a arc with ceter Q that crosses QP ad lie. Label poits ad. Usig the sae copass settig, draw a arc with ceter P. Label poit. op agle raw a arc with radius ad ceter. Usig the sae copass settig, draw a arc with ceter. Label the itersectio. raw parallel lies raw P. This lie is parallel to lie. Theores Theore. lterate Iterior gles overse If two lies are cut b a trasversal so the alterate iterior agles are cogruet, the the lies are parallel. Proof Exaple, p. 0 5 j k j k Theore.7 lterate Exterior gles overse If two lies are cut b a trasversal so the alterate exterior agles are cogruet, the the lies are parallel. Proof Ex., p. Theore.8 osecutive Iterior gles overse If two lies are cut b a trasversal so the cosecutive iterior agles are suppleetar, the the lies are parallel. 5 8 j k j k j k Proof Ex., p. If ad 5 are suppleetar, the j k. Sectio. Proofs with Parallel Lies 9

19 Provig Theores about Parallel Lies Provig the lterate Iterior gles overse Prove that if two lies are cut b a trasversal so the alterate iterior agles are cogruet, the the lies are parallel. SOLUTION Give 5 5 g Prove g h h STTEMENTS RESONS. 5. Give.. Vertical gles ogruece Theore (Theore.). 5. Trasitive Propert of ogruece (Theore.). g h. orrespodig gles overse eteriig Whether Lies re Parallel r s I the diagra, r s ad is cogruet to. Prove p q. p q SOLUTION Look at the diagra to ake a pla. The diagra suggests that ou look at agles,, ad. lso, ou a fid it helpful to focus o oe pair of lies ad oe trasversal at a tie. Pla for Proof a. Look at ad. because r s. b. Look at ad. If, the p q. Pla for ctio a. It is give that r s, so b the orrespodig gles Theore (Theore.),. b. It is also give that. The b the Trasitive Propert of ogruece (Theore.). So, b the lterate Iterior gles overse, p q. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. If ou use the diagra below to prove the lterate Exterior gles overse, what Give ad Prove stateets would ou use? j 8 k. op ad coplete the followig paragraph proof of the lterate Iterior gles overse usig the diagra i Exaple. It is give that 5. the,. The b the Trasitive Propert of ogruece (Theore.),. So, b the, g h. 0 hapter Parallel ad Perpedicular Lies

20 Usig the Trasitive Propert of Parallel Lies Theore Theore.9 Trasitive Propert of Parallel Lies If two lies are parallel to the sae lie, p the the are parallel to each other. q r Proof Ex. 9, p. ; Ex. 8, p. If p q ad q r, the p r. Usig the Trasitive Propert of Parallel Lies The flag of the Uited States has alteratig red ad white stripes. Each stripe is parallel to the stripe iediatel below it. Explai wh the top stripe is parallel to the botto stripe. s s s s s 5 s s 7 s 8 s 9 s 0 s s s SOLUTION You ca ae the stripes fro top to botto as s l, s, s,..., s. Each stripe is parallel to the oe iediatel below it, so s s, s s, ad so o. The s s b the Trasitive Propert of Parallel Lies. Siilarl, because s s, it follows that s s. cotiuig this reasoig, s s. So, the top stripe is parallel to the botto stripe. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co 5. Each step is parallel to the step iediatel above it. The botto step is parallel to the groud. Explai wh the top step is parallel to the groud.. I the diagra below, p q ad q r. Fid 8. Explai our reasoig. s 5 p q 8 r Sectio. Proofs with Parallel Lies

21 . Exercises aic Solutios available at igideasmath.co Vocabular ad ore ocept heck. VOULRY Two lies are cut b a trasversal. Which agle pairs ust be cogruet for the lies to be parallel?. WRITING Use the theores fro Sectio. ad the coverses of those theores i this sectio to write three bicoditioal stateets about parallel lies ad trasversals. Moitorig Progress ad Modelig with Matheatics I Exercises 8, fid the value of x that akes. Explai our reasoig. (See Exaple.). 0 x. (x + 5) 5 I Exercises 8, decide whether there is eough iforatio to prove that. If so, state the theore ou would use. (See Exaple.).. r r (x 5). (80 x) x 5. r. r 7. x x 8. (x + 0) x 7. r s 8. r s I Exercises 9 ad 0, use a copass ad straightedge to costruct a lie through poit P that is parallel to lie. 9. P 0. P ERROR NLYSIS I Exercises 9 ad 0, describe ad correct the error i the reasoig. 9. x x a b c PROVING THEOREM I Exercises ad, prove the theore. (See Exaple.). lterate Exterior gles overse (Theore.7). osecutive Iterior gles overse (Theore.8) 0. oclusio: a b a b oclusio: a b hapter Parallel ad Perpedicular Lies

