{"id":3085,"date":"2023-05-19T17:19:23","date_gmt":"2023-05-19T17:19:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3085"},"modified":"2024-10-18T20:51:32","modified_gmt":"2024-10-18T20:51:32","slug":"introduction-to-geometry-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/introduction-to-geometry-fresh-take\/","title":{"raw":"Introduction to Geometry: Fresh Take","rendered":"Introduction to Geometry: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify and explain points, lines, and planes<\/li>\r\n\t<li>Measure angles using degrees and radians<\/li>\r\n\t<li>Determine the supplement and complement of an angle<\/li>\r\n\t<li>Calculate angles formed by transversals intersecting parallel lines<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Points, Lines, and Planes<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>point<\/strong> is a location in space with no length, width, or height.<\/p>\r\n<p>A <strong>line <\/strong>is an infinite collection of points extending infinitely in both directions.<\/p>\r\n<p>A <strong>line segment<\/strong> is a finite portion of a line with two endpoints.<\/p>\r\n<p>A <strong>ray<\/strong> is a part of a line that starts at an endpoint and extends infinitely in one direction.<\/p>\r\n<p><strong>Parallel lines<\/strong> are lines that lie in the same plane and move in the same direction, but never intersect.<\/p>\r\n<p>Two lines that intersect at a [latex]90^\\circ[\/latex] angle are <strong>perpendicular lines<\/strong>.<\/p>\r\n<p>The properties of <strong>planes <\/strong>include:<\/p>\r\n<ul>\r\n\t<li>Three points including at least one noncollinear point determine a plane.<\/li>\r\n\t<li>A line and a point not on the line determine a plane.<\/li>\r\n\t<li>The intersection of two distinct planes is a straight line.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/k5etrWdIY6o[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Points%2C+Lines%2C+%26+Planes.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - Points, Lines, &amp; Planes\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Angle Basics<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>An <strong>angle<\/strong> is formed when two lines, line segments, or rays meet at a common point called the vertex. The two lines or segments that form the angle are called the sides of the angle.<\/p>\r\n<p>Measurement: Angles are measured in degrees ([latex]\u00b0[\/latex]) or radians. We use a protractor to measure the size of an angle.<\/p>\r\n<p>Types of Angles:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><strong>Right Angle<\/strong>: A right angle measures exactly [latex]90[\/latex] degrees. It forms a perfect \"L\" shape.<\/li>\r\n\t<li><strong>Acute Angle<\/strong>: An acute angle measures less than [latex]90[\/latex] degrees. It is smaller than a right angle. A key to remember acute angles is a small angle is \"a cute\" little angle.<\/li>\r\n\t<li><strong>Obtuse Angle<\/strong>: An obtuse angle measures more than [latex]90[\/latex] degrees but less than [latex]180[\/latex] degrees. It is larger than a right angle.<\/li>\r\n\t<li><strong>Straight Angle<\/strong>: A straight angle measures exactly [latex]180[\/latex] degrees. It forms a straight line.<\/li>\r\n\t<li><strong>Reflex Angle<\/strong>: A reflex angle measures more than [latex]180[\/latex] degrees but less than [latex]360[\/latex] degrees. It is larger than a straight angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p>The following video explores four of the five angle types.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350402&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=AuJQZ8Pusdg&amp;video_target=tpm-plugin-jnhoafn3-AuJQZ8Pusdg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Types+of+Angles+%7C+Obtuse%2C+Acute%2C+Right%2C+%26+Straight+Angles.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTypes of Angles | Obtuse, Acute, Right, &amp; Straight Angles\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Measuring Angles<\/h3>\r\n<p>Watch the following video for more on how to use a protractor.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350403&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=t4xCOUNEInI&amp;video_target=tpm-plugin-8lck4g0b-t4xCOUNEInI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+use+a+protractor.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use a protractor\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Degrees versus Radians<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Degrees <\/strong>are the most common measurement of angles. A circle is divided into [latex]360^\\circ[\/latex]. The symbol for degrees is a small, raised circle: [latex]^\\circ[\/latex].<\/p>\r\n<p><strong>Radians <\/strong>are an alternative unit of angle measurement. In a circle, there are [latex]2\u03c0[\/latex] radians, where [latex]\u03c0[\/latex] (pi) is approximately [latex]3.14[\/latex]. The symbol for radians is [latex]\\text{rad}[\/latex].