Review Cocepts: Equatios of Lies Geoetry Uit Notes Parallel ad Perpedicular Lies Syllabus Objective:. - The studet will differetiate aog parallel, perpedicular, ad skew lies. Lies that DO NOT itersect: o Parallel Lies - coplaar lies that do ot itersect. (sybol: ) Idicate correct diagra arkigs for parallel lies. o Skew Lies - o-coplaar lies that do ot itersect. Lies that DO itersect: o Perpedicular Lies coplaar lies that itersect to for a right agle. (sybol: ) Idicate correct diagra arkigs for perpedicular lies. Reid studets that itersectig lies are always coplaar. Two lies ca be related i differet ways: itersectig, parallel or skew. Itersectig lies ay or ay ot be perpedicular. Two plaes ca oly be related i ways: parallel or itersectig. Itersectig ad parallel lies are coplaar, by defiitio. Parallel Postulate: If there is a lie ad a poit ot o the lie, the there is exactly oe lie through the poit parallel to the give lie. Perpedicular Postulate: If there is a lie ad a poit ot o the lie, the there is exactly oe lie through the poit perpedicular to the give lie. Syllabus Objective:. - The studet will aalyze relatioships whe two lies are cut by a trasversal. gle Pair Relatioships Trasversal - a lie that itersects two or ore coplaar lies. gle pairs created by a trasversal. Correspodig - agles i the sae relative positio, sae side of the trasversal ad o the sae side of the give lies. lterate - opposite sides of the trasversal. Uit Parallel ad Perpedicular Lies Page of 0
Cosecutive - sae side of the trasversal. (also called sae-side) Iterior - betwee two lies cut by a trasversal. Exterior - outside of two lies cut by a trasversal. I the figure to the left, t is a trasversal. I the t figures followig, all lies are trasversals. s 8 7 0 9 gle pairs ca be defied by likig two ters together, such as: lterate Iterior gles a pair of agles o opposite sides of the trasversal ad betwee the two itersected lies. Whe idetifyig these agle pairs, it is helpful to idetify the trasversal first. Ex: ad are correspodig agles, ad are alterate iterior agles, ad 8 are sae-side exterior agles. w 7 8 9 0 7 8 t 7 8 v u Exaples:. Nae two sae-side iterior agles to } ad their trasversals. {, s ad,. Nae two correspodig agles to ad their trasversals. {, s ad, }. Nae the alterate exterior agle to ad its trasversal. {7, t }. Nae two sae-side iterior agles to 0 ad their trasversals. {, w ad, u}. Nae two correspodig agles to ad their trasversals. {9, u ad 7, v}. Nae two alterate exterior agles to 8 ad their trasversals. {, v ad 9, w} Uit Parallel ad Perpedicular Lies Page of 0
7. Nae the type of agle pair of the followig agles: a. ad {alterate iterior agles} b. ad {correspodig agles} c. ad 7 {vertical agles} d. ad 0 {o relatioship} e. 0 ad 7 {correspodig agles} f. ad 7 {sae-side iterior agles} g. 9ad 0 {liear pair or suppleetary agles} Special gle Pairs (Postulate ad Theores) o Correspodig gles Postulate: If two parallel lies are cut by a trasversal, the the correspodig agles are cogruet. {C Post.} o lterate Iterior gles Theore: If two parallel lies are cut by a trasversal, the the pairs of alterate iterior agles are cogruet. {I Th.} o Cosecutive Iterior gles Theore: If two parallel lies are cut by a trasversal, the the pairs of cosecutive iterior agles are suppleetary. {CI Th.} o lterate Exterior gles Theore: If two parallel lies are cut by a trasversal, the the pairs of alterate exterior agles are cogruet. {E Th.} The theores all follow directly fro the postulate. Cosecutive Iterior gles ca also be called Sae Side Iterior gles. However, cosecutive exterior agles would be a isoer as oe of the exterior agles are actually cosecutive. It is a siple exercise to write a proof of Sae Side Exterior gles. This provides a opportuity for studets to develop their ow useful theore. I siple ters, ay two agles fored by the trasversal ad two parallel lies ca have oly oe of two relatioships: a) they are cogruet, or b) they are suppleetary. Uit Parallel ad Perpedicular Lies Page of 0
