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DAY23 GOAL: Use Definitions, Postulates, Properties and Theorems to prove other shortcuts for measuring. GEOMETRIC POSTULATES; Basic rules for measuring Ruler Postulate Segment Addition Postulate Protractor Postulate Angle Addition Postulate Distance = | a – b | If B is between A and C then AB + BC = AC Angle Measure = |a – b| If B is in the interior of ∠ AOC, then m ∠ AOB + m ∠ BOC = m ∠ AOC ALGEBRAIC PROPERTIES CONGRUENCE PROPERTIES Addition, Subtraction, Multiplication, Division, Reflexive, Symmetric, Transitive, Distributive Reflexive, Symmetric, Transitive _______________________________________________________________________ DEFINITIONS Straight Angle Definition Linear Pair Definition Perpendicular Pair Definition Supplementary Angles Definition Complementary Angles Definition Vertical Angles Definition USED IN PROOFS An angle is a straight angle iff its measure is 180 Two angles form a linear pair iff non-shared sides form a straight angle. Two angles form a perpendicular pair iff non-shared sides form a right angle. The measure of two angles adds up to 180 iff the angles are supplementary. The measure of two angles adds up to 90 iff the angles are complementary. Two angles are vertical angles iff their sides form two pairs of opposite rays. THEOREMS: Have to be proven before being used TO 1. 2. 3. PROVE THEOREMS Start with the given information. Use a Definition to explain the given information. Reason back from what you would like to prove using Definitions, Properties, Postulates and Theorems that have already been proven. 4. Conclude with what needs to be proven. ___________________________________________________________________________ LINEAR PAIR THEOREM: If two angles form a linear pair, then the angles are supplementary. GIVEN: ∠1 and ∠2 form a linear pair PROVE: ∠1 and ∠2 are supplementary STATEMENT REASONS 1. ∠1 and ∠2 form a linear pair. Given 2. ∠TAM is a straight angle. Linear Pair Definition (Two angles form a linear pair iff their non-shared sides form a straight angle) ! 3. m∠TAM = 180 Straight Angle Definition (An angle is a straight angle iff its measure is 180) 4. m∠1+ m∠2 = m∠TAM 5. m∠1+ m∠2 = 180 6. Angle Addition Postulate ! Substitution ∠1 and ∠2 are supplementary angles Supplementary Angle Definition (The measure of two angles is 180 iff the angles are supplementary) Given QED Definition of Supplementary The measure of two angles sums to 180° iff the angles are supplementary. Substitution Property of Equality If mÐ 1 + m Ð 2 = mÐ TAM and mÐTAM = 180° Then mÐ1 + m Ð2 = 180° Definition of Linear Pair Two angles form a linear pair f two angles sums 180° re supplementary. m ∠BAT = 180° TO PROVE THEOREMS f Straight Angle a straight angle, 1. Start with the given information. asure is 180°. ition Postulate 2. Use a Definition to explain the given information. es are adjacent, f their individual 3. Reason back from what you would like to prove using the measure of the Theorems that have already been proven. y their non-shared 4.Conclude with what needs to be proven ides . Definitions, Properties, Postulates and AT form linear pair ______________________________________________________________________ If two angles form a linear pair, then the angles are supplementary. AT = 180° _____________________________________________________________________ on Property of uality VERTICAL ANGLES THEOREM: If two angles are vertical angles, then the angles have c = d then a+b = d equal measures. Vertical Angles Theorem form a linear pair If two angles are vertical angles ∠2 m a linear pair ∠1 ∠3 Then they have equal measures, m ∠ 3 = 180° = 180° GIVEN: ∠1 and ∠2 are vertical angles air Theorem ∠ 1 have and ∠equal 2 are vertical Angles ∠1Given and ∠2 measures form a linearPROVE: pair, plementary. Prove that ∠ 1 and ∠ 2 have equal measures. Given LINEAR PAIR THEOREM: ∠BAT = m∠ CAT 180° - m ∠ 3 180° - m ∠ 3 STATEMENT Statements 1. ∠1 and ∠2 are vertical angles 1 = m∠ 2 ve equal measures) of Linear Pair m a linear pair iff 2. sides form a straight are supplementary e supplementary operty of Equality mber is subtracted3. es of an equation, uation is equivalent vertical angles f Supplementary two angles sums to 4. 