<\/p>\r\n<p>The conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/p>\r\n<p>Degrees are commonly used in everyday situations like navigation, construction, and basic geometry, while radians are more prevalent in advanced mathematics, physics, engineering, and other scientific fields.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/l6hSY2Pcch0[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/_Radians+and+Degrees+(NancyPi).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRadians and Degrees (NancyPi)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7048[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7049[\/ohm2_question]<\/section>\r\n<h3>Supplementary and Complementary Angles<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Supplementary Angles<\/strong>: Two angles are considered supplementary when the sum of their measures is equal to 180 degrees. In other words, if you have two angles, and when you add their measures together, the result is 180 degrees, then those angles are supplementary.<\/p>\r\n<p><strong>Complementary Angles<\/strong>: Two angles are considered complementary when the sum of their measures is equal to 90 degrees. In other words, if you have two angles, and when you add their measures together, the result is 90 degrees, then those angles are complementary.<\/p>\r\n<\/div>\r\n<section class=\"textbox proTip\">Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[\/latex] and [latex]180[\/latex] are in numerical order? This can be a helpful mnemonic to remember the difference between complementary and supplementary angles.<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350401&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=E_ulN2FjJac&amp;video_target=tpm-plugin-d8ptj7o9-E_ulN2FjJac\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Complementary+and+Supplementary+Angles+%7C+How+to+Find+Missing+Angles+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cComplementary and Supplementary Angles | How to Find Missing Angles | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the following video we show more examples of how to find the supplement and complement of an angle.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/ZQ_L3yJOfqM[\/embed]<\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+the+Complement+and+Supplement+of+a+Given+Angle.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Complement and Supplement of a Given Angle\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7050[\/ohm2_question]<\/section>\r\n<h3>Vertical Angles<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Vertical angles<\/strong> are a pair of opposite angles that are formed when two lines intersect. These angles are located across from each other and share a common vertex or point of intersection. Vertical angles have equal measures, which means they have the same angle measurement.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350399&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DJRTIGwJlbU&amp;video_target=tpm-plugin-lrt4l9om-DJRTIGwJlbU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+are+Vertical+Angles%3F+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are Vertical Angles? | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Given the two intersecting lines in the figure shown and [latex]m\u22202=67^\\circ[\/latex], find the measure of the remaining angles.<br \/>\r\n<center><img class=\"aligncenter size-medium wp-image-3250\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-300x222.png\" alt=\"Two lines intersect each other. One set of opposite angles is labeled 1 and 3. The other set of opposite angles is labeled 4 and 67 degrees.\" width=\"300\" height=\"222\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n[reveal-answer q=\"57881\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57881\"]<br \/>\r\n[latex]\u200b\u200b\u200bm\u2220 4 = {67^ \\circ }\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 1 = {113^ \\circ }\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 3 = {113^ \\circ }\u200b\u200b\u200b[\/latex][\/hidden-answer]<\/section>\r\n<h3>Transversals<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>transversal <\/strong>is a line that intersects two or more other lines. It cuts across these lines at different points, creating various angles.<\/p>\r\n<p><strong>Alternate interior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the inside of the two other lines. These angles are equal in measure.<\/p>\r\n<p><strong>Alternate exterior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the outside of the two other lines. These angles are equal in measure.<\/p>\r\n<p><strong>Corresponding angles<\/strong> are a pair of angles that are in the same relative position with respect to the transversal, but they are on different intersected lines. These angles are also equal in measure.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350400&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=uYRvUdpL94g&amp;video_target=tpm-plugin-yc4n1chn-uYRvUdpL94g\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Parallel+Lines+cut+by+a+Transversal.