22 I Exercises, are ad F parallel? Explai our reasoig E F E F.. 7 E F E F 5. NLYZING RELTIONSHIPS The ap shows part of ever, olorado. Use the arkigs o the ap. re the ubered streets parallel to oe aother? Explai our reasoig. (See Exaple.) 8. RESONING Use the diagra. Which ras are parallel? Which ras are ot parallel? Explai our reasoig. F E 58º º H G º 59º 9. TTENING TO PREISION Use the diagra. Which theores allow ou to coclude that? Select all that appl. Explai our reasoig. E. 0th ve. orrespodig gles overse (Th..5) E. 9th ve. E. 8th ve. E. 7th ve. Peslvaia St. Pearl St. Washigto St. larkso St. Ogde St.. NLYZING RELTIONSHIPS Each rug of the ladder is parallel to the rug directl above it. Explai wh the top rug is parallel to the botto rug. owig St. Park ve. Frakli St. lterate Iterior gles overse (Th..) lterate Exterior gles overse (Th..7) osecutive Iterior gles overse (Th..8) 0. MOELING WITH MTHEMTIS Oe wa to build stairs is to attach triagular blocks to a agled support, as show. The sides of the agled support are parallel. If the support akes a agle with the floor, what ust be so the top of the step will be parallel to the floor? Explai our reasoig. triagular block 7. MOELING WITH MTHEMTIS The diagra of the cotrol bar of the kite shows the agles fored betwee the cotrol bar ad the kite lies. How do ou kow that is parallel to? STRT RESONING I the diagra, how a agles ust be give to deterie whether j k? Give four exaples that would allow ou to coclude that j k usig the theores fro this lesso. j k t Sectio. Proofs with Parallel Lies

23 . THOUGHT PROVOKING raw a diagra of at least two lies cut b at least oe trasversal. Mark our diagra so that it caot be prove that a lies are parallel. The explai how our diagra would eed to chage i order to prove that lies are parallel. 7. MKING N RGUMENT Your classate decided that based o the diagra. Is our classate correct? Explai our reasoig. PROOF I Exercises, write a proof.. Give = 5, = 5 Prove 8. HOW O YOU SEE IT? re the arkigs o the diagra eough to coclude that a lies are parallel? If so, which oes? If ot, what other iforatio is eeded?. Give ad are suppleetar. Prove p q r s 5. Give, Prove. Give a b, Prove c d c E d a b 9. PROVING THEOREM Use these steps to prove the Trasitive Propert of Parallel Lies Theore (Theore.9). a. op the diagra with the Trasitive Propert of Parallel Lies Theore o page. b. Write the Give ad Prove stateets. c. Use the properties of agles fored b parallel lies cut b a trasversal to prove the theore. 0. MTHEMTIL ONNETIONS Use the diagra. (x + ) (x + 5) ( + 7) ( 7) q a. Fid the value of x that akes p q. b. Fid the value of that akes r s. c. a r be parallel to s ad ca p be parallel to q at the sae tie? Explai our reasoig. r s p Maitaiig Matheatical Proficiec Use the istace Forula to fid the distace betwee the two poits. (Sectio.). (, ) ad (, 9). (, 7) ad (8, ). (5, ) ad (0, 8). (, ) ad (9, ) Reviewig what ou leared i previous grades ad lessos hapter Parallel ad Perpedicular Lies

24 .. What id You Lear? ore Vocabular parallel lies, p. skew lies, p. parallel plaes, p. trasversal, p. 8 correspodig agles, p. 8 alterate iterior agles, p. 8 alterate exterior agles, p. 8 cosecutive iterior agles, p. 8 ore ocepts Sectio. Parallel Lies, Skew Lies, ad Parallel Plaes, p. Postulate. Parallel Postulate, p. 7 Sectio. Theore. orrespodig gles Theore, p. Theore. lterate Iterior gles Theore, p. Sectio. Theore.5 orrespodig gles overse, p. 8 Theore. lterate Iterior gles overse, p. 9 Theore.7 lterate Exterior gles overse, p. 9 Postulate. Perpedicular Postulate, p. 7 gles Fored b Trasversals, p. 8 Theore. lterate Exterior gles Theore, p. Theore. osecutive Iterior gles Theore, p. Theore.8 osecutive Iterior gles overse, p. 9 Theore.9 Trasitive Propert of Parallel Lies, p. Matheatical Practices. raw the portio of the diagra that ou used to aswer Exercise o page 0.. I Exercise 0 o page, explai how ou started solvig the proble ad wh ou started that wa. Stud Skills alzig Your Errors Misreadig irectios What Happes: You icorrectl read or do ot uderstad directios. How to void This Error: Read the istructios for exercises at least twice ad ake sure ou uderstad what the ea. Make this a habit ad use it whe takig tests. 5

25 .. Quiz Thik of each seget i the diagra as part of a lie. Which lie(s) or plae(s) cotai poit G ad appear to fit the descriptio? (Sectio.). lie(s) parallel to EF. lie(s) skew to EF. lie(s) perpedicular to EF. plae(s) parallel to plae E Idetif all pairs of agles of the give tpe. (Sectio.) 5. cosecutive iterior E F G alterate iterior 7. correspodig 8. alterate exterior H Fid ad. Tell which theore ou use i each case. (Sectio.) ecide whether there is eough iforatio to prove that. If so, state the theore ou would use. (Sectio.)... 9 ad 5. ellular phoes use bars like the oes show to idicate how uch sigal stregth a phoe receives fro the earest service tower. Each bar is parallel to the bar directl ext to it. (Sectio.) a. Explai wh the tallest bar is parallel to the shortest bar. b. Iagie that the left side of each bar exteds ifiitel as a lie. If = 58, the what is?. The diagra shows lies fored o a teis court. (Sectio. ad Sectio.) k q p a. Idetif two pairs of parallel lies so that each pair is i a differet plae. b. Idetif two pairs of perpedicular lies. c. Idetif two pairs of skew lies. d. Prove that. hapter Parallel ad Perpedicular Lies

