Geometry. Parallel Lines. Slide 1 / 206. Slide 2 / 206. Slide 3 / 206. Table of Contents

Transcription

1 Slide 1 / 206 Slide 2 / 206 Geoetry Parallel Lies Table of otets Slide 3 / 206 lic o the topic to go to that sectio Lies: Itersectig, Parallel & Sew Lies & Trasversals Parallel Lies & Proofs Properties of Parallel Lies ostructig Parallel Lies PR Saple Questios

2 ostructios Videos Table of otets Slide 4 / 206 lic o the topic to go to that video Parallel Lies - orrespodig gles Parallel Lies - lterate Iterior gles Parallel Lies - lterate Exterior gles Parallel Lies - usig Meu Optios Slide 5 / 206 Lies: Itersectig, Parallel & Sew Retur to Table of otets Euclid's Fifth Postulate Slide 6 / 206 Euclid's Fifth Postulate is perhaps his ost faous. It's bothered atheaticias for thousads of years. Fifth Postulate: That, if a straight lie fallig o two straight lies ae the iterior agles o the sae side less tha two right agles, the two straight lies, if produced idefiitely, eet o that side o which are the agles less tha the two right agles.

3 Euclid's Fifth Postulate Slide 7 / 206 This seeed so atural that the Gree geoeters thought they should be able to prove it, ad would't eed it to be a postulate. They resisted usig it for years. However, they foud that they eeded it. d they could't prove it. They just had to postulate it. Euclid's Fifth Postulate Slide 8 / 206 It says that there are two possible cases if oe lie crosses two others. 1 2 Euclid's Fifth Postulate Slide 9 / 206 The pairs of agles o both sides, (either 1 & 3 or 2 & 4) each add up to 180º, two right agles, ad the two red lies ever eet. Lie this

4 Euclid's Fifth Postulate Slide 10 / 206 Or lie this. 1 2 Or,... Euclid's Fifth Postulate Slide 11 / 206 They add up to less tha 180º o oe side (agles 2 & 4), ad ore tha 180º o the other (agles 1 & 3), i which case the lies eet o the side with the saller agles. Lie this Euclid's Fifth Postulate Slide 12 / 206 Or lie this. 1 2

5 Euclid's Fifth Postulate Slide 13 / 206 They could't prove this fro the other axios ad postulates. ut, without it there were a lot of iportat pieces of geoetry they could't prove. So they gave i ad ade it the fial postulate of Euclidea Geoetry. For the ext thousads of years, atheaticias felt the sae way. They ept tryig to show why this postulate was ot eeded. No oe succeeded. Euclid's Fifth Postulate Slide 14 / 206 I 1866, erhard Riea too the other perspective. For his doctoral dissertatio he desiged a geoetry i which Euclid's Fifth Postulate was ot true, rather tha assuig it was. This led to o-euclidea geoetry. Where parallel lies always eet, rather tha ever eet. Euclid's Fifth Postulate Slide 15 / 206 ut half a cetury later, o- Euclidea geoetry, based o rejectig the fifth postulate, becae the atheatical basis of Eistei's Geeral Relativity. It creates the idea of curved spacetie. This is ow the accepted theory for the shape of our uiverse.

6 Euclid's Fifth Postulate Slide 16 / 206 Lies that are i the sae plae ad ever eet are called parallel. Lies that itersect are called oparallel or itersectig. ll lies that itersect are i a coo plae. Euclid's Fifth Postulate Slide 17 / 206 Lies that are i differet plaes ad ever eet are called sew. Q Lies & i the figure are sew. P The Parallel Postulate Slide 18 / 206 Oe way of restatig Euclid's Fifth Postulate is to say that parallel lies ever eet. extesio of it is the Parallel Postulate: give a lie ad a poit, ot o the lie, there is oe, ad oly oe, lie that ca be draw through the poit which is parallel to the lie. a a you estiate where the parallel lie would be?

7 The Parallel Postulate Slide 19 / 206 a you iagie ay other lie which could be draw through Poit ad still be parallel to lie a? a Parallel, Itersectig ad Sew Slide 20 / 206 Parallel lies are two lies i a plae that ever eet. We would say that lies E ad FG are parallel. Or, sybolically: E FG E F G Idicatig Lies are Parallel Slide 21 / 206 Lies caot be assued to be parallel uless it is idicated that they are. Just looig lie they are parallel is ot sufficiet. There are two ways of idicatig that lies are parallel. The first way is as show o the prior slide: E FG E F G

8 Idicatig Lies are Parallel Slide 22 / 206 The other way to idicate lies are parallel is to label the with arrows, as show below. The lies which share the arrow (show i red to ae it ore visible here) are parallel. If two differet pairs of lies are parallel, the oes with the atchig uber of arrows are parallel, as show o the ext slide. l Idicatig Lies are Parallel Slide 23 / 206 This idicates that lies l ad are parallel to each other. d, lies a ad b are parallel to each other. ut lies l ad are ot parallel to a ad b. a b l Parallel, Itersectig ad Sew Slide 24 / 206 If two differet lies i the sae plae are ot parallel they are itersectig, ad they itersect at oe poit. We also ow that four agles are fored. E F G