180° e supplementary. roperty of Equality 1) for (180° - m ∠ 3) ° - m ∠ 3) ∠1 and ∠3 form a linear pair ∠2 and ∠3 form a linear pair REASONS Reasons Given Linear Pair Definition . form a linear pair iff their non(Two angles shared sides form a straight angle) ∠1 and ∠3 are supplementary ∠2 and ∠3 are supplementary m∠1+ m∠3 = 180! m∠2 + m∠3 = 180 ! ! 5. m∠1 = 180 − m∠3 Linear Pair Theorem (If two angles form a linear pair then they are supplementary) Supplementary Angle Definition (The measure of two angles is 180 iff the angles are supplementary) Subtraction Prop. of Equality m∠2 = 180 − m∠3 ! 6. m∠1 = m∠2 Substitution _______________________________________________________________________________ 2. 4. Proof: Reasons Statements LINEAR PAIR THEOREM: If two angles form a linear pair, then the angles are l. Angle Addition Postulate l. mll + m/-3 = 180; supplementary mL2*m/-3-180 2. Substitution Prop. 2. m/-l + mL3 - mL2 + mL3 VERTICAL ANGLES THEOREM: If two angles are vertical angles, they have equal measures m/-3 3. Reflexive Prop. m13 = 3. _______________________________________________________ 4. Subtraction Prop. of = ON SMALL BOARD 4. mll 2 - mL2 3 GROUP 1-4 Example IGOMILOG- 1. ? '-! 2. 3 5-8 In the diagram, /-4: /-5. Name two other angles congruent to 15' J J Solution L8: /-5 ∠5 L4:congruent L5, 17: 7 : other L4 and In the diagram, ∠4 ≅ ∠5 . Since NameLtwo angles to L5. 35,and and wLUOY==35, bisectsLSOa, LSOa,wLUOY OIbisects diagram,OI thediagram, InInthe angle. ofeach each themeasure measure 120. Findthe m/-YOW= angle. 3. Findofthe value of X Find Exs. m/-YOW 11-14 =120. Classroom Exercrses14. 14.wLZOW wLZOW 13.wLZOY wLZOY 13. 15.nIVOW nIVOW Find the measures 15. 16.m/-SOU m/-SOU and a supplement of LA. of a complement 16. 17.wLTOU wLTOU 17. l.mLA=10 18.mLZOT mLZOT 18. 2.mLA-75 3.mLA=89 4. mLA:/ LM value Findthe thevalue x.x. 5.ofofName Find two right angles. 6. Name two adjacent complementary angles. angles that are not adjacent. 7. Name two complementary (4x+8)" (4x+8)" 8. a. Name a supplement of /- MLQ. , b. Name another pair of supplementary angles. 19. 19. a. b. c. d. e. m/-QIR = m/-PIQ = m/-VIT - WIVIQ = wLSIT: ? ? ? ? /x'-n Exs.S-E ffix O B 9. In the diagram, assume that m/-CDB:90. Name: /\ a. TWo congruent supplementary angles an obtuse angle is ? c. a right angle is angles that are not congruent _______________________________________________________________________________ Two supplementary b. le is ? aresupplements. supplements. 22, /-l and /-2are complementary angles 22 d. A straight G angle have a complement? 22, /-l and /-2 c. Two G arc supplements and /-4 L3 supplements /-4 arc L3 and 10. thitofofnLDFB assume diagram, In the - 90 and FE bi- CDEA andL4. L4. /-2and measures the findthe m/-l=m13 /-2 measures a.a.lflfm/-l -27,find =m13-27, measure. each ofofl-2 termsofofx.x. L4ininterms arrdL4 l-2arrd measures flrndFind themeasures m/-3IAFD. mll ==sects the m/-3 b.b.IfIfmll ==x,x,flrnd Exs. 9, l0 congruent? c. mLBFE be b. wLAFE supplements must their m/-AFD a. are congruent, angles congruent? c. If two be supplements c. If two angles are congruent, must their ? ,/ l" ? your conclusion. Points, Lines, Planes, and Angles the measuresofofthe themeasures andthe findthe thevalue valueofofr rand supplementary,find aresupplementary, LAand andLB LBare ItItLA angles. angles. 