6 6-6 What You ll Learn To determine whether lines are parallel To determine whether lines are And Wh To use parallel and lines to plan a bike path, as in Eample Parallel Lines Parallel and Perpendicular Lines Check Skills You ll Need G for Help Lessons - and 6- What is the reciprocal of each fraction? 7 7 What are the slope and -intercept of each equation? = + 6 = ; - 8 ; 8 7 = 6 6; 0 8 = 6 + 6; New Vocabular parallel lines lines negative reciprocal 6-6 Plan bjectives To determine whether lines are parallel To determine whether lines are Eamples Determining Whether Lines Are Parallel Writing Equations of Parallel Lines Writing Equations for Perpendicular Lines Real-World Problem Solving In the graph at the right, the red and blue lines are parallel Parallel lines are lines in the same plane that never intersect The equation of the red line is = + The equation of the blue line is = - Ke Concepts Propert Slopes of Parallel Lines Nonvertical lines are parallel if the have the same slope and different -intercepts An two vertical lines are parallel Eample The equations = + and = - have the same slope,, and different -intercepts The graphs of the two equations are parallel Math Background Vertical lines have an undefined, or infinite, slope because the denominator in the ratio is zero This is wh the Propert of Slopes of Parallel Lines includes the word nonvertical More Math Background: p 06D Lesson Planning and Resources You can use slope-intercept form to determine whether the lines are parallel See p 06E for a list of the resources that support this lesson Real-World Connection The lanes for competitive swimming are parallel Determining Whether Lines Are Parallel Are the graphs of = + and + 6 = parallel? Eplain Write + 6 = in slope-intercept form Then compare with = + 6 =- + Subtract from each side 6 = 6 6 Divide each side b 6 = + Simplif The lines are parallel The equations have the same slope, different -intercepts Are the graphs of -6 + 8 =- and = es; same slope, different -intercepts, and - 7 parallel? Eplain Lesson 6-6 Parallel and Perpendicular Lines Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Multipling and Dividing Lesson -: Eample 7 Etra Skills and Word Problem Practice, Ch Slope-Intercept Form Lesson 6-: Eample Etra Skills and Word Problem Practice, Ch 6 Special Needs L Some students ma confuse the terms parallel and Have them underline the ll in the word parallel Ask: Do the l s in the word parallel look parallel or? learning stle: visual Below Level L To demonstrate that the signs of the slopes of parallel lines are the same, have students graph = - Then have them compare their graphs with the one on p learning stle: visual
Teach Guided Instruction Alternative Method Write = + and = - on the board Ask: What number do the equations have in common? What does this number represent in each equation? the slope Have students find the -coordinates of the point on each line when = Then find the difference between the -coordinates Tell students that the difference is the vertical distance between the two lines Repeat for = 00, = 0,000 and = 00,000 Compare the vertical distance between the lines for each pair of points The vertical distances are all the same Ask: What geometric term describes lines that are alwas the same distance apart? parallel What might ou conclude about lines that have the same slope? Lines that have the same slope are parallel Additional Eamples Are the graphs of =- - and + = 6 parallel? es Write an equation for the line that contains (-, ) and is parallel to = - ± 8 nline Visit: PHSchoolcom Web Code: ate-077 You can use the fact that the slopes of parallel lines are the same to write the equation of a line parallel to a given line To write the equation, ou use the slope of the given line and the point-slope form of a linear equation Writing Equations of Parallel Lines Write an equation for the line that contains (, ) and is parallel to = - Step Identif the slope of the given line = - c slope Step Perpendicular Lines Write the equation of the line through (, ) using slope-intercept form - = m( - ) point-slope form - = ( ) Substitute (, ) for (, ) and for m - = - () Use the Distributive Propert - = - Simplif = - Add to each side Write an equation for the line that contains (, -6) and is parallel to = + 9 The lines at the right are Perpendicular lines are lines that intersect to form right angles The equation of the red line is = - The equation of the blue