26 . Proofs with Perpedicular Lies Essetial Questio What cojectures ca ou ake about perpedicular lies? Writig ojectures Work with a parter. Fold a piece of paper i half twice. Label poits o the two creases, as show. a. Write a cojecture about ad. Justif our cojecture. b. Write a cojecture about O ad O. Justif our cojecture. O Explorig a Seget isector Work with a parter. Fold ad crease a piece of paper, as show. Label the eds of the crease as ad. a. Fold the paper agai so that poit coicides with poit. rease the paper o that fold. b. Ufold the paper ad exaie the four agles fored b the two creases. What ca ou coclude about the four agles? Writig a ojecture ONSTRUTING VILE RGUMENTS To be proficiet i ath, ou eed to ake cojectures ad build a logical progressio of stateets to explore the truth of our cojectures. Work with a parter. a. raw, as show. b. raw a arc with ceter o each side of. Usig the sae copass settig, draw a arc with ceter o each side of. Label the itersectios of the arcs ad. c. raw. Label its itersectio with as O. Write a cojecture about the resultig diagra. Justif our cojecture. O ouicate Your swer. What cojectures ca ou ake about perpedicular lies? 5. I Exploratio, fid O ad O whe = uits. Sectio. Proofs with Perpedicular Lies 7

27 . Lesso What You Will Lear ore Vocabular distace fro a poit to a lie, p. 8 perpedicular bisector, p. 9 Fid the distace fro a poit to a lie. ostruct perpedicular lies. Prove theores about perpedicular lies. Solve real-life probles ivolvig perpedicular lies. Fidig the istace fro a Poit to a Lie The distace fro a poit to a lie is the legth of the perpedicular seget fro the poit to the lie. This perpedicular seget is the shortest distace betwee the poit ad the lie. For exaple, the distace betwee poit ad lie k is. k distace fro a poit to a lie Fid the distace fro poit to. Fidig the istace fro a Poit to a Lie (, ) (, 0) REMEMER Recall that if (x, ) ad (x, ) are poits i a coordiate plae, the the distace betwee ad is = (x x ) + ( ). (, ) x (, ) SOLUTION ecause, the distace fro poit to is. Use the istace Forula. = ( ) + [ ( )] = ( ) + = 5.7 So, the distace fro poit to is about 5.7 uits. Moitorig Progress. Fid the distace fro poit E to FH. Help i Eglish ad Spaish at igideasmath.co F(0, ) G(, ) H(, ) x E(, ) 8 hapter Parallel ad Perpedicular Lies

28 ostructig Perpedicular Lies Use a copass ad straightedge to costruct a lie perpedicular to lie through poit P, which is ot o lie. SOLUTION ostructig a Perpedicular Lie Step Step Step P P Q P Q P raw arc with ceter P Place the copass at poit P ad draw a arc that itersects the lie twice. Label the itersectios ad. raw itersectig arcs raw a arc with ceter. Usig the sae radius, draw a arc with ceter. Label the itersectio of the arcs Q. raw perpedicular lie raw PQ. This lie is perpedicular to lie. P M Q The perpedicular bisector of a lie seget PQ is the lie with the followig two properties. PQ passes through the idpoit M of PQ. ostructig a Perpedicular isector Use a copass ad straightedge to costruct the perpedicular bisector of. SOLUTION Step Step Step c i. 5 M raw a arc Place the copass at. Use a copass settig that is greater tha half the legth of. raw a arc. raw a secod arc Keep the sae copass settig. Place the copass at. raw a arc. It should itersect the other arc at two poits. isect seget raw a lie through the two poits of itersectio. This lie is the perpedicular bisector of. It passes through M, the idpoit of. So, M = M. Sectio. Proofs with Perpedicular Lies 9

29 Provig Theores about Perpedicular Lies Theores Theore.0 Liear Pair Perpedicular Theore If two lies itersect to for a liear pair of cogruet agles, the the lies are perpedicular. If l, the g h. Proof Ex., p. 5 g h Theore. Perpedicular Trasversal Theore I a plae, if a trasversal is perpedicular to oe of two parallel lies, the it is perpedicular to the other lie. If h k ad j h, the j k. Proof Exaple, p. 50; Questio, p. 50 j h k Theore. Lies Perpedicular to a Trasversal Theore I a plae, if two lies are perpedicular to the sae lie, the the are parallel to each other. If p ad p, the. Proof Ex., p. 5; Ex. 7, p. p Provig the Perpedicular Trasversal Theore Use the diagra to prove the Perpedicular Trasversal Theore. SOLUTION Give h k, j h Prove j k j h k STTEMENTS RESONS. h k, j h. Give. = 90. efiitio of perpedicular lies.. orrespodig gles Theore (Theore.). =. efiitio of cogruet agles 5. = Trasitive Propert of Equalit. j k. efiitio of perpedicular lies Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. Prove the Perpedicular Trasversal Theore usig the diagra i Exaple ad the lterate Exterior gles Theore (Theore.). 50 hapter Parallel ad Perpedicular Lies