9 Parallel, Itersectig ad Sew Slide 25 / 206 Fro these four agles, there are four pairs of liear agles that are fored or liear pairs. Liear pairs are adjacet agles fored by itersectig lies, the agles are suppleetary. 1 & 3 are oe liear pair F 1 2 E G List the other liear pairs. Perpedicular Lies Slide 26 / 206 If the adjacet agles fored by itersectig lies are cogruet, the lies are perpedicular. E Sybolically, this is stated as E FG F G Sew Lies Slide 27 / 206 If two lies itersect, tha they defie a plae, so are co-plaar. Two lies that do ot itersect ca either be parallel if they are i the sae plae or sew if they are i differet plaes. P Lies & i the figure are sew. Q

10 Sew Lies Slide 28 / 206 H G Usig the followig diagra, ae a lie which is sew with Lie HG: a lie that does ot lie i a coo plae. E F Sew Lies Slide 29 / 206 H G Usig the followig diagra, ae a lie which is sew with Lie HG: a lie that does ot lie i a coo plae. E F 1 re lies a ad b sew? Slide 30 / 206 Yes a No G b

11 2 re lies a ad b sew? Slide 31 / 206 Yes a No G b 3 How ay lies ca be draw through ad parallel to Lie? Slide 32 / Nae all lies parallel to EF. Slide 33 / 206 H G H E F E HG

12 Slide 34 / Nae lies sew to EF. E H G E H F G 6 Two itersectig lies are always coplaar. Slide 35 / 206 True False 7 Two sew lies are coplaar. Slide 36 / 206 True False

13 8 oplete this stateet with the best appropriate word: Slide 37 / 206 Two sew lies are parallel. always ever soeties Slide 38 / 206 Lies & Trasversals Retur to Table of otets Trasversals Slide 39 / 206 Trasversal is a lie that itersects two or ore coplaar lies. (This is the ae of the lie that Euclid used to itersect two lies i his fifth postulate.) I the iage, trasversal, Lie, is show itersectig Lie ad Lie. Lie ad Lie ay or ay ot be parallel.

14 gles Fored by a Trasversal Slide 40 / 206 Whe a trasversal itersects two lies, eight agles are fored. These agles are give special aes. 6 5 Iterior gles are the 4 agles that lie betwee the two lies. gles Fored by a Trasversal Slide 41 / 206 Whe a trasversal itersects two lies, eight agles are fored. These agles are give special aes. 2 1 Exterior gles are the 4 agles that lie outside the two lies. 9 Nae all of the iterior agles. Slide 42 / E 5 F G 7 H 8

15 10 Nae all of the exterior agles. Slide 43 / E 5 2 F G 7 H 8 orrespodig gles Slide 44 / 206 orrespodig gles are pairs of agles that lie i the sae positio relative to the trasversal, as show above. There are four pairs of correspodig agles fored whe a trasversal itersects two lies. 11 Which agle correspods with 1? Slide 45 / 206

16 12 Which agle correspods with 7? Slide 46 / Which agle correspods with 6? Slide 47 / Which agle correspods with 4? Slide 48 / 206

17 lterate Iterior gles Slide 49 / 206 lterate Iterior gles are iterior agles that lie o opposite sides of the trasversal. 6 5 There are two pairs fored by the trasversal; they are show above i red ad blue. lterate Exterior gles Slide 50 / 206 lterate Exterior gles are exterior agles that lie o opposite sides of the trasversal. 2 1 There are two pairs fored by the trasversal; they are show above i red ad blue. 15 Which is the alterate iterior agle that is paired with 3? E 5 F 6 G 7 H 8 Slide 51 / 206

18 16 Which is the alterate exterior agle that is paired with 7? Slide 52 / E 5 F 6 G 7 H 8 17 Which is the alterate exterior agle that is paired with 2? Slide 53 / E 5 F 6 G 7 H 8 18 Which is the alterate iterior agle that is paired with 6? Slide 54 / E 5 F 6 G 7 H 8

19 Sae Side Iterior gles Slide 55 / 206 Sae Side Iterior gles are iterior agles that lie o the sae side of the trasversal. 6 5 There are two pairs fored by the trasversal; they are show above i red ad blue. Sae Side Exterior gles Slide 56 / 206 Sae Side Exterior gles are exterior agles that lie o the sae side of the trasversal. 2 1 There are two pairs fored by the trasversal; they are show above i red ad blue. 19 Which is the sae side iterior agle that is paired with 6? Slide 57 / E 6 F 7 G 8

20 20 Which is the sae side exterior agle that is paired with 7? Slide 58 / E 6 F 7 G 8 lassifyig gles Slide each word ito the appropriate square to classify each pair of agles. Slide 59 / 206 a. 1 ad 2 b. 1 ad 3 c. 1 ad 5 d. 3 ad 6 e. 3 ad 5 swer f. 3 ad 8 Liear Pair Vertical Sae-Side Exterior lterate Iterior orrespodig Sae Side Iterior lterate Exterior 21 3 ad 6 are... Slide 60 / 206 orrespodig gles lterate Exterior gles Sae-Side Exterior gles Vertical gles E Noe of these 1 2 l 5 6 t