Syllabus Objective:. The studet will solve probles which ivolve parallel or perpedicular lies usig algebraic techiques. Postulates: Parallel Postulate: If there is a lie ad a poit ot o the lie, the there is exactly oe lie through the poit parallel to the give lie. Perpedicular Postulate: If there is a lie ad a poit ot o the lie, the there is exactly oe lie through the poit perpedicular to the give lie. Exaples: Give the iforatio, fid the easure of each ubered agle. Which theore or postulate did you use to calculate your solutios?. = {,,, 8 = ad,,, 7 = 7 }. 7 = 7 {,,, 8 = ad,,, 7 = 7 }. 8 = (x + ), = (x-) x + = x, x = {,,, 8 = ad,,, 7 = } 8. = (x + ), = (x-) 7 x + + x = 80, x = 8 {,,, 8 = 97 ad,,, 7 = 8 } Syllabus Objective:. The studet will write proofs relatig to parallel ad perpedicular lies. t Theores of Perpedicular Lies o If two lies are perpedicular, the they itersect to for four right agles. This follows directly fro the defiitio of perpedicular lies, after applyig the liear pair postulate ad the vertical agle theore. o Perpedicular Trasversal Theore If a trasversal is perpedicular to oe of two parallel lies, the it is perpedicular to the other. {P.T. Th.} Uit Parallel ad Perpedicular Lies Page of 0
Give: Prove: ad t t (Note: diagra is NOT draw to scale) 7 8 t Stateets Reasos ) t ) Give ) ) C. Post. ) ) V.. Th. ) {proof of: I Th.} ) Substitutio ) 8 ) C.s Post. ) 8 {proof of: E Th.} ) Substitutio 7) is suppleetary to 7) L. P. Post. 8) is suppleetary to {proof of: CI Th.} 8) Substitutio 9) t 9) Give 0) = 90 0) Def. of lies ) = 90 ) Substitutio ) t {proof of: P.T. Th.} ) Def. of lies Uit Parallel ad Perpedicular Lies Page of 0
dditioal Theores bout Perpedicular Lies o Theores: If two lies itersect to for a liear pair of cogruet, agles, the the lies are perpedicular. C D We are give that the two itersectig lies, B ad CD, for cogruet adjacet agles. Therefore CD DCB. So by the defiitio of cogruet, CD = DCB. B CB = 80 by defiitio of a straight agle. The gle dditio Postulate says that CD + DCB = CB hece, CD + DCB = 80. Usig substitutio we obtai CD + CD = 80, which siplifies to CD = 80. d the divisio property of equality allows us to coclude that CD = 90. If two sides of two adjacet acute agles are perpedicular, the the agles are copleetary. We are give B BC, hece, BC = 90. B D C The gle dditio Postulate says that BD + DBC = BC hece BD + DBC = 90 ad therefore, the two agles ust be copleetary. Uit Parallel ad Perpedicular Lies Page of 0
Exaples: I the figure, B EC, FB BD ad FB =.. Fid EBF, BD ad DBC. { EBF =, BD =, DBC = }. Which theores did you use to fid your aswers? { If sides of adj. s are,the the s are cop.}. Usig the diagra above, coplete the followig proof: E F B D C Give: B EC, FB BD Prove: EBF BD Stateets Reasos ) B EC, FB BD ) Give ) sides of adj. s are, the s are ) BD is copleetary to FB cop. ) sides of adj. s are, the s are ) EBF is copleetary to FB. cop. ) EBF BD ) cop. Th. Provig Lies re Parallel - Special gle Pairs Coverses (Postulate ad Theores) Correspodig gles Coverse: If two lies are cut by a trasversal so that correspodig agles are cogruet, the the lies are parallel. {C Post. Coverse} lterate Iterior gles Coverse: If two lies are cut by a trasversal so that alterate iterior agles are cogruet, the the lies are parallel. {I Th. Coverse} Cosecutive Iterior gles Coverse: If two lies are cut by a trasversal so that cosecutive iterior agles are suppleetary, the the lies are parallel. {CI Th. Coverse} lterate Exterior gles Coverse: If two lies are cut by a trasversal so that alterate exterior agles are cogruet, the the lies are parallel. {E Th. Coverse} Uit Parallel ad Perpedicular Lies Page 7 of 0
The followig are proofs of the coverse theores: they all use the diagra below. 8 7 Give: t Stateets Reasos ) ) Give ) ) V.. Th. ) ) Substitutio ) {proof of: IE Th. Coverse} ) C., Post. Coverse Give: is suppleetary to Stateets Reasos ) is suppleetary to ) Give ) is suppleetary to ) L. P. Post. ) ) Supp. Th. ) {proof of: CI Th. Coverse} ) C.. Post. Coverse Give: 8 Stateets Reasos ) 8 ) Give ) ) V.. Th. ) 8 ) Substitutio ) {proof of: E Th. Coverse} ) C.. Post. Coverse Uit Parallel ad Perpedicular Lies Page 8 of 0