30 Solvig Real-Life Probles Provig Lies re Parallel The photo shows the laout of a eighborhood. eterie which lies, if a, ust be parallel i the diagra. Explai our reasoig. s t u p q SOLUTION Lies p ad q are both perpedicular to s, so b the Lies Perpedicular to a Trasversal Theore, p q. lso, lies s ad t are both perpedicular to q, so b the Lies Perpedicular to a Trasversal Theore, s t. So, fro the diagra ou ca coclude p q ad s t. Moitorig Progress Use the lies arked i the photo. Help i Eglish ad Spaish at igideasmath.co a b c d. Is b a? Explai our reasoig.. Is b c? Explai our reasoig. Sectio. Proofs with Perpedicular Lies 5

31 . Exercises aic Solutios available at igideasmath.co Vocabular ad ore ocept heck. OMPLETE THE SENTENE The perpedicular bisector of a seget is the lie that passes through the of the seget at a agle.. IFFERENT WORS, SME QUESTION Which is differet? Fid both aswers. Fid the distace fro poit X to lie WZ. X(, ) Z(, ) Fid XZ. Y(, ) Fid the legth of XY. x W(, ) Fid the distace fro lie to poit X. Moitorig Progress ad Modelig with Matheatics I Exercises ad, fid the distace fro poit to XZ. (See Exaple.). Z(, 7) ONSTRUTION I Exercises 5 8, trace lie ad poit P. The use a copass ad straightedge to costruct a lie perpedicular to lie through poit P. 5. P. P Y(0, ) (, 0) x X(, ) 7. P 8. P. (, ) X(, ) x Z(, ) Y(,.5) ONSTRUTION I Exercises 9 ad 0, trace. The use a copass ad straightedge to costruct the perpedicular bisector of hapter Parallel ad Perpedicular Lies

32 ERROR NLYSIS I Exercises ad, describe ad correct the error i the stateet about the diagra.. I Exercises 7, deterie which lies, if a, ust be parallel. Explai our reasoig. (See Exaple.) v 7. z w a 8. x x b c Lies ad z are parallel. 9.. b c p c a 0. d q 8 c The distace fro poit to is cetieters.. p z. v w PROVING THEOREM I Exercises ad, prove the theore. (See Exaple.). Liear Pair Perpedicular Theore (Th..0) x k. Lies Perpedicular to a Trasversal Theore (Th..). USING STRUTURE Fid all the ukow agle PROOF I Exercises 5 ad, use the diagra to write a proof of the stateet. easures i the diagra. Justif our aswer for each agle easure. 5. If two itersectig lies are perpedicular, the the itersect to for four right agles. Give a b Prove,,, ad are right agles b a. If two sides of two adjacet acute agles are perpedicular, the the agles are copleetar. Give Prove ad are copleetar.. MKING N RGUMENT Your fried clais that because ou ca fid the distace fro a poit to a lie, ou should be able to fid the distace betwee a two lies. Is our fried correct? Explai our reasoig. 5. MTHEMTIL ONNETIONS Fid the value of x whe a b ad b c. b a (9x + 8) c [5(x + 7) + 5] Sectio. hs_geo_pe_00.idd 5 Proofs with Perpedicular Lies 5 /9/5 9:5 M

33 . HOW O YOU SEE IT? You are trig to cross a strea fro poit. Which poit should ou jup to i order to jup the shortest distace? Explai our reasoig. 9. ONSTRUTION ostruct a square of side legth. E 0. NLYZING RELTIONSHIPS The paited lie segets that for the path of a crosswalk are usuall perpedicular to the crosswalk. Sketch what the segets i the photo would look like if the were perpedicular to the crosswalk. Which tpe of lie seget requires less pait? Explai our reasoig. 7. TTENING TO PREISION I which of the followig diagras is ad? Select all that appl. E. STRT RESONING Two lies, a ad b, are perpedicular to lie c. Lie d is parallel to lie c. The distace betwee lies a ad b is x eters. The distace betwee lies c ad d is eters. What shape is fored b the itersectios of the four lies?. MTHEMTIL ONNETIONS Fid the distace betwee the lies with the equatios = x + ad x + =. 8. THOUGHT PROVOKING The postulates ad theores i this book represet Euclidea geoetr. I spherical geoetr, all poits are poits o the surface of a sphere. lie is a circle o the sphere whose diaeter is equal to the diaeter of the sphere. I spherical geoetr, how a right agles are fored b two perpedicular lies? Justif our aswer.. WRITING escribe how ou would fid the distace fro a poit to a plae. a ou fid the distace fro a lie to a plae? Explai our reasoig. Maitaiig Matheatical Proficiec Siplif the ratio. (Skills Review Hadbook). ( ) ( ) 7 ( ) Idetif the slope ad the -itercept of the lie. (Skills Review Hadbook) Reviewig what ou leared i previous grades ad lessos 7. ( ) 8. = x = x = x 8. = 8x 5 hapter Parallel ad Perpedicular Lies