21 22 1 ad 6 are. Slide 61 / 206 orrespodig gles lterate Exterior gles Sae-Side Exterior gles Vertical gles E Noe of these 1 2 l 5 6 t 23 2 ad 7 are. Slide 62 / 206 orrespodig gles lterate Iterior gles Sae-Side Iterior gles Vertical gles E Noe of these l t 24 4 ad 8 are. Slide 63 / 206 orrespodig gles lterate Exterior gles Sae-Side Exterior gles Vertical gles E Noe of these 1 2 l 5 6 t

22 25 1 ad 7 are. orrespodig gles lterate Exterior gles Sae-Side Exterior gles Vertical gles E Noe of these 1 2 l 5 6 t Slide 64 / ad 8 are. Slide 65 / 206 orrespodig gles lterate Exterior gles Sae-Side Exterior gles Vertical gles E Noe of these 1 2 l 5 6 t 27 2 ad 5 are. Slide 66 / 206 orrespodig gles lterate Iterior gles Sae-Side Iterior gles Vertical gles E Noe of these l t

23 Slide 67 / 206 Parallel Lies & Proofs Retur to Table of otets Properties of ogruece ad Equality Slide 68 / 206 I additio to the postulates ad theores used so far, there are three essetial properties of cogruece upo which we will rely as we proceed. There are also four properties of equality, three of which are closely related to atchig properties of cogruece. Properties of ogruece ad Equality Slide 69 / 206 They all represet the sort of coo sese that Euclid would have described as a oo Uderstadig, ad which we would ow call a xio. The cogruece properties are true for all cogruet thigs: lie segets, agles ad figures. The equality properties are true for all easures of thigs icludig legths of lies ad easures of agles.

24 Reflexive Property of ogruece Slide 70 / 206 thig is always cogruet to itself. While this is obvious, it will be used i provig theores as a reaso. For istace, whe a lie seget serves as a side i two differet triagles, you ca state that the sides of those triagles are cogruet with the reaso: Reflexive Property of ogruece I the diagra, Reflexive Property of Equality Slide 71 / 206 The easures of agles or legths of sides ca be tae to be equal to theselves, eve if they are parts of differet figures, with the reaso: Reflexive Property of Equality The Lie Seget dditio Postulate tell us that = + ad = + The Reflexive Property of Equality idicates that the legth is equal to itself i both equatios Syetric Property of ogruece Slide 72 / 206 If oe thig is cogruet to aother, the secod thig is also cogruet to the first. gai, this is obvious but allows you to reverse the order of the stateets about cogruet properties with the reaso: Syetric Property of ogruece For exaple: is cogruet to EF that EF is cogruet to,

25 Syetric Property of Equality Slide 73 / 206 If oe thig is equal to aother, the secod thig is also equal to the first. gai, this is obvious but allows you to reverse the order of the stateets about equal properties with the reaso: Syetric Property of Equality For exaple: If = EF, the EF =, Trasitive Property of ogruece Slide 74 / 206 If two thigs are cogruet to a third thig, the they are also cogruet to each other. So, if Δ is cogruet to ΔEF ad ΔLMN is also cogruet to ΔEF, the we ca say that Δ is cogruet to ΔLMN due to the With the reaso: Trasitive Property of ogruece Trasitive Property of Equality Slide 75 / 206 If two thigs are equal to a third thig, the they are also equal to each other. If = ad =, the = This is idetical to the trasitive property of cogruece except it deals with the easure of thigs rather tha the thigs. Trasitive Property of Equality

26 Substitutio Property of ogruece Slide 76 / 206 If oe thig is equal to aother, the oe ca be substituted for aother. This is a coo step i a proof where oe thig is prove equal to aother ad replaces that other i a expressio usig the reaso: Substitutio Property of ogruece For istace if x + y = 12, ad x = 2y We ca substitute 2y for x to get 2y + y = 12 ad use the divisio property to get y = 4 orrespodig gles Theore Slide 77 / 206 If parallel lies are cut by a trasversal, the the correspodig agles are cogruet. ccordig to the orrespodig gles which of the above agles are cogruet? orrespodig gles Proof Slide 78 / 206 To eep the arguet clear, let's just prove oe pair of those agles equal here. You ca follow the sae approach to prove the other three pairs of agles equal. We could pic ay pair of correspodig agles: 2 & 6; 3 & 7; 1 & 5; or 4 & 8. Together, let's prove that 2 & 6 are cogruet.

27 orrespodig gles Proof Slide 79 / 206 Give: Lie ad Lie are parallel ad itersected by lie Prove: 2 = 6 orrespodig gles Proof Slide 80 / 206 Stateet 1 Lie ad Lie are parallel ad itersected by lie Reaso 1 Give orrespodig gles Proof Slide 81 / 206 Reeber Euclid's Fifth Postulate. The oe that o oe lies but which they eed. This is where it's eeded. Fifth Postulate: That, if a straight lie fallig o two straight lies ae the iterior agles o the sae side less tha two right agles, the two straight lies, if produced idefiitely, eet o that side of which are the agles less tha the two right agles.