line is = + Ke Concepts Propert Slopes of Perpendicular Lines Two lines are if the product of their slopes is - A vertical and a horizontal line are also Eample The slope of is The slope of = + is Since? =-, the graphs of the two equations are The product of two numbers is - if one number is the negative reciprocal of the other Here is how to find the negative reciprocal of a number Start with Find its Write the a fraction: S reciprocal: S negative reciprocal: Since? =-, is the negative reciprocal of Start with Find its Write its an integer: S reciprocal: S negative reciprocal: Since Q =-, R is the negative reciprocal of Chapter 6 Linear Equations and Their Graphs Advanced Learners L Draw a square and its diagonals on a coordinate grid Challenge students to prove that the diagonals are learning stle: verbal English Language Learners ELL In Eample, have students undo negative reciprocal one word at a time Ask: What do ou multipl b to make a fraction negative? Discuss the meaning of reciprocal learning stle: verbal
A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Test-Taking Tip After finding the slopes of the lines, multipl the two slopes as a check ( ) You can use the negative reciprocal of the slope of a given line to write an equation of a line to that line Writing Equations for Perpendicular Lines Multiple Choice Which equation describes a line that contains (0, -) and is to = +? = - = = = Step Identif the slope of Step Find the negative reciprocal the given line of the slope = + The negative reciprocal of is c slope Step Use the slope-intercept form to write an equation = m + b = + (-) Substitute for m, and for b = - Simplif The equation is = - So D is the correct answer Write an equation of the line that contains (, 8) and is to = + ± 9 You can use equations of parallel and lines to solve some real-world problems Real-World Problem Solving Urban Planning A bike path for a new cit park will connect the park entrance to Park Road The path will be to Park Road Write an equation for the line representing the bike path Error Prevention Students ma think that a negative reciprocal is alwas negative Tell students that the phrase negative reciprocal actuall means the negative of the reciprocal So the are reall just finding the opposite of the reciprocal Additional Eamples Find an equation of the line that contains (6, ) and is to =- + 7 The line in the graph represents the street in front of a new house The point is the front door The sidewalk from the front door will be to the street Write an equation representing the sidewalk 6 front door street Park park entrance Park Road -intercept new path to Park Road (, ) (, ) Resources Dail Notetaking Guide 6-6 L Dail Notetaking Guide 6-6 Adapted Instruction L Closure Real-World Connection Careers Urban planners use mathematics to make decisions on communit problems such as traffic congestion and air pollution Step Find the slope m of Park Road m = = = = Points (, ) and (, ) are on Park Road Step Find the negative reciprocal of the slope The negative reciprocal of is So the slope of the bike path is The -intercept is The equation for the bike path is = + A second bike path is planned It will be parallel to Park Road and will also contain the park entrance Write an equation for the line representing this bike path ± Lesson 6-6 Parallel and Perpendicular Lines Ask: Compare the equations of non-vertical parallel lines Parallel lines have the same slope, but different -intercepts Compare the equations of lines Perpendicular lines have slopes that are negative reciprocals of each other The will have the same -intercept onl if that is where the two lines intersect
Practice Assignment Guide A B -8, 0,,,, -60 A B 9-9,,, 6-0, 6-6 C Challenge 6-70 Test Prep 7-76 Mied Review 77-89 Homework To check students understanding of ke skills and concepts, go over Eercises 8, 6,, 7-9, Eercise Suggest to students that the graph the line = 6 EXERCISES Practice and Problem Solving A G Practice b Eample for Help Eample (page ) Eample (page ) For more eercises, see Etra Skill and Word Problem Practice Find the slope of a line parallel to the graph of each equation = + = - = = 6 0 + = 6 7 - = 7 Are the graphs of the lines in each pair parallel? Eplain 7 See margin 7 = + 8 = + 9 = + - + = + = 8 - = 6 0 = + =- = - - 0 = + 7 = - + = 8 Write an equation for the line that is parallel to the given line and that passes through the given point = 6 - ; (0, 0) 8 See margin =-; (, 0) =- + ; (-, ) 6 = 7 + 6; (-, -6) 7 = 0-8; (8, -) 8 = + ; (, -) Eample (page ) Find the slope of a line to the graph of each equation 9 = 0 =- = 7-7 = - 7 + = =-8undefined Write an equation for the line that is to the given line and that passes through the given point GPS Guided Problem Solving L Enrichment Reteaching Adapted Practice L L L Practice Pearson Education, Inc All rights reserved Name Class Date Practice 6- Parallel and Perpendicular Lines Find the slope of a line parallel to the graph of each equation = + = + =-9 - = 7 + 6 + = 6 - = 0 7 - + = 8 9 - = 9 - + = 6 0 6-7 = 0 =- - - = 6 Write an equation for the line that is to the given line and that passes through the given point (6, ); = - (-, ); =- + 9 (-, -); = + 6 6 (, ) ; = + 7 7 (, -6); = + 8 (0, -); = - 7 9 0 Write an equation for the line that is parallel to the given line and that passes through the given point (, ); = - 7 (, ); =- + (, -); = - (,0); = + 9 6 (-8, -); = + 7 (9, -7); -7 - = 8 9 0 Tell whether the lines for each pair of equations are parallel,, or neither = - 8 + =- = + L B Eample (page ) Appl Your Skills = + 7; (0, 0) 6 = - ; (, 6) ± 0 7 = + ; (, ) 0 8 + = 7; (-, ) ± 9-0 + 8 = ; (, ) 0 - = 9; (8, -) ± ± Maps A cit s civil engineer is planning a new parking garage and a new street The new street will go from the entrance of the parking garage to Handel St It will be to Handel St What is the equation of the line representing the new street? ± Entrance Tell whether the lines for each pair of equations are parallel,, or neither = + = + = - 6, =, + 6 parallel =- +, = + =, =- + 7 neither 6 = -, = + parallel 7 =, = 9 8 + =, + = parallel 9 - =, - + = 8 neither Handel St - =- = + 6 - + = 0 9 + = 6 =- 6 = 0 + 9 = 6 = =- 0 Multiple Choice Which pair of equations graph as lines? C = + = + = - 9 = - 9 =- + = - = + 8 = + pages 6 9 Eercises 7 no, different slopes 8 es, same slopes and different -intercepts 9 es, same slopes and different -intercepts 6 6 Chapter 6 Linear Equations and Their Graphs 0 no, different slopes es, same slopes and different -intercepts es, same slopes and different -intercepts 6 ± 9 7 6 0 7 0 9 8 ±
Real-World Connection A skier s fastest speed occurs when the skis are parallel Find the equation for each line 9 ; ± ; ± ; ± 7 ± ; Teaching Tip Eercises 9 Ask students to describe equations of lines that are neither parallel nor The slopes are neither the same nor negative reciprocals of each other Error Prevention! Eercise 6 Suggest that students first sketch a graph of the triangle to determine which two sides ma be 6 ; ; ± GPS Maps Use the map below for Eercises 7 9 New Hampshire Ave Massachusetts Ave Pennslvania Ave Washington, DC G nline Homework Help Visit: PHSchoolcom Web Code: ate-0606 7 What is the slope of New Hampshire Avenue? about 8 Show that the parts of Pennslvania Avenue and Massachusetts Avenue near New Hampshire Avenue are parallel Answers ma var Sample: same slope of 9 Show that New Hampshire Avenue is not to Pennslvania Avenue Answers ma var Sample:? Q R u 0 a The graphs of = and =-are shown on the standard screen at the right The product of the slopes is - Eplain wh the lines do not appear to be a b See margin b Graphing Calculator Graph = and =- on a graphing calculator In the feature, choose the square screen What do ou notice? pen-ended Write an equation for a line parallel to the graph of - = Answers ma var Sample: ± Lesson 6-6 Parallel and Perpendicular Lines 7 0a The units on the aes are different, so the screen is not square b The lines appear 7
Assess & Reteach Lesson Quiz Find the slope of a line parallel to - = Find the slope of a line to + = 7 Tell whether the lines for each pair of equations are parallel,, or neither = -, =- + = +, + =- neither =- -, + = 6 parallel 6 Write an equation of the line that contains (, ) and is to =- + Alternative Assessment Write an equation in slopeintercept form on the board Have students first write an equation of a line that is parallel to the given line, and then write an equation of a line that is to the given equation Let students compare their answers Repeat No; the slopes are not equal No; the slopes are not neg reciprocals Problem Solving Hint For Eercises 7, sketch a graph to help ou understand the statement in each eercise False; the product of two positive numbers can t be 6 True; ± and ± are parallel 7 False; all direct variations go through the point (0, 0) If the have the same slope, the are the same line, not