34 .5 Equatios of Parallel ad Perpedicular Lies Essetial Questio Essetial Questio How ca ou write a equatio of a lie that is parallel or perpedicular to a give lie ad passes through a give poit? Writig Equatios of Parallel ad Perpedicular Lies Work with a parter. Write a equatio of the lie that is parallel or perpedicular to the give lie ad passes through the give poit. Use a graphig calculator to verif our aswer. What is the relatioship betwee the slopes? a. b. (0, ) (0, ) = x = x c. d. = x + = x + (, ) (, ) e. f. = x + = x + (0, ) (, 0) Writig Equatios of Parallel ad Perpedicular Lies Work with a parter. Write the equatios of the parallel or perpedicular lies. Use a graphig calculator to verif our aswers. a. b. MOELING WITH MTHEMTIS To be proficiet i ath, ou eed to aalze relatioships atheaticall to draw coclusios. ouicate Your swer. How ca ou write a equatio of a lie that is parallel or perpedicular to a give lie ad passes through a give poit?. Write a equatio of the lie that is (a) parallel ad (b) perpedicular to the lie = x + ad passes through the poit (, ). Sectio.5 Equatios of Parallel ad Perpedicular Lies 55

35 .5 Lesso What You Will Lear ore Vocabular directed lie seget, p. 5 Previous slope slope-itercept for -itercept Use slope to partitio directed lie segets. Idetif parallel ad perpedicular lies. Write equatios of parallel ad perpedicular lies. Use slope to fid the distace fro a poit to a lie. Partitioig a irected Lie Seget directed lie seget is a seget that represets ovig fro poit to poit. The followig exaple shows how to use slope to fid a poit o a directed lie seget that partitios the seget i a give ratio. Partitioig a irected Lie Seget REMEMER Recall that the slope of a lie or lie seget through two poits, (x, ) ad (x, ), is defied as follows. = x x chage i = chage i x = rise ru You ca choose either of the two poits to be (x, ). Fid the coordiates of poit P alog the directed lie seget so that the ratio of P to P is to. SOLUTION I order to divide the seget i the ratio to, thik of dividig, or partitioig, the seget ito +, or 5 cogruet pieces. Poit P is the poit that is of the wa fro poit to poit. 5 Fid the rise ad ru fro poit to poit. Leave the slope i ters of rise ad ru ad do ot siplif. slope of : = 8 = = rise ru To fid the coordiates of poit P, add of the ru 5 (, 8) to the x-coordiate of, ad add 8 of the rise to the 5 -coordiate of. ru: 5 of = P(.8, 5.) 5 =.8. rise: 5 of = 5 =. (, ).8 So, the coordiates of P are ( +.8, +.) = (.8, 5.). The ratio of P to P is to. 8 (, ) (, 8) 8 x 8 x Moitorig Progress Help i Eglish ad Spaish at igideasmath.co Fid the coordiates of poit P alog the directed lie seget so that P to P is the give ratio.. (l, ), (8, ); to. (, ), (, 5); to 7 5 hapter Parallel ad Perpedicular Lies

36 Idetifig Parallel ad Perpedicular Lies I the coordiate plae, the x-axis ad the -axis are perpedicular. Horizotal lies are parallel to the x-axis, ad vertical lies are parallel to the -axis. Theores Theore. Slopes of Parallel Lies I a coordiate plae, two distict overtical lies are parallel if ad ol if the have the sae slope. two vertical lies are parallel. x REING If the product of two ubers is, the the ubers are called egative reciprocals. Proof p. 9; Ex., p. Theore. Slopes of Perpedicular Lies I a coordiate plae, two overtical lies are perpedicular if ad ol if the product of their slopes is. = Horizotal lies are perpedicular to vertical lies. x Proof p. 0; Ex., p. = Idetifig Parallel ad Perpedicular Lies eterie which of the lies are parallel ad which of the lies are perpedicular. SOLUTION Fid the slope of each lie. Lie a: = 0 ( ) = Lie b: = 0 ( ) = 0 ( 5) Lie c: = = ( ) 0 Lie d: = ( ) = d (, ) (, 0) a (0, ) b (, 0) x (0, ) c (, 5) (, ) ecause lies b ad c have the sae slope, lies b ad c are parallel. ecause ( ) =, lies b ad d are perpedicular ad lies c ad d are perpedicular. Moitorig Progress. eterie which of the lies are parallel ad which of the lies are perpedicular. Help i Eglish ad Spaish at igideasmath.co (0, ) (, 0) a b c (, ) (, ) (, ) (0, ) x (, ) d Sectio.5 Equatios of Parallel ad Perpedicular Lies 57

37 Writig Equatios of Parallel ad Perpedicular Lies You ca appl the Slopes of Parallel Lies Theore ad the Slopes of Perpedicular Lies Theore to write equatios of parallel ad perpedicular lies. Writig a Equatio of a Parallel Lie REMEMER The liear equatio = x is writte i slope-itercept for = x + b, where is the slope ad b is the -itercept. Write a equatio of the lie passig through the poit (, ) that is parallel to the lie = x. SOLUTION Step Fid the slope of the parallel lie. The lie = x has a slope of. the Slopes of Parallel Lies Theore, a lie parallel to this lie also has a slope of. So, =. Step Fid the -itercept b b usig = ad (x, ) = (, ). = x + b Use slope-itercept for. = ( ) + b Substitute for, x, ad. = b Solve for b. heck = x + (, ) ecause = ad b =, a equatio of the lie is = x +. Use a graph to check that the lie = x is parallel to the lie = x +. Writig a Equatio of a Perpedicular Lie x = x Write a equatio of the lie passig through the poit (, ) that is perpedicular to the lie x + =. SOLUTION Step Fid the slope of the perpedicular lie. The lie x + =, or = x +, has a slope of. Use the Slopes of Perpedicular Lies Theore. = The product of the slopes of lies is. = ivide each side b. heck = x + = x + (, ) Step Fid the -itercept b b usig = ad (x, ) = (, ). = x + b Use slope-itercept for. = () + b Substitute for, x, ad. = b Solve for b. x ecause = ad b =, a equatio of the lie is = x +. heck that the lies are perpedicular b graphig their equatios ad usig a protractor to easure oe of the agles fored b their itersectio. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. Write a equatio of the lie that passes through the poit (, 5) ad is (a) parallel to the lie = x 5 ad (b) perpedicular to the lie = x How do ou kow that the lies x = ad = are perpedicular? 58 hapter Parallel ad Perpedicular Lies