28 Euclid's Fifth Postulate Recall that we leared early i this uit that this eas that... If the pairs of iterior agles o both sides of the trasversal, (both 1 & 3 or 2 & 4) each add up to 180º, the two red lies are parallel...ad ever eet. Slide 82 / So, i this case, which agles ust add up to 180º based o Euclid's Fifth Postulate? Slide 83 / & 4 6 & 8 4 & 5 3 & 6 E ll of the above orrespodig gles Proof Slide 84 / 206 Stateet 2 3 & 6 are suppleetary 4 & 5 are suppleetary Reaso 2 Euclid's Fifth Postulate Which other agle is suppleetary to 3, because together they for a straight agle? How about to agle 6?

29 orrespodig gles Proof Slide 85 / 206 Stateet 3 2 & 3 are suppleetary Reaso 3 gles that for a liear pair are suppleetary What do we ow about agles who have the sae suppleets? orrespodig gles Proof Slide 86 / 206 Stateet 4 2 = 6 Reaso 4 Two agles suppleetary to the sae agle are equal orrespodig gles Proof Give: Lie ad Lie are parallel ad itersected by Lie Prove: 2 = 6 Slide 87 / Stateet Lie ad Lie are parallel ad itersected by Lie 4 & 5 are suppleetary 3 & 6 are suppleetary 3 3 & 2 are suppleetary 4 2 = 6 Reaso Give Euclid's Fifth Postulate gles that for a liear pair are suppleetary Two agles suppleetary to the sae agle are equal

30 Properties of Parallel Lies Slide 88 / 206 This is a iportat result, which was oly ade possible by Euclid's Fifth Postulate. It leads to soe other pretty iportat results. It allows us to prove soe pairs of agles cogruet ad soe other pairs of agles suppleetary. d, it wors i reverse, if ay of these coditios are et we ca prove that lies are parallel. overses of Parallel Lie Proofs Slide 89 / 206 We proved that if two lies are parallel, their correspodig agles are equal. The coverse ust also be true: If two lies are cut by a trasversal ad the correspodig agles are cogruet, the the lies are parallel. overses of Parallel Lie Proofs Slide 90 / 206 The sae reaso: orrespodig gles of Parallel Lies are Equal is used i each case. To prove the relatioship betwee certai agles if we ow the lies are parallel OR To prove that the lies are parallel if we ow the relatioship betwee those agles.

31 overses of Parallel Lie Proofs Slide 91 / 206 This patter will be true of each theore we prove about the agles fored by the trasversal itersectig the parallel lies. They prove the relatioship betwee agles of lies ow to be parallel, or they prove that the lies are parallel. lterate Iterior gles Theore Slide 92 / 206 If parallel lies are cut by a trasversal, the the alterate iterior agles are cogruet. ccordig to the lterate Iterior gles Theore which of these agles are cogruet? lterate Iterior gles Proof Slide 93 / 206 Give: Lie ad Lie are parallel ad itersected by lie Prove: 3 5 ad 4 6

32 lterate Iterior gles Proof Slide 94 / 206 Stateet 1 Lie ad Lie are parallel ad itersected by lie Reaso 1 Give ccordig to the orrespodig gles Theore which of the above agles are cogruet? lterate Iterior gles Proof Slide 95 / 206 Stateet Reaso 2 Whe two parallel lies are cut by a trasversal, the correspodig agles are cogruet. Which other agle is cogruet to 1? Which other agle is cogruet to 2? lterate Iterior gles Proof Slide 96 / 206 Stateet Reaso 3 Vertical agles are cogruet. What do we ow about agles that are cogruet to the sae agle?

33 lterate Iterior gles Proof Slide 97 / 206 Stateet Reaso 4 Trasitive property of cogruece ut those are the pairs of alterate iterior agles which we set out to prove are cogruet. So, our proof is coplete: lterate Iterior gles of Parallel Lies are ogruet lterate Iterior gles Proof Give: Lie ad Lie are parallel ad itersected by Lie Prove: Slide 98 / Stateet Lie ad Lie are parallel ad itersected by Lie ad 2 6 Reaso Give If two parallel lies are cut by a trasversal, the the correspodig agles are ad 2 4 Vertical gles are ad 4 6 Trasitive Property of ogruece overse of lterate Iterior gles Theore Slide 99 / 206 If two lies are cut by a trasversal ad the alterate iterior agles are cogruet, the the lies are parallel.

34 lterate Exterior gles Theore Slide 100 / 206 If two parallel lies are cut by a trasversal, the the alterate exterior agles are cogruet. ccordig to the lterate Exterior gles Theore which agles are cogruet? lterate Exterior gles Theore Slide 101 / 206 Sice the proof for the lterate Exterior gles Theore is very siilar to the lterate Iterior gles Theore, you will be copletig this proof as a part of your Hoewor for this lesso. overse of lterate Exterior gles Theore Slide 102 / 206 If two lies are cut by a trasversal ad the alterate exterior agles are cogruet, the the lies are parallel.