23.mLA-2x,mLB-x-15 BB 23.mLA-2x,mLB-x-15 nd a supplement l. mLB mLB-- 2x2x-- 1616 16,mLB m/-A==xx** 16, 24,m/-A 24, the themeasures measuresofofthe andthe findthe thevalue valueofofyyand complementary, find arecomplementary, andlD lDare LCand of LB.lflfLC - iz.s angles. angles. 4. m/B - 3x 25.mLC mLC=3/ +5,mLD=2y =2y 25. =3/+5,mLD +2 26.m/-C m/-C=/ -3y+2 26. =/ -8,mLD -8,mLD-3y d complementary. Find their measures. findthe the equationtotofind theequation d supplementary. FindUse Solvethe equation.Solve their writeananequation. measures. giveninformation informationtotowrite Usethe the given described. two angles measures of the measures of the two angles described. ry angles. entary angles.' hat may or may not P angle' theangle' largeasasthe twiceasaslarge anangle angleisistwice 27.AAsupplement supplementofofan 27. angle. theangle. largeasasthe hvetimes timesasaslarge an,angleisishve 28.AAcomplement complementofofan,angle 28. twicethe the lessthan thantwice sixless anglesisissix complementaryangles twocomplementary oneofoftwo 29.The Themeasure measureofofone 29. the other. of measure measure of the other. 42. anglesisis42. supplementaryangles twosupplementary themeasures measuresofoftwo betweenthe 30. Thedifference differencebetween 30. The itssupplement. supplement. andits itscomplement, complement,and theangle, angle,its measuresofofthe Findthe themeasures Find the complementofofthe largeasasaacomplement timesasaslarge sixtimes angleisissix anangle supplementofofan 31.AAsupplement 31. angle. angle. the timesthe eighttimes angleisiseight anangle supplementofofan themeasure measureofofaasupplement timesthe 32.Three Threetimes 32. angle. theangle. measure of a complementofofthe / 3l l.3.i isin /isinX;misiny;Xlly Z;misinZ. andmm 2.4.I tand. doare notcoplanar. intersect. (Def. of ll planes) Given l.3.Given Def. ofplanes coplanar 2.4.Parullel do not intersect. s. tllm (Def. (See steps 2 and, 5. Def. of ll Hnes of ll planes) 4.) Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, 3. /isin Z;misinZ. 3. Given Same Side Interior Angles, Same Side Exterior Angles 4. t and m are coplanar. Def. of coplanar The following terms, which are needed for 4. future theorems about parallel s.lines, tllm (See steps 2 and, 4.) 5. Def.coplanar of ll Hnes lines. onlyby to acoplanar lines. line and two Angles apply formed transversal A transversal is a line that intersects two or more coplanar lines in different points. [n the next diagram, is a needed transversal of n aia *. Theabout angles formed The following terms, whichI are for future theorems parallel have special names. lines, apply only to coplanar lines. Interior angles: angles 3, 4, 5,6 angles: angles 7, 8 A transversal is a line that intersects twoExterior or more coplanar lines l,in2,different points. [n the Alternate next angles diagram, interior (alt.I is a transversal int. A-) are two of nonadjacent n aia *. The in- angles formed have special terior anglesnames. on opposite sides of the transversal. Interior angles: L3 and /-6 angles 3, 4, 5,6L4 and L5 Exterior angles: angles l, 2, 7, 8 Alternate interior (alt.