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cParallel Lines cut by a Transversal\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Using the figure below and given that angle [latex]2[\/latex] measures [latex]23^\\circ[\/latex], find the measure of the remaining angles and state the reason for your solution.<br \/>\r\n<center><img class=\"aligncenter wp-image-3253\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-300x111.png\" alt=\"Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 23 degrees, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 23 degrees, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.\" width=\"365\" height=\"135\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n[reveal-answer q=\"57882\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57882\"]<br \/>\r\n[latex]m\u22202=m\u22203=23^\\circ[\/latex] by vertical angles, because [latex]\u22202[\/latex] and [latex]\u22203[\/latex] are the opposite angles formed by two intersecting lines. [latex]m\u22201=157^\\circ[\/latex] by supplementary angles to [latex]\u22202[\/latex] or [latex]\u22203[\/latex]. We see that [latex]\u22201[\/latex] and [latex]\u22202[\/latex] form a straight angle as does [latex]\u22201[\/latex] and [latex]\u22203[\/latex]. A straight angle measures [latex]180^\\circ[\/latex], so [latex]180^\\circ\u221223^\\circ=157^\\circ[\/latex]. [latex]m\u22204=m\u22201=157^\\circ[\/latex] by vertical angles, because [latex]\u22204[\/latex] and [latex]\u22201[\/latex] are the two opposite angles formed by two intersecting lines. [latex]m\u22205=m\u22201=157^\\circ[\/latex] by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior. [latex]m\u22208=m\u22205=157^\\circ[\/latex] by vertical angles because [latex]\u22208[\/latex] and [latex]\u22205[\/latex] are the two opposite angles formed by two intersecting lines. [latex]m\u22207=m\u22202=23^\\circ[\/latex] by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines. [latex]m\u22206=m\u22207=23^\\circ[\/latex] by vertical angles because these are the opposite angles formed by two intersecting lines.[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">In the provided figure given that the [latex]m\u22202=48^\\circ[\/latex], find [latex]m\u22201[\/latex], and [latex]m\u22205[\/latex].<br \/>\r\n<center><img class=\"aligncenter wp-image-3257\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-300x143.png\" alt=\"Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.\" width=\"342\" height=\"163\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n[reveal-answer q=\"57884\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57884\"]<br \/>\r\n[latex]\u200b\u200b\u200bm\u2220 1 = {132^ \\circ}\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 5 = {132^ \\circ}\u200b\u200b\u200b[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify and explain points, lines, and planes<\/li>\n<li>Measure angles using degrees and radians<\/li>\n<li>Determine the supplement and complement of an angle<\/li>\n<li>Calculate angles formed by transversals intersecting parallel lines<\/li>\n<\/ul>\n<\/section>\n<h2>Points, Lines, and Planes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>point<\/strong> is a location in space with no length, width, or height.<\/p>\n<p>A <strong>line <\/strong>is an infinite collection of points extending infinitely in both directions.<\/p>\n<p>A <strong>line segment<\/strong> is a finite portion of a line with two endpoints.<\/p>\n<p>A <strong>ray<\/strong> is a part of a line that starts at an endpoint and extends infinitely in one direction.<\/p>\n<p><strong>Parallel lines<\/strong> are lines that lie in the same plane and move in the same direction, but never intersect.<\/p>\n<p>Two lines that intersect at a [latex]90^\\circ[\/latex] angle are <strong>perpendicular lines<\/strong>.<\/p>\n<p>The properties of <strong>planes <\/strong>include:<\/p>\n<ul>\n<li>Three points including at least one noncollinear point determine a plane.<\/li>\n<li>A line and a point not on the line determine a plane.<\/li>\n<li>The intersection of two distinct planes is a straight line.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Math Antics - Points, Lines, &amp; Planes\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/k5etrWdIY6o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Points%2C+Lines%2C+%26+Planes.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; Points, Lines, &amp; Planes\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Angle Basics<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>An <strong>angle<\/strong> is formed when two lines, line segments, or rays meet at a common point called the vertex. The two lines or segments that form the angle are called the sides of the angle.<\/p>\n<p>Measurement: Angles are measured in degrees ([latex]\u00b0[\/latex]) or radians. We use a protractor to measure the size of an angle.<\/p>\n<p>Types of Angles:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>Right Angle<\/strong>: A right angle measures exactly [latex]90[\/latex] degrees. It forms a perfect &#8220;L&#8221; shape.