38 Fidig the istace fro a Poit to a Lie Recall that the distace fro a poit to a lie is the legth of the perpedicular seget fro the poit to the lie. Fidig the istace fro a Poit to a Lie REMEMER = x + (, 0) Recall that the solutio of a sste of two liear equatios i two variables gives the coordiates of the poit of itersectio of the graphs of the equatios. There are two special cases whe the lies have the sae slope. Whe the sste has o solutio, the lies are parallel. Whe the sste has ifiitel a solutios, the lies coicide. x Fid the distace fro the poit (, 0) to the lie = x +. SOLUTION Step Fid a equatio of the lie perpedicular to the lie = x + that passes through the poit (, 0). First, fid the slope of the perpedicular lie. The lie = x + has a slope of. Use the Slopes of Perpedicular Lies Theore. = The product of the slopes of lies is. = ivide each side b. The fid the -itercept b b usig = ad (x, ) = (, 0). = x + b Use slope-itercept for. 0 = () + b Substitute for x,, ad. = b Solve for b. ecause = ad b =, a equatio of the lie is = x. Step Use the two equatios to write ad solve a sste of equatios to fid the poit where the two lies itersect. = x + Equatio = x Equatio Substitute x + for i Equatio. = x Equatio x + = x Substitute x + for. x = Solve for x. Substitute for x i Equatio ad solve for. = x + Equatio = + Substitute for x. = Siplif. So, the perpedicular lies itersect at (, ). Step Use the istace Forula to fid the distace fro (, 0) to (, ). (, ) x distace = ( ) + (0 ) = ( ) + ( ) =. (, 0) = x + = x So, the distace fro the poit (, 0) to the lie = x + is about. uits. Moitorig Progress Help i Eglish ad Spaish at igideasmath.co. Fid the distace fro the poit (, ) to the lie = x Fid the distace fro the poit (, ) to the lie = x. Sectio.5 Equatios of Parallel ad Perpedicular Lies 59

39 .5 Exercises aic Solutios available at igideasmath.co Vocabular ad ore ocept heck. OMPLETE THE SENTENE lie seget is a seget that represets ovig fro poit to poit.. WRITING How are the slopes of perpedicular lies related? Moitorig Progress ad Modelig with Matheatics I Exercises, fid the coordiates of poit P alog the directed lie seget so that P to P is the give ratio. (See Exaple.). (8, 0), (, ); to. (, ), (, ); to 5. (, ), (, ); 5 to. (, ), (5, ); to I Exercises 7 ad 8, deterie which of the lies are parallel ad which of the lies are perpedicular. (See Exaple.) (, ) d (, ) a (5, ) (, ) b (, ) c (, ) x (, ) (, 0) d (0, ) (, ) (, ) (, ) (, 0) a x (, 0) (, ) b (0, ) I Exercises 9, tell whether the lies through the give poits are parallel, perpedicular, or either. Justif our aswer. 9. Lie : (, 0), (7, ) Lie : (7, 0), (, ) c 0. Lie : (, ), ( 7, ) Lie : (, ), (8, ). Lie : ( 9, ), ( 5, 7) Lie : (, ), ( 7, ). Lie : (0, 5), ( 8, 9) Lie : (, ), (, ) I Exercises, write a equatio of the lie passig through poit P that is parallel to the give lie. Graph the equatios of the lies to check that the are parallel. (See Exaple.). P(0, ), = x +. P(, 8), = (x + ) 5 5. P(, ), x = 5. P(, 0), x + = I Exercises 7 0, write a equatio of the lie passig through poit P that is perpedicular to the give lie. Graph the equatios of the lies to check that the are perpedicular. (See Exaple.) 7. P(0, 0), = 9x 8. P(, ), = 9. P(, ), = (x + ) 0. P( 8, 0), x 5 = I Exercises, fid the distace fro poit to the give lie. (See Exaple 5.). (, 7), = x. ( 9, ), = x. (5, ), 5x + =. (, 5 ), x + = 0 hapter Parallel ad Perpedicular Lies