35 Sae-Side Iterior gles Theore ccordig to Sae-Side Iterior gles Theore: If two parallel lies are cut by a trasversal, <3+<6=180 the the sae-side 0 iterior agles are suppleetary. <4+<5=180 0 Slide 103 / 206 ccordig to the Sae-Side Iterior gles Theore which pairs of agles are suppleetary? Sae-Side Iterior gles Proof Slide 104 / 206 Give: Lies ad are parallel ad itersected by lie Prove: 3 & 6 are suppleetary ad 4 & 5 are suppleetary 29 Which reaso applies to step 1? efiitio of suppleetary Euclid's Fifth Postulate Give lterate Iterior s are E orrespodig s are swer Slide 105 / Stateet Lies ad are parallel ad itersected by lie = 180º = 180º Reaso 3? efiitio of suppleetary s??

36 30 Which reaso applies to step 2? Slide 106 / 206 efiitio of suppleetary Euclid's Fifth Postulate Give lterate Iterior s are E orrespodig s are swer 1 2 Stateet Lies ad are parallel ad itersected by lie = 180º = 180º Reaso?? 3? efiitio of suppleetary s 31 Which stateet should be i step 3? Slide 107 / ad 6 are suppleetary 6 ad 5 are suppleetary 2 ad 6 are suppleetary 4 ad 5 are suppleetary E 3 ad 5 are suppleetary swer 1 2 Stateet Lie ad Lie are parallel ad itersected by Lie The sus of 3 ad 6 ad of 4 ad 5 are 180º. 3? Reaso?? efiitio of suppleetary agles Sae Side Iterior gles Proof Slide 108 / 206 Give: Lie ad Lie are parallel ad itersected by Lie Prove: 3 & 6 are suppleetary ad 4 & 5 are suppleetary Stateet Lies ad are parallel ad itersected by lie = 180º = 180º 3 ad 6 are suppleetary 4 ad 5 are suppleetary Reaso Give Euclid's Fifth Postulate efiitio of suppleetary s

37 overse of Sae-Side Iterior gles Theore Slide 109 / 206 If two lies are cut by a trasversal ad the sae-side iterior agles are suppleetary, the the lies are parallel. Sae-Side Exterior gles Theore If two parallel lies are cut by a trasversal, the the sae-side exterior agles are suppleetary. Slide 110 / 206 ccordig to the Sae- Side Exterior gles Theore which agles are suppleetary? Sae Side Exterior gles Proof Slide 111 / 206 Give: Lies ad are parallel ad itersected by Lie Prove: 2 & 7 are suppleetary I provig that 2 & 7 are suppleetary we are thereby provig that 1 & 8 are suppleetary as the sae arguets apply to both pairs of agles.

38 32 Which reaso applies to step 1? efiitio of suppleetary s Substitutio property of equality Give s that for a liear pair are suppleetary E s suppleetary to the sae are 1 Stateet Lie ad Lie are parallel ad itersected by Lie 2? 3? Reaso Sae-side iterior agles are suppleetary? gles that for a liear pair are suppleetary ad 3 7? 5 2 & 7 are suppleetary? Slide 112 / 206 swer 33 Which stateet is ade i step 2? 2 & 1 are suppleetary 7 & 8 are suppleetary 3 & 6 are suppleetary 4 & 5 are suppleetary E 5 & 8 are suppleetary Slide 113 / Stateet Lie ad Lie are parallel ad itersected by Lie 2? 3? Reaso? Sae-side iterior agles are suppleetary gles that for a liear pair are suppleetary ad 3 7? 5 2 & 7 are suppleetary? swer 34 Which stateet is ade i step 3? E 2 & 3 are suppleetary 1 & 3 are suppleetary 6 & 8 are suppleetary 6 & 7 are suppleetary 7 & 1 are suppleetary 1 Stateet Lie ad Lie are parallel ad itersected by Lie 2? 3? Reaso? Sae-side iterior agles are suppleetary gles that for a liear pair are suppleetary ad 3 7? 5 2 & 7 are suppleetary? swer Slide 114 / 206

39 35 Which reaso applies to step 4? efiitio of suppleetary s Substitutio property of equality Give s that for a liear pair are suppleetary E s suppleetary to the sae are 1 Stateet Lie ad Lie are parallel ad itersected by Lie 2? 3? Reaso? Sae-side iterior agles are suppleetary gles that for a liear pair are suppleetary ad 3 7? 5 2 & 7 are suppleetary? Slide 115 / 206 swer 36 Which reaso applies to step 5? efiitio of suppleetary s Substitutio property of equality Give gles that for a liear pair are suppleetary E s suppleetary to the sae are 1 Stateet Lies ad are parallel ad itersected by lie 2? 3? Reaso? Sae-side iterior agles are suppleetary gles that for a liear pair are suppleetary ad 3 7? 5 2 & 7 are suppleetary? Slide 116 / 206 swer Sae Side Exterior gles Proof Give: Lie ad Lie are parallel ad itersected by Lie Prove: 2 & 7 are suppleetary (ad thereby that 1 & 8 are as well) 1 Stateet Lies ad are parallel ad itersected by lie 2 3 & 6 are suppleetary 3 2 & 3 are suppleetary 6 & 7 are suppleetary ad 3 7 Reaso Give Sae-side iterior agles are suppleetary gles that for a liear pair are suppleetary gles suppleetary to the sae agle are cogruet 6 2 & 7 are suppleetary Substitutio Property of Equality Slide 117 / 206

40 overse of Sae Side Exterior gles Theore Slide 118 / 206 If two lies are cut by a trasversal ad the sae side exterior agles are suppleetary, the the lies are parallel. Slide 119 / 206 Properties of Parallel Lies Retur to Table of otets Properties of Parallel Lies Slide 120 / 206 Exaple: If 4 = 54º, fid the 8. Explai your aswer.