(s-s. angles Same-side interior int.int. angles A-)A-) are are twotwo nonadjacent ininterior angles terior angles on the opposite sameonside of the sides transversal. of the transversal. L3 L3and and,/-6 L5 L4 L4and andL516 Same-side Corresponding interior (s-s.d") angles (corr. int.areA-)two angles areangles two interior angles in correspondoning thepositions same side of thetotransversal. relative the twb lines. L3 1l and, andL5 15 L2 and L4 /-6 and 16L3 and, /-7 /-4 and /-8 Corresponding angles (corr. d") are two angles in corresponding positions relative Angles to the twb ∠1 lines. ∠2 and ∠7 Alternate Exterior and ∠8 Classroom Exercises 1l and 15 Exterior and /-6 L2Angles Same Side L3 ∠7 and, /-7∠2 and /-4 ∠1 and ∠8and /-8 1. The blue line is a transversal. a. Name four pairs of corresponding angles. b. Name Exercises two pairs of alternate interior angles. Classroom c. Name two pairs of same-side interior angles. 1. The blue line is a transversal. 56 / Chapter 2 a. Name four pairs of corresponding angles. b. Name two pairs of alternate interior angles. c. Name two pairs of same-side interior angles. 56 2 Chapter each / Classify pair of angles as alternate interior angles, same'side interior angles, corresponding angles, or none of these. 2. 12 and L4 4. 17 and Ll5 6. L7 and LIO 8. Lll and 114 3. L6 and /-10 5. 17 and /-12 7. Ll4 and Ll5 9. 1l and Lll 10. Name alternate exterior alternate exterior angles, yorr 10. Although we have not definedangles. two pairs of them. whatside theyexterior are. Name guess 11. can Name same angles. 11. Name two pairs of angles we would call same-side exterior angles. t2. Suppose one pair of alternate interior angles are congruent (say, /-2: /-7). Explain why the other pair of alternate 13. interior angles must also be congruent. Suppose a pair of same-side interior angles (say, /-2 arrd L3) are supplementary. What must be true of any pair of corresponding angles? eac!-pair of lines as intersectrlQ parallel, or ske[_ c. AB and ID b. AB and FK a. AE and Ei e <+€ <+e f. CN and FG e. EF and NM d. EF and IH 14. Classify --.4, -.+__! nt. CORRESPONDING ANGLES POSTULATE Exs. 1-11 IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL Lines o 15. THEN CORRESPONDING ANGLES ARE CONGRUENT andangles? a protractor in the last rsversals, numbered Given: Parallel cut by a Transversal Conclude: Alternate Interior Angles are Congruent that corresponding angles Lines are conhe did not inHowever, our previous postulates and theo∠1 ≅ ∠2 If k || l, then (Corresponding Angles Postulate) m. we Lines will accept it as a postulate. llel allel corresversals, and a protractor in the last d that corresponding angles are conin ourthen previous postulatesangres and theorsal, corresponding are em. we will accept it as a postulate. ________________________________________________________________________________ ALTERNATE INTERIOR ANGLE THEOREM ersal, then IFcorresponding TWOtheorems. PARALLEL LINES ARE CUT BY A TRANSVERSAL angres are ove the following n THEN ALTERNATE INTERIOR ANGLES ARE CONGRUENT parallel lines are cut by a transverhen alt. int. A- are :. Given: Parallel Lines cut by a Transversal Conclude: Alternate Interior Angles are Congruent arc:. al,Athen alternate interior angles are If k || l,theorems. then ∠1 ≅ ∠2 (Alternate Interior Angle Theorem) rove following sitive the Prop. of =z m∠1 = m∠3(VAT ) m∠3 = m∠2(CAP) al, then alternate interior angles are . m∠1 = m∠2(substitution) ALTERNATE EXTERIOR ANGLE THEOREM /-1.Reasons IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL /-1. l. Given THEN ALTERNATE INTERIOR ANGLES ARE CONGRUENT 2. Yert. A- arc:3.Given: If two parallel are cut by a transReasons Parallellines Lines cut by a Transversal Conclude: Alternate Exterior Angles are Congruent versal, then corr. A- arc o. 4. Transitive Property l.