Exaples: Give the iforatio, what other iforatio would esure the lies ad were parallel? Discuss with studets: Which theore or postulate did you use to arrive at your aswers?. = {,,, 8 = ad,,, 7 = 7 }. 7 = 7 {,,, 8 = ad,,, 7 = 7 } What value of x will esure lies ad are parallel? Fid the agle easures.. 8 = (x + ), = (x-) { x =,,,, 8 = ad,,, 7 = }. = (x + ), = (x-) { x = 8,,,, 8 = 97 ad,,, 7 = 8 } 7 t 8 More Lie Theores o If two lies are parallel to the sae lie, they are parallel to each other. Costruct a acillary trasversal, apply correspodig agles postulate trasitive property, ad correspodig agle coverse. o I a plae, if two lies are perpedicular to the sae lie, they are parallel to each other. pply correspodig agle coverse. These provide the perfect opportuity to review coverse stateets ad bicoditioal stateets. siple exercise ight iclude writig the Theores ad their coverses as bicoditioal stateets. Exaple: Two lies cut by a trasversal are parallel if ad oly if they for cogruet correspodig agles. Uit Parallel ad Perpedicular Lies Page 9 of 0
Syllabus Objective:. The studet will costruct parallel ad perpedicular lies. (requires suppleetal aterial) cceptable ethods of costructio iclude patty paper or copass/straightedge costructios. PERPENDICULR LINES (perpedicular to a give lie fro a poit o the lie) Copass/straightedge: Costruct a perpedicular to B C B through C. Usig C as the ceter, draw two arcs with equal radii, itersectig B at two poits (D ad E). D C E B Usig poits D ad E as ceters, draw arcs with equal radii itersectig at a poit (F). Note, the radii used ust be greater tha the legth of DC or CE. F D C E B Draw CF. CF B F D C E B *Review cocept by cuttig fraes out ad havig studets reorder the (with or without diagras). Uit Parallel ad Perpedicular Lies Page 0 of 0
Patty Paper: Fold or draw a lie o a patty paper. Place a poit o the lie. Fold the patty paper so that the crease goes through the poit ad oe side of the lie lies o top of the other side of the lie. PERPENDICULR LINES (perpedicular to a give lie fro a poit NOT o the lie) Copass/straightedge: Costruct a perpedicular to B through C. B C Usig C as the ceter, draw a arc itersectig B at two poits (D ad E). Note, the radii used ust be greater tha the distace fro C to B. D E B C Usig poits D ad E as ceters, draw arcs with equal radii that itersect at a poit (F) o the opposite side of the lie. Note, the radii eed ot be chaged fro the previous settig. F D E B C Uit Parallel ad Perpedicular Lies Page of 0
Draw CF. CF B. F D E B C *Review cocept by cuttig fraes out ad havig studets reorder the (with or without diagras). Patty Paper: Fold or draw a lie o a patty paper. Place a dot o your patty paper to represet the give poit. (ot o the lie) Fold the lie o top of itself so that the fold passes through the give poit. PRLLEL LINES (parallel to a give lie thru ay poit): The easiest ethod for studets to uderstad is the applicatio of the postulate: I a plae, if two lies are perpedicular to the sae lie, they are parallel to each other. Studets are able to perfor a cobiatio of the perpedicular lie costructios; st fro a poit NOT o the lie, d fro a poit o the lie. Syllabus Objective:. The studet will copare strategies for deteriig the slope of a lie. Slope ca be itroduced as a rate of chage. {ex.: iles per hour, dollars per hour, feet per secod, etc } Lookig at the ueric sequece, 7, 0,, we otice a costat icrease of (called the coo differece ), which is copared to a slope of. y give ter x of this sequece ca be foud by the expressio x +. Uit Parallel ad Perpedicular Lies Page of 0
Siilarly, a table of values ca be used to deostrate the slope idea. t.. gai there is a costat decrease of. or a slope of is foud by t = +.. The th ter of the sequece Most studets are failiar with slope beig referred to as rise, it would be ru helpful to itroduce the cocepts of y, x chage i y vertical chage or chage i x horizotal chage to ecourage greater uderstadig ad likage with higher level sciece/atheatic cocepts. Deteriig the slope of a lie: Give the equatio of a lie solve for y. (slope-itercept for) y = x + b, is the slope. Give two poits use slope forula. vertical chage y y = = horizotal chage x x Give a graph of the lie or a graph of the poits use lattice poits. Exaple: (cout rise ru or vertical chage horizotal chage ) llows likage to distace forula ad Pythagorea Theore! rise of uits ru of uits Slope of seget: - - Uit Parallel ad Perpedicular Lies Page of 0