40 5. ERROR NLYSIS escribe ad correct the error i deteriig whether the lies are parallel, perpedicular, or either. Lie : (, 5), (, ) Lie : (0, ), (,7) ( 5) = = = 7 0 = Lies ad are perpedicular.. ERROR NLYSIS escribe ad correct the error i writig a equatio of the lie that passes through the poit (, ) ad is parallel to the lie = x +. = x +, (, ) = () + = The lie = x + is parallel to the lie = x +. I Exercises 7 0, fid the idpoit of PQ. The write a equatio of the lie that passes through the idpoit ad is perpedicular to PQ. This lie is called the perpedicular bisector. 7. P(, ), Q(, ) 8. P( 5, 5), Q(, ). RESONING triagle has vertices L(0, ), M(5, 8), ad N(, ). Is the triagle a right triagle? Explai our reasoig.. MOELING WITH MTHEMTIS ew road is beig costructed parallel to the trai tracks through poit V. equatio of the lie represetig the trai tracks is = x. Fid a equatio of the lie represetig the ew road. V x 5. MOELING WITH MTHEMTIS bike path is beig costructed perpedicular to Washigto oulevard through poit P(, ). equatio of the lie represetig Washigto oulevard is = x. Fid a equatio of the lie represetig the bike path. 9. P(0, ), Q(, ) 0. P( 7, 0), Q(, 8). MOELING WITH MTHEMTIS Your school lies directl betwee our house ad the ovie theater. The distace fro our house to the school is oe-fourth of the distace fro the school to the ovie theater. What poit o the graph represets our school? (5, ). PROLEM SOLVING gazebo is beig built ear a ature trail. equatio of the lie represetig the ature trail is = x. Each uit i the coordiate plae correspods to 0 feet. pproxiatel how far is the gazebo fro the ature trail? (, ) x gazebo (, ) 8 x. RESONING Is quadrilateral QRST a parallelogra? Explai our reasoig. Q(, ) R(, ) T(, ) S(5, ) x 7. RITIL THINKING The slope of lie is greater tha 0 ad less tha. Write a iequalit for the slope of a lie perpedicular to. Explai our reasoig. Sectio.5 Equatios of Parallel ad Perpedicular Lies

41 8. HOW O YOU SEE IT? eterie whether quadrilateral JKLM is a square. Explai our reasoig. K(0, ) L(, ) MTHEMTIL ONNETIONS I Exercises ad, fid a value for k based o the give descriptio.. The lie through (, k) ad ( 7, ) is parallel to the lie = x +.. The lie through (k, ) ad (7, 0) is perpedicular to the lie = x 8 5. J(0, 0) M(, 0) x 5. STRT RESONING Make a cojecture about how to fid the coordiates of a poit that lies beod poit alog. Use a exaple to support our cojecture. 9. RITIL THINKING Suppose poit P divides the directed lie seget XY so that the ratio of XP to PY is to 5. escribe the poit that divides the directed lie seget YX so that the ratio of YP to PX is 5 to. 0. MKING N RGUMENT Your classate clais that o two overtical parallel lies ca have the sae -itercept. Is our classate correct? Explai.. MTHEMTIL ONNETIONS Solve each sste of equatios algebraicall. Make a cojecture about what the solutio(s) ca tell ou about whether the lies itersect, are parallel, or are the sae lie. a. = x + 9 x = b. + x = x = 8 c. = 5x + 0x + =. THOUGHT PROVOKING Fid a forula for the distace fro the poit (x 0, 0 ) to the lie ax + b = 0. Verif our forula usig a poit ad a lie.. PROLEM SOLVING What is the distace betwee the lies = x ad = x + 5? Verif our aswer. PROVING THEOREM I Exercises 7 ad 8, use the slopes of lies to write a paragraph proof of the theore. 7. Lies Perpedicular to a Trasversal Theore (Theore.): I a plae, if two lies are perpedicular to the sae lie, the the are parallel to each other. 8. Trasitive Propert of Parallel Lies Theore (Theore.9): If two lies are parallel to the sae lie, the the are parallel to each other. 9. PROOF Prove the stateet: If two lies are vertical, the the are parallel. 50. PROOF Prove the stateet: If two lies are horizotal, the the are parallel. 5. PROOF Prove that horizotal lies are perpedicular to vertical lies. Maitaiig Matheatical Proficiec Plot the poit i a coordiate plae. (Skills Review Hadbook) 5. (, ) 5. (0, ) 5. (5, 0) 55. (, ) op ad coplete the table. (Skills Review Hadbook) Reviewig what ou leared i previous grades ad lessos 5. x x 0 = x + 9 = x hapter Parallel ad Perpedicular Lies

42 ..5 What id You Lear? ore Vocabular distace fro a poit to a lie, p. 8 perpedicular bisector, p. 9 directed lie seget, p. 5 ore ocepts Sectio. Fidig the istace fro a Poit to a Lie, p. 8 ostructig Perpedicular Lies, p. 9 Theore.0 Liear Pair Perpedicular Theore, p. 50 Theore. Perpedicular Trasversal Theore, p. 50 Theore. Lies Perpedicular to a Trasversal Theore, p. 50 Sectio.5 Partitioig a irected Lie Seget, p. 5 Theore. Slopes of Parallel Lies, p. 57 Theore. Slopes of Perpedicular Lies, p. 57 Writig Equatios of Parallel ad Perpedicular Lies, p. 58 Fidig the istace fro a Poit to a Lie, p. 59 Matheatical Practices. opare the effectiveess of the arguet i Exercise o page 5 with the arguet You ca fid the distace betwee a two parallel lies. What flaw(s) exist i the arguet(s)? oes either arguet use correct reasoig? Explai.. Look back at our costructio of a square i Exercise 9 o page 5. How would our costructio chage if ou were to costruct a rectagle?. I Exercise o page, a classate tells ou that our aswer is icorrect rect because ou should have divided the seget ito four cogruet pieces. Respod to our classate s ate arguet b justifig our origial aswer. Navajo Rugs Navajo rugs use atheatical properties to ehace their beaut. How ca ou describe these creative works of art with geoetr? What properties of lies ca ou see ad use to describe the patters? To explore the aswers to this questio ad ore, go to igideasmath.co. Perforace Task