________________________________________________________________________________ Given 2. Yert. A- arc:SAME SIDE INTERIOR ANGLE THEOREM 3. If two parallel lines are cut by a transIF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL versal, then corr. A- arc l, then same-side interior angles areo. SAME SIDE INTERIOR ANGLES ARE SUPPLEMENTARY 4.THEN Transitive Property Given: Parallel Lines cut by a Transversal al, then same-side interior angles are If k || l, then m∠1+ m∠4 = 180 Conclude: Same Side Interior Angles are Supplementary (Same Side Interior Angle Theorem) m∠1 = m∠2(AIAT ) m∠4 = m∠2(LinearPT / sup.∠def ) m∠1 = m∠2(substitution) SAME SIDE EXTERIOR ANGLE THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL THEN SAME SIDE INTERIOR ANGLES ARE SUPPLEMENTRY Given: Parallel Lines cut by a Transversal Conclude: Same Side Exterior Angles are Supplementary PROVING PARALLEL LINES THEOREMS If parallel lines are cut by transversal, then ….. Corresponding Angles Postulate (CAP) If two parallel lines are cut by a transversal, then 10/16/16, 1'48 PM corresponding angles are congruent. Alternate Interior Angle Theorem (AIAT) If two parallel lines are cut by a transversal, then Alternate interior angles are congruent. ClassLink PDF Viewer https://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D Alternate Exterior Angle Theorem (AEAT) Page 3 of 10 If two parallel lines are cut by a transversal, then Alternate exterior angles are congruent. https://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D Page 4 of 10 sLink PDF Viewer Same Side Interior Angle Theorem (SSIAT) 10/16/16, 1'48 PM If two parallel lines are cut by a transversal, then Same side interior angles are supplementary. ://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D Page 5 of 10 Same Side Exterior Angle Theorem (SSEAT) If two parallel lines are cut by a transversal, then Same side exterior angles are supplementary. ://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D Page 6 of 10 WAYS TO PROVE TWO LINES ARE PARALLEL 1. Show that a pair of Corresponding Angles are congruent. (Converse of Corresponding Angles Postulate; Converse of CAP) 2. Show that pair of Alternate Interior Angles are congruent. (Converse of Alternate Interior Angle Theorem: Converse of AIAT) 3. Show that pair of Alternate Exterior Angles are congruent. (Converse of Alternate Exterior Angle Theorem: Converse of AEAT) correasure: 4. Show that a pair of Same-Side Interior Angles are supplementary. Exam.ple Which segments are parallel? (Converse of Same-Side Interior Angles Theorem: Converse ofI SSIAT) Solution (ll HI and 7N are parallel sinc sponding angles have the same m Exam.ple I Which 5. Show thatsegments a pairare ofparallel? Same-Side Exterior Angles are supplementary. mlHIL-23+61 -84 (ll HI of Solution (Converse Exterior Angles Theorem: Converse of SSEAT) andSame-Side 7N are parallel since correm/TNI - 22 + 62 :84 sponding angles have the same measure: (2) W and .lN are not paralle mlHIL-23+61 -84 6t # 62. m/TNI - 22 + 62 :84 CLASS DISCUSSION (2) W and .lN are not parallel since Exampl,e 2 Find the values of x and 7 that 6t # 62. ACll DF and trll ar. 1. Which segments are parallel? 2. Find the values of x and y that make Solution ,qCll opitmLCBF = mlBFE. (w Exampl,e 2 Find the values of x and 7 that make 3*+20-n+50 ACll DF and trll ar. Solution ,qCll opitmLCBF = mlBFE. 3*+20-n+50 x-15 AEll BF ake if /AEF and /-F are supplemenrary. (Why?) (2y-5)+(x+50)=180 (2y-s)1(15+s0):180 2y = /- The following theorems can be proved using pr rems. we state the theorems without proof, howev 120 work. The following theorems can be proved using previous postulates and theorems. we state the theorems without proof, however, for you to use in future work. emenrary. (Why?) 120 0 Theorem 2-9 Through a point outside a line, there is exactly on Through a point outside a line, there is exactly one line parallel to the given line. given line. Theorem 2-9 vious postulates Through and theo- outside