{"id":904,"date":"2022-02-26T18:07:47","date_gmt":"2022-02-26T18:07:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=904"},"modified":"2026-01-21T17:04:24","modified_gmt":"2026-01-21T17:04:24","slug":"2-1-2-parallel-and-perpendicular-lines","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-1-2-parallel-and-perpendicular-lines\/","title":{"raw":"2.1.2: Parallel and Perpendicular Lines","rendered":"2.1.2: Parallel and Perpendicular Lines"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Determine the slope between two points on a line<\/li>\r\n \t<li>Determine if two lines are parallel<\/li>\r\n \t<li>Determine if two lines are perpendicular<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Parallel and Perpendicular Lines<\/h2>\r\nTwo lines are said to be <em><strong>parallel<\/strong><\/em> if their slopes are the same. This means that the lines are going in exactly the same direction. Two lines are said to be <em><strong>perpendicular<\/strong> <\/em>if they cross at right angles. Perpendicular lines have slopes that multiply to \u20131. This means that the slopes are the negative reciprocals of each other.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\" colspan=\"2\">\r\n<div class=\"mceTemp\">Parallel Versus Perpendicular Lines<\/div>\r\n<div class=\"mceTemp\"><\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1117\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1117 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-300x300.png\" alt=\"parallel lines\" width=\"300\" height=\"300\" \/> Figure 1. Parallel Lines[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1118\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1118 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-300x300.png\" alt=\"perpendicular lines\" width=\"300\" height=\"300\" \/> Figure 2. Perpendicular Lines[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>parallel and perpendicular lines<\/h3>\r\nTwo lines are parallel if they have the same slope:\u00a0 [latex]m_1=m_2[\/latex]\r\n\r\nTwo lines are perpendicular if the product of their slopes = \u20131:\u00a0 [latex]m_1 \\cdot m_2 = -1[\/latex]\u00a0 \u00a0 i.e.\u00a0 [latex]m_1=\\large -\\frac{1}{m_2}[\/latex] and [latex]m_2=\\large -\\frac{1}{m_1}[\/latex]\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine if the lines are parallel, perpendicular or neither.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1119 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-298x300.png\" alt=\"two lines\" width=\"298\" height=\"300\" \/><\/p>\r\n\r\n<h4>Solution<\/h4>\r\nThe lines look parallel but let's find the slopes to be sure.\r\n\r\nBlue line passes through (\u20131, 1) and (0, 3) so has a rise of 2 and a run of 1. Therefore, the slope of the blue line is [latex]m=\\frac{\\text{rise}}{\\text{run}}=2[\/latex].\r\n\r\nBlack line passes through (3, 1) and (5, 5) so has a rise of 4 and a run of 2. Therefore, the slope of the black line is [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex].\r\n\r\nSince the slopes are identical, the lines are parallel.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine if the lines are parallel, perpendicular or neither.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1120 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-300x257.png\" alt=\"Two lines\" width=\"300\" height=\"257\" \/><\/p>\r\n[reveal-answer q=\"hjm929\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm929\"]\r\n\r\nNeither.\r\n\r\nSlope of the blue line = 3\r\n\r\nSlope of the black line = [latex]-\\dfrac{10}{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples 2<\/h3>\r\nDetermine if the line that passes through the two points is parallel, perpendicular, or neither to a line with slope \u20134.\r\n<ol>\r\n \t<li>(2, 3) and (4, \u20135)<\/li>\r\n \t<li>(0, 0) and (1, 4)<\/li>\r\n \t<li>(\u20132, 4) and (2, 3)<\/li>\r\n \t<li>(3, \u20132) and (7, \u20131)<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWe need to find the slope between the pair of points. If the slope = \u20134, the line is parallel. If the slope = [latex]\\frac{1}{4}[\/latex], the line is perpendicular. Otherwise, the line is neither parallel nor perpendicular to a line with slope \u20134.\r\n<ol>\r\n \t<li>Rise from 3 to \u20135 = \u20138. Run from 2 to 4 = 2.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{-8}{2}=-4[\/latex]. The line is parallel since it has the same slope of \u20134.<\/li>\r\n \t<li>Rise from 0 to 4 = 4. Run from 0 to 1 = 1.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{1}=4[\/latex]. The line is neither parallel nor perpendicular.<\/li>\r\n \t<li>Rise from 4 to 3 = \u20131. Run from \u20132 to 2 = 4. [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{-1}{4}[\/latex]. The line is neither parallel nor perpendicular since [latex]\\frac{-1}{4} \\cdot (-4) = 1 \\ne -1[\/latex]<\/li>\r\n \t<li>Rise from \u20132 to \u20131 = 1. Run from 3 to 7 = 4.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{1}{4}[\/latex]. The line is perpendicular since [latex]\\frac{1}{4} \\cdot (-4) = -1[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine if the line that passes through the two points is parallel, perpendicular, or neither to a line with slope [latex]\\frac{2}{3}[\/latex].\r\n<ol>\r\n \t<li>(1, 3) and (4, 5)<\/li>\r\n \t<li>(0, 0) and (\u20133, \u20132)<\/li>\r\n \t<li>(\u20132, 7) and (0, 4)<\/li>\r\n \t<li>(3, \u20135) and (5, \u20132)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm386\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm386\"]\r\n<ol>\r\n \t<li>Parallel<\/li>\r\n \t<li>Parallel<\/li>\r\n \t<li>Perpendicular<\/li>\r\n \t<li>Neither parallel nor perpendicular<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Determine the slope between two points on a line<\/li>\n<li>Determine if two lines are parallel<\/li>\n<li>Determine if two lines are perpendicular<\/li>\n<\/ul>\n<\/div>\n<h2>Parallel and Perpendicular Lines<\/h2>\n<p>Two lines are said to be <em><strong>parallel<\/strong><\/em> if their slopes are the same. This means that the lines are going in exactly the same direction. Two lines are said to be <em><strong>perpendicular<\/strong> <\/em>if they cross at right angles. Perpendicular lines have slopes that multiply to \u20131. This means that the slopes are the negative reciprocals of each other.