Reteching Lines nd Angles Not ll lines nd plnes intersect. prllel plnes. prllel. } shows tht lines or plnes re prllel: < > < > A } ens Line A is prllel to line. skew. A } plne EFHG A plne FH } plne AEG plne HG } plne AFE Exples of prllel lines: < > } < > < > < > A } EF} GH Exples of skew lines: < > is skew to < > < > < > < > F, AE, EG, nd FH. G E H F In, use the figure t the right. Answers y vry. Sple:. Shde one set of prllel plnes.. Trce one set of prllel lines with solid line.. Trce one set of skew lines with dshed line. In 4 7, use the digr to ne ech of the following. 4. line tht is prllel to N O < > RS Q P Answers y vry. Sple: < 5. line tht is skew to NO > < > QU Answers y vry. Sple: < > RS 6. plne tht is prllel to NRTP plne OSUQ 7. three lines tht re prllel to < > R S OQ < > < > < > NP, RT, nd SU T U In 8, descrie the stteent s true or flse. E If flse, explin. F < > < > H G 8. plne HIKJ } plne IEGK 9. H} GK I Flse; these plnes intersect. true < > < > < > < > J K 0. HJ nd H re skew lines.. FG} KI Flse; the lines re in Flse; the lines intersect. different plnes, nd since they do not intersect they re skew.
Reteching (continued) Lines nd Angles The digr shows lines nd intersected y line x. Line x is trnsversl. A trnsversl is line tht intersects two or ore lines found in the se plne. The ngles fored re either interior ngles or exterior ngles. x 4 7 8 Interior Angles etween the lines cut y the trnsversl, 4, 5, nd 6 in digr ove Four types of specil ngle pirs re lso fored. Exterior Angles outside the lines cut y the trnsversl,, 7, nd 8 in digr ove Angle Pir efinition Exples lternte interior lternte exterior se-side interior inside ngles on opposite sides of the trnsversl, not liner pir outside ngles on opposite sides of the trnsversl, not liner pir inside ngles on the se side of the trnsversl j nd j6 j4 nd j5 j nd j8 j nd j7 j nd j5 j4 nd j6 corresponding in tching positions ove or elow the trnsversl, ut on the se side of the trnsversl j nd j5 j nd j7 j nd j6 j4 nd j8 Use the digr t the right to nswer 5.. Ne ll pirs of corresponding ngles. j nd j5; j nd j6; j nd j8; j4 nd j7. Ne ll pirs of lternte interior ngles. j4 nd j6; j nd j5 4. Ne ll pirs of se-side interior ngles. j4 nd j5; j nd j6 5. Ne ll pirs of lternte exterior ngles. j nd j8; j nd j7 t 4 s 7 8 c Use the digr t the right to nswer 6 nd 7. ecide whether the ngles re lternte interior ngles, se-side interior ngles, corresponding, or lternte exterior ngles. 6. nd 5 7. 4 nd 6 lternte exterior ngles se-side interior ngles 5 8 6 7 4
Reteching Properties of Prllel Lines When trnsversl intersects prllel lines, specil suppleentry nd congruent ngle pirs re fored. Suppleentry ngles fored y trnsversl intersecting prllel lines: 4 + 5 = 80 + 6 = 80 ongruent ngles fored y trnsversl intersecting prllel lines: 4 6 5 5 6 4 7 8 4 7 8 l q 8 7 Identify ll the nuered ngles congruent to the given ngle. Explin.. j6; vert... 4 c re@; j; corresp. re 5 d @; j4; lt. 7 ext. re@. 4 5 6 6 7 6 7 f j; vert. re@; j5; x lt. int. re@; j7; corresp. re@. 4. Supply the issing resons in the two-colun proof. i Given: g } h, i } j Prove: is suppleentry to 6. Stteents ) )? Resons ) g } h; i}j ) Given ) )? 4) nd 6 re suppleentry. 4)? 5) nd 6 re suppleentry. 5)? j 4 i j 9 0 e 5 95 4 7 8 Verticl ngles re congruent. 4 orresponding ngles re congruent. Se-side interior ngles re suppleentry. Sustitution property j; vert. re @; j4; lt. ext. re@. g h