a line, there is exactly one line perpendicular a point r, for yougiven to useline. in future P a Then line m exists and is unique. parallel to the given line. Given this: line perpendicular to the Theorem 2-8 Through a point outside a line, there is exactly one li Theorem 2-8 Given this: if /AEF and /-F are sup (2y-5)+(x+50)= (2y-s)1(15+s0): 2y /-60 y?) 80 80 x-15 AEll BF 2x-30 ' since 2x-30 ' (why?) to the Given this: P a P Given this: PC Then line ft exists Th an Th an CONVERSE OF AIAT: If two lines are cut by a transversal to that Alternate Interior Angles are congruent, then the lines are parallel. CONVERSE OF AEAT: If two lines are cut by a transversal to that Alternate Exterior Angles are congruent, then the lines are parallel. ://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D CONVERSE OF SSIAT: Page 7 of 10 If two lines are cut by a transversal to that Same-Side Interior Angles are supplementary, then the lines are parallel. CONVERSE OF SSEAT: If two lines are cut by a transversal to that Same-Side Interior Angles are supplementary, then the lines are parallel s://myfiles.classlink.com/resources/pdfjs/minified/web/view…ion%3Dattachment%26Signature%3DHewHMSjsDyMOZxdUFtLKSmjpBe4%253D HW#25 Worksheet / Fill Up the Blank Proofs Page 9 of 10 GROUP WORK Classroom Exercises 'ru State which segments (if any) are parallel. State the postulate or theorem that justilies your answer. 1. W 300 Use the given information to name the segments that must be parallel. If there are no such segments, say so. 4. m/-l : mL8 5. L5: L3 8.mL5+mL6=m13+mL4 6. 9. mLAPL + mLPAR = L2: 'l A P L7 7. mL5 =m/-4 4 I L R 180 10. Reword Theorem 2-8 as two statements, one describing existence and the other describing uniqueness. " How many lines can be drawn through P parallel ,o E 12. How many lines can be drawn through Q parallel to PR? 13. H ow ma ny lines can be drawn through P perpendicular to 11. € a. QR? b. oR:? \ \ -- \ \-\ a Parallel Lines and Planes R / 67 b. In a plane two lines perpendicular to a third line must be parallel. 17. In a plane, why i tll/ and klln. Use the diagram to explain ll n. More Practice Written Exercises HW#26 Use the given information to name the segments that must be parallel. If there are no such segments, write nonc. A t. /_t: 2. m12 /_4 3. m15 = m17 5. m16 : m/-9 - 90 7, m/-7 - m/-10 - m/-l 9. 12:15 4. L5: - m1l0 L8 : mL3 - 90 m/-2 = m15 - m18 6, m16 s.AUtOr,Nrtor 10. 11. Write the reasons. Given: Tiansversal / cuts lines / and n; 1l - /_2 Prove: / ll n Proof: Statements l. Tiansversal / cuts / 4. 11/-2: /-2: 5. tll n 2. J. Reasons l. and n. /-3 /-1 13 2. J. 4. 5 12. Restate Theorem 2-9 as two statements, one describing existence and the othe.r uniqueness. pairs of parallel lines in each figure. Which congruent or supplemenName ,nro describing figure. Which congruent or supplemenName ,nro pairs of parallel lines in each tary angles did you use to determine the parallel lines? parallel lines? or supplementheWhich you use toindetermine angles of didparallel congruent each figure. lines Name ,nro pairs 68 Chapter / tary or supplemencongruent parallel in each figure. Which pairs lines Name ,nro Name F 2of,nro 14. or supplemencongruent 13. parallel in each figure. Which pairs of lines the parallel lines? 14.H tary13. angles did you use to determinethe parallel lines? you determine use to tary 12. anglesF did H parallel lines? 13. the tary angles did you use to determine 14. H 13. F 14. H or supplemen13. F,nro pairs 14. H 13.ofFparallel lines in each figure. Which congruent Name tary angles did you use to determine the parallel lines? 