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\" colspan=\"2\">\n<div class=\"mceTemp\">Parallel Versus Perpendicular Lines<\/div>\n<div class=\"mceTemp\"><\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1117\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1117\" class=\"wp-image-1117 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-300x300.png\" alt=\"parallel lines\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel.png 1204w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1117\" class=\"wp-caption-text\">Figure 1. Parallel Lines<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1118\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1118\" class=\"wp-image-1118 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-300x300.png\" alt=\"perpendicular lines\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Perpendicular-Lines.png 1202w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1118\" class=\"wp-caption-text\">Figure 2. Perpendicular Lines<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>parallel and perpendicular lines<\/h3>\n<p>Two lines are parallel if they have the same slope:\u00a0 [latex]m_1=m_2[\/latex]<\/p>\n<p>Two lines are perpendicular if the product of their slopes = \u20131:\u00a0 [latex]m_1 \\cdot m_2 = -1[\/latex]\u00a0 \u00a0 i.e.\u00a0 [latex]m_1=\\large -\\frac{1}{m_2}[\/latex] and [latex]m_2=\\large -\\frac{1}{m_1}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine if the lines are parallel, perpendicular or neither.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1119 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-298x300.png\" alt=\"two lines\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex-350x352.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Parallel-Ex.png 666w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The lines look parallel but let&#8217;s find the slopes to be sure.<\/p>\n<p>Blue line passes through (\u20131, 1) and (0, 3) so has a rise of 2 and a run of 1. Therefore, the slope of the blue line is [latex]m=\\frac{\\text{rise}}{\\text{run}}=2[\/latex].<\/p>\n<p>Black line passes through (3, 1) and (5, 5) so has a rise of 4 and a run of 2. Therefore, the slope of the black line is [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex].<\/p>\n<p>Since the slopes are identical, the lines are parallel.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine if the lines are parallel, perpendicular or neither.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1120 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-300x257.png\" alt=\"Two lines\" width=\"300\" height=\"257\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-300x257.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-768x658.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-1024x877.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-65x56.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-225x193.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp-350x300.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/not-quite-perp.png 1252w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm929\">Show Answer<\/span><\/p>\n<div id=\"qhjm929\" class=\"hidden-answer\" style=\"display: none\">\n<p>Neither.<\/p>\n<p>Slope of the blue line = 3<\/p>\n<p>Slope of the black line = [latex]-\\dfrac{10}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples 2<\/h3>\n<p>Determine if the line that passes through the two points is parallel, perpendicular, or neither to a line with slope \u20134.<\/p>\n<ol>\n<li>(2, 3) and (4, \u20135)<\/li>\n<li>(0, 0) and (1, 4)<\/li>\n<li>(\u20132, 4) and (2, 3)<\/li>\n<li>(3, \u20132) and (7, \u20131)<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>We need to find the slope between the pair of points. If the slope = \u20134, the line is parallel. If the slope = [latex]\\frac{1}{4}[\/latex], the line is perpendicular. Otherwise, the line is neither parallel nor perpendicular to a line with slope \u20134.<\/p>\n<ol>\n<li>Rise from 3 to \u20135 = \u20138. Run from 2 to 4 = 2.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{-8}{2}=-4[\/latex]. The line is parallel since it has the same slope of \u20134.<\/li>\n<li>Rise from 0 to 4 = 4. Run from 0 to 1 = 1.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{1}=4[\/latex]. The line is neither parallel nor perpendicular.<\/li>\n<li>Rise from 4 to 3 = \u20131. Run from \u20132 to 2 = 4. [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{-1}{4}[\/latex]. The line is neither parallel nor perpendicular since [latex]\\frac{-1}{4} \\cdot (-4) = 1 \\ne -1[\/latex]<\/li>\n<li>Rise from \u20132 to \u20131 = 1. Run from 3 to 7 = 4.\u00a0 [latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{1}{4}[\/latex]. The line is perpendicular since [latex]\\frac{1}{4} \\cdot (-4) = -1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine if the line that passes through the two points is parallel, perpendicular, or neither to a line with slope [latex]\\frac{2}{3}[\/latex].<\/p>\n<ol>\n<li>(1, 3) and (4, 5)<\/li>\n<li>(0, 0) and (\u20133, \u20132)<\/li>\n<li>(\u20132, 7) and (0, 4)<\/li>\n<li>(3, \u20135) and (5, \u20132)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm386\">Show Answer<\/span><\/p>\n<div id=\"qhjm386\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Parallel<\/li>\n<li>Parallel<\/li>\n<li>Perpendicular<\/li>\n<li>Neither parallel nor perpendicular<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-904\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Slope; Parallel and Perpendicular Lines. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Slope; 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