Reteching (continued) Properties of Prllel Lines You cn use the specil ngle pirs fored y prllel lines nd trnsversl to find issing ngle esures. If = 00, wht re the esures of through 8? Suppleentry ngles: = 80-00 = 80 4 = 80-00 4 = 80 Verticl ngles: = = 00 Alternte exterior ngles: = 7 7 = 00 Alternte interior ngles: = 5 5 = 00 orresponding ngles: = 6 6 = 80 Se-side interior ngles: + 8 = 80 8 = 80 00 4 8 7 Wht re the esures of the ngles in the figure? (x + 0) + (x - 5) = 80 Se-Side Interior Angles Postulte 5x + 5 = 80 oine like ters. 5x = 75 Sutrct 5 fro ech side. x = 5 ivide ech side y 5. Find the esure of these ngles y sustitution. x + 0 = (5) + 0 = 80 x - 5 = (5) - 5 = 00 x - 0 = (5) - 0 = 50 To find, use the Se-Side Interior Angles Postulte: 50 + = 80, so = 0 (x 0) (x 0) (x 5) Find the vlue of x. Then find the esure of ech leled ngle. 5. 6. 7. (4x 0) x 50; 50; 90 5; 0; 50 (x 0) (8x 0) x (x 0) (x 5) 40; 00; 80; 65
Reteching Proving Lines Prllel Specil ngle pirs result when set of prllel lines is intersected y trnsversl. The converses of the theores nd postultes in Lesson - cn e used to prove tht lines re prllel. Theore -4: onverse of orresponding Angles Theore If 5, then }. 4 7 8 Theore -5: onverse of the Alternte Interior Angles Theore If 6, then }. Theore -6: onverse of the Se-Side Interior Angles Postulte If is suppleentry to 5, then }. Theore -7: onverse of the Alternte Exterior Angles Theore If 7, then }. For wht vlue of x is } c? The given ngles re lternte exterior ngles. If they re congruent, then } c. (x ) x - = 8 x = 40 x = 70 Which lines or line segents re prllel? Justify your nswers. OP } QN ecuse the@ ngles re lt. int.... W X. N A Y Z Q A} ecuse the@ WX}YZ ecuse the@ re lt. ext. ngles. re lt. int.. Find the vlue of x for which g } h. Then find the esure of ech leled ngle. 4. 5. 6. 6 g g (6x) (x 6) (0x ) (x 6) h h 55; 6 7; 4; 8 8 P g c O h (4x 8) 4; 78; 78
Reteching (continued) Proving Lines Prllel A flow proof is wy of writing proof nd type of grphic orgnizer. Stteents pper in oxes with the resons written elow. Arrows show the logicl connection etween the stteents. Write flow proof for Theore -: If trnsversl intersects two prllel lines, then lternte interior ngles re congruent. Given: / } Prove: t l l Given j _ j If lines, then corresponding re _. j _ j Verticl ngles re _. j _ j Trnsitive Property of _ oplete flow proof for ech. 7. oplete the flow proof for Theore - using the following steps. Then write the resons for ech step.. nd re suppleentry.. c. /} d. nd re suppleentry. Theore -: If trnsversl intersects two prllel lines, then se side interior ngles re suppleentry. t l Given: / } Prove: nd re suppleentry. l Given j _ j corr. _ j nd j re suppleentry. liner pir j nd j re suppleentry. Sustitution 8. Write flow proof for the following: Given: Prove: } j _ j Given j _ j Verticl re _. j _ j Sustitution If _ corresponding, then lines re prllel.