Syllabus Objective:.7 The studet will aalyze slopes i a coordiate plae. - j k - - I the figure, j k because the slope of both lies are equal to. The lies j ad k are both perpedicular to lie because their slopes are opposite reciprocals or, i other words, the product of their slopes is -. Slope of j ad k: Slope of : = Parallel Lies i the Coordiate Plae o I the coordiate plae, two o-vertical lies are parallel if ad oly if they have equal slopes. ll vertical lies are parallel. o Slope is the rate of chage i y to the chage i x. Perpedicular Lies i the Coordiate Plae o I the coordiate plae, two lies are perpedicular if ad oly if the product of their slopes is -. Vertical ad horizotal lies are perpedicular. o Slopes of perpedicular lies are opposite reciprocals. o Exaples: Choose if the lie pairs are parallel, perpedicular or either. ) x = ) y = x + y = Perpedicular y x = 0 ) x + y = x + y = Parallel ) x y = 8 Neither y = x Perpedicular Uit Parallel ad Perpedicular Lies Page of 0
Syllabus Objective:.8 The studet will forulate strategies to write the equatio of a lie give certai data. Review cocepts: Forats for equatios of lies o Stadard For: x + By = C { ust be a whole uber} o Slope-itercept For: y = x + b { = slope, b = y-itercept} o Poit-slope For: y y = x ( x) { = slope, ( x, y ) = x ad y coordiates of a give poit} o Through the poit (h, k): Horizotal lie: y = k Vertical lie: x = h Exaples: (all aswers give i slope-itercept for) - j k - -. Write the equatios of the lies j, k ad i slope-itercept for. Lie j : y = x + Lie k : y = x Lie : y = x +. Write the equatio of the lie perpedicular to j through (-, 0). 9 y = x. Write the equatio of the lie parallel to through (0, ). y = x + Uit Parallel ad Perpedicular Lies Page of 0
B ( -, ) C (, ) ( -, 0 ) D (, 0 ) - - E ( 0, - ) - -. Write the equatios of lies C, CE ad BD i slope-itercept for. C : y = x + CE : y = x BD : y = x +. Write the equatio of the lie parallel to BD through. y = x. Write the equatio of the lie perpedicular to BD through. y = x + 0 7. Write the equatio of the lie parallel to C through B. y = x + 8. Write the equatio of the lie perpedicular to CE through E. y = x Uit Parallel ad Perpedicular Lies Page of 0
This uit is desiged to follow the Nevada State Stadards for Geoetry, CCSD syllabus ad bechark caledar. It loosely correlates to Chapter of McDougal Littell Geoetry 00, sectios..7. The followig questios were take fro the st seester coo assesset practice ad operatioal exas for 008-009 ad would apply to this uit. Multiple Choice # Practice Exa (08-09) Operatioal Exa (08-09). I the figure below, lie is a trasversal. I the figure below lie is a trasversal. Which best describes the pair of agles: ad?. alterate exterior B. alterate iterior C. correspodig D. vertical. I the figure below, ad l is a trasversal. ( x ) Which best describes the pair of agles ad?. alterate exterior B. alterate iterior C. correspodig D. vertical I the figure below, ad l is a trasversal. ( x ) What is the value of x?. B. 9 C. 0 D. l What is the value of x?. B. C. D. l Uit Parallel ad Perpedicular Lies Page 7 of 0
. I the figure below, ad l is a trasversal. x I the figure below, ad l is a trasversal. 7 x l What is the value of x?. 80 B. 7 C. D.. I the figure below, FGH =. l H What value of x would ake lie l parallel to lie?. B. 9 C. D. 7. I the figure below, lies l ad are parallel. l Which stateet is true?. ad are cogruet. B. ad 8 are suppleetary. C. ad are suppleetary. D. ad 7 are cogruet. ( x 7) G 8 7 F l What is the value of x?. B. C. 8 D. I the figure below, FGH = 8. H l What value of x would ake lie l parallel to lie?. 8 B. 90 C. 9 D. 00 I the figure below, lies l ad are parallel. l F ( x ) Which stateet is true?. ad are suppleetary B. ad are suppleetary C. ad are cogruet D. ad are cogruet 8 8 7 G Uit Parallel ad Perpedicular Lies Page 8 of 0
Saple Nevada High School Proficiecy Exa questio(s): Take fro the 009 Istructioal Materials for the NHSPE provided by the Nevada Departet of Educatio. I which graph do lie ad p represet a syste of equatios that has o solutio?. I the diagra below, PF ad MH are parallel lies itersected by EK. What is the easure of 8 B 8 C 70 D 7 GJH?. I the figure below, TV ZW, UY VX, UX VW, ad TZ VW. What is the easure of 80 B 00 C 0 D 0 ZYU? Uit Parallel ad Perpedicular Lies Page 9 of 0
Saple ST Questio(s): Take fro College Board olie practice probles.. I the figure above, l. Which of the followig ust equal 80? () k + + r (B) k + p+ s (C) + p+ s (D) + p+ t (E) r+ s+ t. Three parallel lies i a plae are itersected by a fourth lie, forig twelve agles. If oe of the agles has easure 8, how ay of the other eleve agles have easure 8? Grid-I Uit Parallel ad Perpedicular Lies Page 0 of 0