parallel lines C Challenge Are the graphs of + 7 = 6 and 7 = + 6 parallel? Eplain See left Are the graphs of 8 + = 6 and 8 - = 6? Eplain See left Writing Are all horizontal lines parallel? Eplain See margin Tell whether each statement is true or false Eplain our choice 7 See below left Two lines with positive slopes can be 6 Two lines with positive slopes can be parallel 7 The graphs of two different direct variations can be parallel 8 60 See Geometr A quadrilateral with both pairs of opposite sides parallel is a margin parallelogram Use slopes to determine whether each figure is a parallelogram 8 B 9 K 60 P J A Q C S 6 L D M R Geometr A quadrilateral with four right angles is a rectangle Use slopes to determine whether each figure is a rectangle 6 6 See margin 6 B 6 6 K A L Q P R C N S D M 6 Geometr A quadrilateral with two pairs of parallel sides and with diagonals that are is a rhombus Quadrilateral ABCD has vertices A(-, ), B(, 6), C(6, 6), and D(, ) Show that ABCD is a rhombus See margin 6 Geometr A triangle with two sides that are to each other is a right triangle Triangle PQR has vertices P(, ), Q(, -), and R(0, ) Determine whether PQR is a right triangle Eplain See margin es; same slopes and different -intercepts 8 The slopes of AD and BC are both undefined, so the are parallel The slopes of AB and CD are both, so the are parallel The quadrilateral is a parallelogram 9 The slope of is The JK slope of KL is The slope of LM is The slope of JM 6 is The quadrilateral is a parallelogram 8 7 68 8 8 ; 8 ± 7 69 ; ± 7 8 Chapter 6 Linear Equations and Their Graphs Tell whether the lines in each pair are parallel,, or neither 66 a - b = ; -a + b = 67 a + b = 8; b- a = parallel Assume the two lines are Find an equation for each line 68 69 See left 68 69 70 For what value of k are the graphs of + = 8 and 6 = k - parallel? Perpendicular? ; 60 The slopes of PQ and RS 6 The slopes of AB and CD 6 The slopes of KL and MN are both The slopes of QR are both The slopes of and SP are both BC are both The slopes and AD are both of LM 6 and KN are both The quadrilateral is The product is, so The product is not, so a parallelogram the quadrilateral is a the quadrilateral is not a rectangle rectangle
Test Prep Multiple Choice Short Response Standardized Test Prep 7 Which equation has as its graph a line to a line that has a slope of? D A = + B = - C = - D + = 8 7 Which equation has as its graph a line parallel to the graph of - - =? F F = + G = - 6 H =- + J = - 7 A parallelogram has vertices A(0, ), B(, -), C(6, ), and D(p, q) Which of the following ordered pairs has possible values for (p, q)? C A (0, 6) B (6, ) C (, 6) D (6, ) 7 Which equation has as its graph a line to the graph of the line that passes through the points (0, -) and (8, 9)? F F ( ) G 0 H 0 J 7 7 Which equation has as its graph a line parallel to the graph of the line that passes through the points (-8, ) and (-, )? C A ( ) ( 8) B 0 C 6 D 7 76 Suppose the line through points (, 6) and (, ) is parallel to the graph of + = Find the value of Show our work See margin Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p 69 Test-Taking Strategies, p 6 Test-Taking Strategies with Transparencies Connection to Geometr Eercise 7 Tell students that the can eliminate some possible answers b first sketching the three points in the eercise on a coordinate plane A-head Mied Review G for Help Lesson 6- Write an equation for the line through the given point with the given slope 77 (0, ); m = ± 78 (-, 0); m =- 8 79 (, -); m = ± ( ) 80 (-, -9); m = ± 9 ( ± ) 8 (-6, ); m = ( ± 6) 8 (7, ); m = ( 7) Lesson 6- Lesson - Find the slope of each line 8 8 8 lesson quiz, PHSchoolcom, Web Code: ata-0606 Determine whether each relation is a function 86 {(, ), (, ), (, )} es 87 {(, ), (, ), (, )} es 88 {(, ), (, ), (, )} no 89 {(, ), (, ), (, )} es Lesson 6-6 Parallel and Perpendicular Lines 9 6 The slopes of PQ and RS 6 BC and AD both have a BD has a slope of are both The slopes of PS and QR slope of zero and are both AD BC are parallel and The product is, so CD AB The diagonal AC has a slope of The diagonals are both have a slope of the quadrilateral is a rectangle AB and CD ~ABCD is a rhombus are parallel The diagonal 6 RP has a slope of has a slope of RP RQ is the neg reciprocal of RQ, so kpqr is a right triangle 76 [] Find slope of ± : ±, therefore slope is 6 Find :,, ±, (R equivalent eplanation) [] correct value of but no work R minor computation error in work 9