Reteching Prllel nd Perpendiculr Lines You cn use ngle pirs to prove tht lines re prllel. The postultes nd theores you lerned re the sis for other theores out prllel nd perpendiculr lines. Theore -8: Trnsitive Property of Prllel Lines If two lines re prllel to the se line, then they re prllel to ech other. If } nd }c, then }c. Lines,, nd c cn e in c different plnes. Theore -9: If two lines re perpendiculr to the se line, then those two lines re prllel to ech other. This is only true if ll the lines re in the se plne. If #d nd #d, then }. Theore -0: Perpendiculr Trnsversl Theore d If line is perpendiculr to one of two prllel lines, then it is lso perpendiculr to the other line. This is only true if ll the lines re in the se plne. If } nd c, nd #d, then #d, nd c#d. c d. oplete this prgrph proof of Theore -8. Given: d}e, e}f Prove: d}f Proof: ecuse it is given tht d}e, then is suppleentry to y the Se-Side Int. Angles Postulte. ecuse it is given tht e}f, then y the orresponding Angles Theore. So, y sustitution, is suppleentry to. y the onverse of the Se-Side Int. Angles Postulte, d}f. h d e f. Write prgrph proof of Theore -9. Given: t#n, t#o Prove: n}o Given tht t#n nd t#o, j = 90 nd j = 90, y def. of perpendiculr lines. Thus j@j. So, n}o ecuse of the onverse of the orr. Th. n o t
Reteching (continued) Prllel nd Perpendiculr Lines A crpenter is uilding cinet. A decortive door will e set into n outer fre. 5. If the lines on the door re perpendiculr to the top of the outer fre, wht ust e true out the lines?. The outer fre is de of four seprte pieces of olding. Ech piece hs ngled corners s shown. When the pieces re fitted together, will ech set of sides e prllel? Explin. 55 55 c. According to Theore -9, lines tht re perpendiculr to the se line re prllel to ech other. So, since ech line is perpendiculr to the top of the outer fre, ll the lines re prllel. 5 The ngles for the top nd otto pieces re 5. The ngles for the sides re 55. eterine whether ech set of sides will e prllel. rw the pieces s fitted together to deterine the esures of the new ngles fored. Use this to decide if ech set of sides will e prllel. The new ngle is the su of the ngles tht coe together. Since 5 + 55 = 90, the pieces for right ngles. Two lines tht re perpendiculr to the se line re prllel. So, ech set of sides is prllel.. An rtist is uilding osic. The osic consists of the repeting pttern shown t the right. Wht ust e true of nd to ensure tht the sides of the osic re prllel? = 50 nd = 5 40 40 65 65 4. Error Anlysis A student sys tht ccording to Theore -0, if < A > } < F > nd < A > # < A >, then < F > # < A >. Explin the student s error. < > < > A nd F re in different plnes. E F A
Reteching Prllel Lines nd Tringles Tringle Angle-Su Theore: The esures of the ngles in tringle dd up to 80. In the digr t the right, A is right tringle. Wht re nd? A Step + A = 90 Angle Addition Postulte + 0 = 90 Sustitution Property = 60 Sutrction Property of Equlity Step + + A = 80 Tringle Angle-Su Theore 60 + + 60 = 80 Sustitution Property + 0 = 80 Addition Property of Equlity = 60 Sutrction Property of Equlity 0 60 Find j.. 56. 55. 75 54 5 5 4 4. 59 5. 80 6. 9 50 50 8 Alger Find the vlue of ech vrile. 7. 6 8. X Y 9. 7 95 Z 6 7 7; 6; 4; 8; 6 X Y Z 8 7 X Y Z 90; 90; 58
Reteching (continued) Prllel Lines nd Tringles In the digr t the right, is n exterior ngle of the tringle. An exterior ngle is n ngle fored y one side of polygon nd n extension of n djcent side. For ech exterior ngle of tringle, the two interior ngles tht re not next to it re its reote interior ngles. In the digr, nd re reote interior ngles to. The Exterior Angle Theore sttes tht the esure of n exterior ngle is equl to the su of its reote interior ngles. So, = +. Wht re the esures of the unknown ngles? A + A + A = 80 Tringle Angle-Su Theore 45 + + = 80 Sustitution Property = 04 Sutrction Property of Equlity A + A = Exterior Angle Theore 45 + = Sustitution Property 76 = Sutrction Property of Equlity 45 A Wht re the exterior ngle nd the reote interior ngles for ech tringle? 0... P O 4 M L K M N J exterior: j4 exterior: jnop exterior: jjlm interior: j, j interior: jomn, jmno interior: jjkl, jljk Find the esure of the exterior ngle. G H. 5 4. U 4 5. 7 70 90 45 X W V T E 46 5 7 F
Reteching (continued) onstructing Prllel nd Perpendiculr Lines onstruct line prllel to line nd through point Y. Sple nswers shown. Z. Y.. Y Z Y Z Perpendiculr Postulte Through point not on line, there is exctly one line perpendiculr to the given line. Given: Point not on < > onstruct: line perpendiculr to < > through Step onstruct n rc centered t tht intersects < > t two points. Lel those points G nd H. Step onstruct two rcs of equl length centered t points G nd H. Step onstruct the line through point nd the intersection of the rcs fro Step. G H G H G Step Step Step H onstruct line perpendiculr to line n nd through point X. Sple nswers shown. 4. X 5. 6. n X n X n