41 Properties of Parallel Lies Slide 121 / 206 Exaple: If 3 = 125º, fid the 5. Explai your aswer. Properties of Parallel Lies Slide 122 / 206 Exaple: If 2 = 78º, fid the 8. Explai your aswer. Properties of Parallel Lies Slide 123 / 206 Exaple: If 3 = 163º, fid 6. Explai your aswer.

42 Properties of Parallel Lies Slide 124 / 206 Nae all of the agles cogruet to 1. Properties of Parallel Lies Slide 125 / 206 Nae all of the agles suppleetary to Fid all of the agles cogruet to all of the above l Slide 126 / l 5 6

43 38 Fid the value of x. Slide 127 / 206 l 5x+30 l 120º 39 Fid the value of x. Slide 128 / 206 l 1.5x+40 l 110º 40 If the 4 = 116º the 9 = º? Slide 129 / 206 p p

44 41 If the 15 = 57º, the the 2 = º. Slide 130 / oe of the above p p Properties of Parallel Lies Slide 131 / 206 There are several theores ad postulates related to parallel lies. t this tie, please go to the lab titled, "Properties of Parallel Lies". lic here to go to the lab titled, "Properties of Parallel Lies" Extedig Lies to Mae Trasversals Slide 132 / 206 Fid º 1 41º With the give diagra, o trasversal exists but we ca exted oe of the lies to ae a trasversal.

45 Extedig Lies to Mae Trasversals Slide 133 / 206 Fid º 1 41º The fill i the agle which is correspodig to the 131º agle. Extedig Lies to Mae Trasversals Slide 134 / 206 Fid º 1 131º 41º The fill i the suppleetary agle to that ewly labeled agle. Extedig Lies to Mae Trasversals Slide 135 / 206 Fid º 1 41º 131º 49º s you ay recall, the third agle i the triagle ust ae the su of the agles equal to 180º.

46 Extedig Lies to Mae Trasversals Slide 136 / 206 Fid º 1 90º 131º 49º 41º d, fially that agle 1 is suppleetary to that 90º agle. Extedig Lies to Mae Trasversals Slide 137 / 206 Fid º 1 90º 131º 49º 41º gle 1 = 90º ouble Trasversals Slide 138 / 206 Fid the values of x ad y. 132º (4y+12)º xº

47 Trasversals ad Perpedicular Lies Slide 139 / 206 Fid the values of x, y, ad z. (14x+6)º 66º 2zº (3y-6)º 42 Fid the 1. Slide 140 / º 110º 43 Fid the value of x Slide 141 / º (3x)º

48 44 Fid the value of x. Slide 142 / 206 (2x-3)º (4x-61)º 45 Fid the value of x. Slide 143 / º (16x+10)º Provig Lies are Parallel Slide 144 / 206 If 3 = 56º, fid the 7 that aes lies ad parallel. Explai your aswer.

49 Provig Lies are Parallel Slide 145 / 206 If 4 = 110º, fid the 6 that aes lies ad parallel. Explai your aswer. Provig Lies are Parallel Slide 146 / 206 If 1 = 48º, fid the 7 that aes lies ad parallel. Explai your aswer. Provig Lies are Parallel Slide 147 / 206 If 5 = 54º, fid the 4 that aes lies ad parallel. Explai your aswer.

50 46 Which stateet would show lies ad parallel? 2 = =180º Slide 148 / = =90º I this diagra, which of the followig is true? Slide 149 / 206 e f f g h i e g e f g 123º 64º 57º h 132º i 48 If lies a ad b are cut by a trasversal which of the followig would NOT prove that they are parallel? Slide 150 / 206 orrespodig agles are cogruet. lterate iterior agles are cogruet. Sae-side iterior agles are copleetary. Sae-side iterior agles are suppleetary. E ll of the above.

51 49 Fid the value of x for which a b. Slide 151 / 206 a b 115º xº 50 Fid the value of x which aes a b. Slide 152 / 206 (6x-20)º a 2xº b 51 Fid the value of x for which. Slide 153 / 206 (14x - 10)º (5x)º

52 52 If a b, how ca we prove 1 = 4? Slide 154 / 206 orrespodig agles theore overse of correspodig agles theore lterate Iterior agles theore overse of alterate iterior agles theore a b c 53 If 1 = 3, how ca we prove a b? Slide 155 / 206 orrespodig agles theore overse of correspodig agles theore lterate Iterior agles theore overse of alterate iterior agles theore a b c 54 Give 1 = 2, 3 = 4, what ca we prove? (choose all that apply) a b c d lie a is perpedicular to lie c lie b is perpedicular to lie d Slide 156 / 206 a c b d

53 55 Give a b, what ca we prove? 1 = 2 1 = 4 2 = = 180º Slide 157 / 206 a 2 b c Slide 158 / 206 ostructig Parallel Lies Retur to Table of otets Parallel Lie ostructio Slide 159 / 206 ostructig geoetric figures eas you are costructig lies, agles, ad figures with basic tools accurately. We use a copass, ad straightedge for costructios, but we also use soe paper foldig techiques. lic here to see a aiated costructio of a parallel lie through a poit. ostructio by: MathIsFu