43 hapter Review. Pairs of Lies ad gles (pp. 5 0) aic Solutios available at igideasmath.co Thik of each seget i the figure as part of a lie. a. Which lie(s) appear perpedicular to?,, H, ad G appear perpedicular to. b. Which lie(s) appear parallel to?, GH, ad EF appear parallel to. c. Which lie(s) appear skew to? F, E, F, FH, ad EG appear skew to. G H E F d. Which plae(s) appear parallel to plae? Plae EFG appears parallel to plae. K L Thik of each seget i the figure as part of a lie. Which lie(s) or plae(s) appear to fit the descriptio?. lie(s) perpedicular to QR. lie(s) skew to QR. lie(s) parallel to QR. plae(s) parallel to plae LMQ J N P M R Q. Parallel Lies ad Trasversals (pp. ) Fid the value of x. the Vertical gles ogruece Theore (Theore.), = 50. (x 5) + = 80 osecutive Iterior gles Theore (Th..) (x 5) + 50 = 80 Substitute 50 for. x + 5 = 80 obie like ters. x = 5 Subtract 5 fro each side. 50 So, the value of x is 5. Fid the values of x ad. (x 5) 5. xº 5º º. (5x 7)º 8º º º 58º xº (5 )º (x + )º º hapter Parallel ad Perpedicular Lies

44 . Proofs with Parallel Lies (pp. 7 ) Fid the value of x that akes. the lterate Iterior gles overse (Theore.), whe the arked agles are cogruet. (5x + 8) = 5 5x = 5 x = 9 The lies ad are parallel whe x = 9. Fid the value of x that akes. (5x + 8) 5 9. x (x + ). (x + 0) x. (7x ) (x + 58). Proofs with Perpedicular Lies (pp. 7 5) eterie which lies, if a, ust be parallel. Explai our reasoig. Lies a ad b are both perpedicular to d, so b the Lies Perpedicular to a Trasversal Theore (Theore.), a b. lso, lies c ad d are both perpedicular to b, so b the Lies Perpedicular to a Trasversal Theore (Theore.), c d. c d a b eterie which lies, if a, ust be parallel. Explai our reasoig.. x z. x w z 5. a. a b b c hapter hapter Review 5

45 .5 Equatios of Parallel ad Perpedicular Lies (pp. 55 ) a. Write a equatio of the lie passig through the poit (, ) that is parallel to the lie = 5x 7. Step Fid the slope of the parallel lie. The lie = 5x 7 has a slope of 5. the Slopes of Parallel Lies Theore (Theore.), a lie parallel to this lie also has a slope of 5. So, = 5. Step Fid the -itercept b b usig = 5 ad (x, ) = (, ). = x + b Use slope-itercept for. = 5( ) + b Substitute for, x, ad. = b Solve for b. ecause = 5 ad b =, a equatio of the lie is = 5x +. b. Write a equatio of the lie passig through the poit (, ) that is perpedicular to the lie x + = 9. Step Fid the slope of the perpedicular lie. The lie x + = 9, or = x + 9, has a slope of. Use the Slopes of Perpedicular Lies Theore (Theore.). = The product of the slopes of lies is. = ivide each side b. Step Fid the -itercept b b usig = ad (x, ) = (, ). = x+ b Use slope-itercept for. = () + b Substitute for, x, ad. = b Solve for b. ecause = ad b =, a equatio of the lie is = x. Write a equatio of the lie passig through the give poit that is parallel to the give lie. 7. (, ), = x (, 5), = x 7 9. (, 0), = x 5 0. (, ), = x + 0 Write a equatio of the lie passig through the give poit that is perpedicular to the give lie.. (, ), = x + 8. (0, ), = x. (8, ), = x 7. (, 5), = 7 x + Fid the distace fro poit to the give lie. 5. (, ), = x +. (, ), = x + hapter Parallel ad Perpedicular Lies

46 hapter Test Fid the values of x ad. State which theore(s) ou used.... x 8x ( + 9) 9 (8x + ) [( )] Fid the distace fro poit to the give lie.. (, ), = x 5. (, 7), = x Fid the value of x that akes.. x (x + ) 8x 8. (x ) (x + ) Write a equatio of the lie that passes through the give poit ad is (a) parallel to ad (b) perpedicular to the give lie. 9. ( 5, ), = x 0. (, 9), = x + j k. studet sas, ecause j k, j. What issig iforatio is the studet assuig fro the diagra? Which theore is the studet trig to use?. You ad our fail are visitig soe attractios while o vacatio. You ad our o visit the shoppig all while our dad ad our sister visit the aquariu. You decide to eet at the itersectio of lies q ad p. Each uit i the coordiate plae correspods to 50 ards. a. Fid a equatio of lie q. b. Fid a equatio of lie p. c. What are the coordiates of the eetig poit? d. What is the distace fro the eetig poit to the subwa? q shoppig all aquariu subwa p x I K E F G J L H M. Idetif a exaple o the puzzle cube of each descriptio. Explai our reasoig. a. a pair of skew lies b. a pair of perpedicular lies c. a pair of parallel lies d. a pair of cogruet correspodig agles e. a pair of cogruet alterate iterior agles hapter hapter Test 7