<\/li>\n<li><strong>Acute Angle<\/strong>: An acute angle measures less than [latex]90[\/latex] degrees. It is smaller than a right angle. A key to remember acute angles is a small angle is &#8220;a cute&#8221; little angle.<\/li>\n<li><strong>Obtuse Angle<\/strong>: An obtuse angle measures more than [latex]90[\/latex] degrees but less than [latex]180[\/latex] degrees. It is larger than a right angle.<\/li>\n<li><strong>Straight Angle<\/strong>: A straight angle measures exactly [latex]180[\/latex] degrees. It forms a straight line.<\/li>\n<li><strong>Reflex Angle<\/strong>: A reflex angle measures more than [latex]180[\/latex] degrees but less than [latex]360[\/latex] degrees. It is larger than a straight angle.<\/li>\n<\/ol>\n<\/div>\n<p>The following video explores four of the five angle types.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350402&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=AuJQZ8Pusdg&amp;video_target=tpm-plugin-jnhoafn3-AuJQZ8Pusdg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Types+of+Angles+%7C+Obtuse%2C+Acute%2C+Right%2C+%26+Straight+Angles.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTypes of Angles | Obtuse, Acute, Right, &amp; Straight Angles\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Measuring Angles<\/h3>\n<p>Watch the following video for more on how to use a protractor.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350403&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=t4xCOUNEInI&amp;video_target=tpm-plugin-8lck4g0b-t4xCOUNEInI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+use+a+protractor.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use a protractor\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Degrees versus Radians<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Degrees <\/strong>are the most common measurement of angles. A circle is divided into [latex]360^\\circ[\/latex]. The symbol for degrees is a small, raised circle: [latex]^\\circ[\/latex].<\/p>\n<p><strong>Radians <\/strong>are an alternative unit of angle measurement. In a circle, there are [latex]2\u03c0[\/latex] radians, where [latex]\u03c0[\/latex] (pi) is approximately [latex]3.14[\/latex]. The symbol for radians is [latex]\\text{rad}[\/latex].<\/p>\n<p>The conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/p>\n<p>Degrees are commonly used in everyday situations like navigation, construction, and basic geometry, while radians are more prevalent in advanced mathematics, physics, engineering, and other scientific fields.<\/p>\n<\/div>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Radians and Degrees (NancyPi)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/l6hSY2Pcch0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/_Radians+and+Degrees+(NancyPi).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRadians and Degrees (NancyPi)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7048\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7048&theme=lumen&iframe_resize_id=ohm7048&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7049\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7049&theme=lumen&iframe_resize_id=ohm7049&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Supplementary and Complementary Angles<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Supplementary Angles<\/strong>: Two angles are considered supplementary when the sum of their measures is equal to 180 degrees. In other words, if you have two angles, and when you add their measures together, the result is 180 degrees, then those angles are supplementary.<\/p>\n<p><strong>Complementary Angles<\/strong>: Two angles are considered complementary when the sum of their measures is equal to 90 degrees. In other words, if you have two angles, and when you add their measures together, the result is 90 degrees, then those angles are complementary.<\/p>\n<\/div>\n<section class=\"textbox proTip\">Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[\/latex] and [latex]180[\/latex] are in numerical order? This can be a helpful mnemonic to remember the difference between complementary and supplementary angles.<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350401&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=E_ulN2FjJac&amp;video_target=tpm-plugin-d8ptj7o9-E_ulN2FjJac\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Complementary+and+Supplementary+Angles+%7C+How+to+Find+Missing+Angles+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cComplementary and Supplementary Angles | How to Find Missing Angles | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the following video we show more examples of how to find the supplement and complement of an angle.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Determine the Complement and Supplement of a Given Angle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZQ_L3yJOfqM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+the+Complement+and+Supplement+of+a+Given+Angle.