54 Parallel Lie ostructio Slide 160 / 206 Give: Lie ad poit, ot o the lie, draw a secod lie that is parallel to ad goes through poit. There are three differet ethods to achieve this. Method 1: orrespodig gles Parallel Lie ostructio: Method 1 Slide 161 / 206 The theory of this costructio is that the correspodig agles fored by a trasversal ad parallel lies are equal. To use this theory, we will draw a trasversal through that creates a acute agle with lie. The we will create a cogruet agle at, o the sae side of the trasversal as the acute agle fored with lie. Sice these are cogruet correspodig agles, the lies are parallel. Parallel Lie ostructio: Method 1 Slide 162 / 206 Step 1: raw a trasversal to through poit that itersects at poit. acute agle with poit as a vertex is fored (the easure of the agle is ot iportat). The agle is the agle we will replicate at poit o the sae side of the trasversal.

55 Parallel Lie ostructio: Method 1 Step 2: eter the copass at poit ad draw a arc that itersects both lies. Usig the sae radius of the copass, ceter it at poit ad draw aother arc. Label the poit of itersectio o the secod arc F. Slide 163 / 206 We are followig the procedure we used previously to costruct a cogruet agle. This step is to ar the sae distaces fro ad fro. E F Parallel Lie ostructio: Method 1 Slide 164 / 206 Step 3: Set the copass radius to the distace betwee the two itersectio poits of the first arc. This replicates the distace betwee where the arc itersects the two legs of the agle at the sae distace fro the vertex. F Whe that is replicated at the agle costructed will be cogruet with the origial agle. Parallel Lie ostructio: Method 1 Slide 165 / 206 Step 4: eter the copass at the poit F where the secod arc itersects lie ad draw a third arc. This assures that the arc legth for each agle is idetical. F

56 Parallel Lie ostructio: Method 1 Slide 166 / 206 Step 5: Mar the arc itersectio poit E ad use a straight edge to joi ad E. FE therefore E F Parallel Lie ostructio: Method 1 Slide 167 / 206 Here are y parallel lies without the costructio lies. Slide 168 / 206 Video eostratig ostructig Parallel Lies with orrespodig gles usig yaic Geoetric Software lic here to see video

57 Parallel Lie ostructio: Method 2 Slide 169 / 206 The theory of this costructio is that the alterate iterior agles fored by a trasversal ad parallel lies are equal. To use this theory, we will draw a trasversal through that creates a acute agle with lie. The we will create a cogruet agle at, o the opposite side of the trasversal as the acute agle fored with lie. Sice these are cogruet alterate iterior agles the lies are parallel. Method 2: lterate Iterior gles Slide 170 / 206 Give ad poit, ot o the lie, draw a secod lie that is parallel to ad goes through poit. Method 2: lterate Iterior gles Slide 171 / 206 Step 1: raw a trasversal to lie through poit that itersects lie at poit. acute agle with poit as a vertex is fored. The agle is the agle we will replicate at poit o the opposite side of the trasversal.

58 Method 2: lterate Iterior gles Step 2: eter the copass at poit ad draw a arc that itersects both lies, at poits E ad at F. We are followig the procedure we used previously to costruct a cogruet agle. This step is to ar the sae distace fro o both lies. Slide 172 / 206 F E Method 2: lterate Iterior gles Step 3: Usig the sae radius, ceter the copass at poit ad draw a arc that passes through lie at poit G. Slide 173 / 206 This replicates the sae distace alog the trasversal ad the ew lie that will be draw fro as was doe for the distaces fro. G F E Method 2: lterate Iterior gles Slide 174 / 206 Step 4: gai, with the sae radius, ceter the copass at poit G ad draw a third arc which itersects the earlier oe, at H. This ow fids that sae distace fro where the arc itersects the trasversal ad the ew lie as was the case for the trasversal ad the origial lie. H G F E

59 Method 2: lterate Iterior gles Step 5: raw lie H, which will be parallel to lie sice their alterate iterior agles are cogruet. Slide 175 / 206 Sice agles HG ad are cogruet ad are alterate iterior agles, the lies are parallel. H G F E Method 2: lterate Iterior gles Slide 176 / 206 Here are the lies without the costructio steps show. H Slide 177 / 206 Video eostratig ostructig Parallel Lies with lterate Iterior gles usig yaic Geoetric Software lic here to see video

60 Method 3: lterate Exterior gles Give lie ad poit, ot o the lie, draw a secod lie that is parallel to lie ad goes through poit. Slide 178 / 206 Method 3: lterate Exterior gles Step 1: raw a trasversal to lie through poit that itersects lie at poit. acute agle with poit as a vertex is fored. Slide 179 / 206 Method 3: lterate Exterior gles Slide 180 / 206 Step 2: eter the copass at poit ad draw a arc to itersect lies ad o the opposite side of poit at ad E. E