14. H 13. F BB B Find the values of x and y that make the red lines parallel and the blue lines parallel. Find the values of x and y that make the red lines parallel and the blue lines paralleland andthe theblue bluelines lines Find thatmake makethe thered redlines linesparallel thevalues valuesofofx xand andyythat Find parallel. the and the blue lines lines parallel 15. Find the values of x and y that make the red 16. Bparallel. parallel. parallel. parallel and the y Find14. 16.blue lines 15. values of x and that make the red lines 15. B the 15. parallel. 15. B 16. 16. 15. 16. 16. 15. Civen: Ll - /-2: 14: /-5 abou/-5 fQ and R-S? Be preWhat can you /-2: prove 17. 14: Civen: Ll/-2: 17.Civen: 14: /-5 Civen: Ll /-2: 17. 14: /-5 Ll 17. 14: Civen: Ll /-2: your class, if asked. to reasons in /-5 17. Civen:pared 14: Llcan /-2:give -/-5abou - can abou fQ andBe R-S? What you prove pre-Be prefQabou and R-S? What youprove prove abou fQ and R-S? What can you Be pre-Be fQ and R-S? What can you prove fQ andpreR-S? Be preWhat canabou you prove 18. L3 L6 Given: pared give your in class, if asked. to reasons pared give your in class, if asked. to reasons paredtotogive give your inabout class, ifinasked. reasons pared your in class, if asked. reasons pared give your class, if asked. reasons Be preother angles? What can to you prove 18. L3 L6 Given: 18. L3 L6 Given: 18. Given: L3 L6 18. L3 L6 Given: in class, if asked. reasons to giveL3your pared 18. Given: L6 -can -about Beprepre-Be preabout otherangles? angles? What can you prove Be preabout otherother angles? What can you prove about other angles? What you prove Be What can you prove Exs.17,lE a about other angles? Be preWhat can you prove in class, if asked. reasons to give your pared in class, if asked. reasons to give your pared in class, if asked. reasons to give your in class, if asked. reasons to give your paredpared pared to give your reasons in class, if asked. Exs.17,lE Exs.17,lE a Exs.17,lE aaExs.17,lE aa in a proof Exs.17,lE 19. Copy what is shown for Theorem 2-6 on page 65. Then write 17. 14. H 13. F Find the values of x and y that make the red lines parallel and the blue lines parallel. e- B Find the values of x and y 16. 15. that make the red lines parallel and the blue lines parallel. e- B 16. 15. a 5. Then Exs.17,lE 17. write 17. a proof in Civen: Ll - /-2: 14: /-5 write a proof in What can you prove abou fQ and R-S? Be prepared to give your reasons in class, if asked. 18. Given: L3 - L6 What can you prove about other angles? Be prepared to give your reasons in class, if asked. 17. 5. Then C 18. Civen: Ll - /-2: 14: /-5 What can you prove abou fQ and R-S? Be pre18. pared to give your reasons in class, if asked. Given: L3 - L6 What can you prove about other angles? Be prepared to give your reasons in class, if asked. a Exs.17 >\* I .Zls-\ I \ Df E 19. Copy what is shown for Theorem 2-6 on page 65. Then write a proof in form. atwo-columnExs.17,lE 20. Copy what is shown for Theorem 2-7 on page 65. Then write a proof in A two-column in a proof form. 19. Copy what is shown for Theorem 2-6 on page 65. Then write Exs.2l,22 C two-column form. parallel RX and gh S 20. a proof is shown for Theorem 2-7 on page 65. Then Copytowhat 21.write nn inOl; CO Given: two-column form. Prove: t O,q. t >\* /-2 I .Zls-\ C 22. Given: :C: 1l: nf t O,4 f I \ A D E >\* Prove: COtO,q' Exs.2l,22 I .Zls-\ \ A I the measure Df FindE of LRST. (Hint: Draw a lin'e through S parallel to RX and Ll t Ol; CO t O,q. - /-2 : 50"\ 1l: nf t O,4 . 22. tGiven: :C: 21. Given: nn Prove: Ll YT Prove: COtO,q' Exs.2l,22 rv.) Parallel Lines and Planes / 69 23.RX24.XR Find the measure of LRST. (Hint: Draw a lin'e through S parallel to RX and 2 rv.) : 23.RX24.XR : . t YT 50"\ Parallel Lines and Planes . t YT 50"\ Parallel Lines and / 69 2