61 Method 3: lterate Exterior gles Slide 181 / 206 Step 3: Keepig the radius the sae draw a arc cetered o that itersects lie above, at F. F E Method 3: lterate Exterior gles Slide 182 / 206 Step 4: Still eepig the radius the sae draw a arc cetered o F that itersects the arc cetered o, at H. F G E Method 3: lterate Exterior gles Slide 183 / 206 Step 5: raw lie E, which is parallel to lie sice the alterate exterior agles fored by the trasversal are cogruet. F E G FE therefore E G

62 Method 3: lterate Exterior gles Here are the lies without the costructio lies. Slide 184 / 206 E Slide 185 / 206 Video eostratig ostructig Parallel Lies with lterate Exterior gles usig yaic Geoetric Software lic here to see video Parallel Lie ostructio Usig Patty Paper Slide 186 / 206 Step 1: raw a lie o your patty paper. Label the lie g. raw a poit ot o lie g ad label the poit. g

63 Parallel Lie ostructio Usig Patty Paper Slide 187 / 206 Step 2: Fold your patty paper so that the two parts of lie g lie exactly o top of each other ad poit is i the crease. g Parallel Lie ostructio Usig Patty Paper Step 3: Ope the patty paper ad draw a lie o the crease. Label this lie h. Slide 188 / 206 g h Parallel Lie ostructio Usig Patty Paper Slide 189 / 206 Step 4: Through poit, ae aother fold that is perpedicular to lie h. g h

64 Parallel Lie ostructio Usig Patty Paper Slide 190 / 206 Step 5: Ope the patty paper ad draw a lie o the crease. Label this lie i. i g h ecause lies i ad g are perpedicular to lie h they are parallel to each other. Therefore lie i lie g. Slide 191 / 206 Video eostratig ostructig a Parallel Lie usig Meu Optios of yaic Geoetric Software lic here to see video 1 lic here to see video 2 56 The lies i the diagra below are parallel because of the: lterate Iterior gles Theore lterate Exterior gles Theore Sae-Side gles Theore orrespodig gles Postulate Slide 192 / 206 F E G

65 57 The lies below are show parallel by the: lterate Iterior gles Theore lterate Exterior gles Theore Sae-Side gles Theore orrespodig gles Postulate Slide 193 / 206 F E G 58 The below lies are show parallel by the: lterate Iterior gles Theore lterate Exterior gles Theore Sae-Side gles Theore orrespodig gles Postultate Slide 194 / 206 E F G PR Saple Test Questios Slide 195 / 206 The reaiig slides i this presetatio cotai questios fro the PR Saple Test. fter fiishig uit 2, you should be able to aswer these questios. Good Luc! Retur to Table of otets

66 PR Saple Test Questios Slide 196 / 206 I the figure show, Lie F itersects lies ad EH at poits ad F, respectively. Slide 197 / 206 Give: FE Prove: F FE E 59 FE Give efiitio of cogruet agles Vertical agles are cogruet Reflexive property of cogruece E Syetric property of cogruece F Trasitive property of cogruece H F G Slide 198 / 206 I the figure show, Lie F itersects lies ad EH at poits ad F, respectively. Give: FE Prove: F FE E F 60 F H G Give efiitio of cogruet agles Vertical agles are cogruet Reflexive property of cogruece E Syetric property of cogruece F Trasitive property of cogruece

67 Slide 199 / 206 I the figure show, Lie F itersects lies ad EH at poits ad F, respectively. Give: FE Prove: F FE E F 61 F FE Give efiitio of cogruet agles Vertical agles are cogruet Reflexive property of cogruece E Syetric property of cogruece F Trasitive property of cogruece H G Slide 200 / 206 I the figure show, Lie F itersects lies ad EH at poits ad F, respectively. Give: FE Prove: F FE E F opleted proof show below. H G Stateet Reaso 1 FE Give 2 F Vertical gles are cogruet 3 F FE Trasitive property of cogruece PR Saple Test Questios Slide 201 / 206 ircle the reaso that supports each lie of the proof.

68 Slide 202 / 206 I the figure show Lie F itersects lies ad EH at poits ad F, respectively. Give: = FE Prove: FE + F = 180º E F H G 62 = FE Give gles that for a liear pair are suppleetary gles that are adjacet are suppleetary Reflexive property of equality E Substitutio property of equality F Trasitive property of equality I the figure show Lie F itersects lies ad EH at poits ad F, respectively. Slide 203 / 206 Give: = FE Prove: FE + F = 180º E F H G 63 + F = 180º Give gles that for a liear pair are suppleetary gles that are adjacet are suppleetary Reflexive property of equality E Substitutio property of equality F Trasitive property of equality Slide 204 / 206 I the figure show Lie F itersects lies ad EH at poits ad F, respectively. Give: = FE Prove: FE + F = 180º F E H G 64 FE + F = 180º Give gles that for a liear pair are suppleetary gles that are adjacet are suppleetary Reflexive property of equality E Substitutio property of equality F Trasitive property of equality

69 I the figure show Lie F itersects lies ad EH at poits ad F, respectively. Slide 205 / 206 Give: = FE Prove: FE + F = 180º E F H G Stateet Reaso 1 = FE Give 2 + F = 180º gles that for a liear pair are suppleetary 3 FE + F = 180º Substitutio Property of Equality Slide 206 / 206