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Complement and Supplement of a Given Angle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7050\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7050&theme=lumen&iframe_resize_id=ohm7050&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Vertical Angles<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Vertical angles<\/strong> are a pair of opposite angles that are formed when two lines intersect. These angles are located across from each other and share a common vertex or point of intersection. Vertical angles have equal measures, which means they have the same angle measurement.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350399&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DJRTIGwJlbU&amp;video_target=tpm-plugin-lrt4l9om-DJRTIGwJlbU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+are+Vertical+Angles%3F+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are Vertical Angles? | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Given the two intersecting lines in the figure shown and [latex]m\u22202=67^\\circ[\/latex], find the measure of the remaining angles.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-3250\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-300x222.png\" alt=\"Two lines intersect each other. One set of opposite angles is labeled 1 and 3. The other set of opposite angles is labeled 4 and 67 degrees.\" width=\"300\" height=\"222\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-300x222.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-65x48.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-225x166.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118-350x259.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190205\/Screenshot-2023-05-22-150118.png 472w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57881\">Show Solution<\/button><\/p>\n<div id=\"q57881\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\u200b\u200b\u200bm\u2220 4 = {67^ \\circ }\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 1 = {113^ \\circ }\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 3 = {113^ \\circ }\u200b\u200b\u200b[\/latex]<\/div>\n<\/div>\n<\/section>\n<h3>Transversals<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>transversal <\/strong>is a line that intersects two or more other lines. It cuts across these lines at different points, creating various angles.<\/p>\n<p><strong>Alternate interior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the inside of the two other lines. These angles are equal in measure.<\/p>\n<p><strong>Alternate exterior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the outside of the two other lines. These angles are equal in measure.<\/p>\n<p><strong>Corresponding angles<\/strong> are a pair of angles that are in the same relative position with respect to the transversal, but they are on different intersected lines. These angles are also equal in measure.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350400&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=uYRvUdpL94g&amp;video_target=tpm-plugin-yc4n1chn-uYRvUdpL94g\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Parallel+Lines+cut+by+a+Transversal.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cParallel Lines cut by a Transversal\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Using the figure below and given that angle [latex]2[\/latex] measures [latex]23^\\circ[\/latex], find the measure of the remaining angles and state the reason for your solution.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3253\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-300x111.png\" alt=\"Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 23 degrees, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 23 degrees, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.\" width=\"365\" height=\"135\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-300x111.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-65x24.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-225x84.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317-350x130.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190349\/Screenshot-2023-05-22-150317.png 705w\" sizes=\"(max-width: 365px) 100vw, 365px\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57882\">Show Solution<\/button><\/p>\n<div id=\"q57882\" class=\"hidden-answer\" style=\"display: none\">\n[latex]m\u22202=m\u22203=23^\\circ[\/latex] by vertical angles, because [latex]\u22202[\/latex] and [latex]\u22203[\/latex] are the opposite angles formed by two intersecting lines. [latex]m\u22201=157^\\circ[\/latex] by supplementary angles to [latex]\u22202[\/latex] or [latex]\u22203[\/latex]. We see that [latex]\u22201[\/latex] and [latex]\u22202[\/latex] form a straight angle as does [latex]\u22201[\/latex] and [latex]\u22203[\/latex]. A straight angle measures [latex]180^\\circ[\/latex], so [latex]180^\\circ\u221223^\\circ=157^\\circ[\/latex]. [latex]m\u22204=m\u22201=157^\\circ[\/latex] by vertical angles, because [latex]\u22204[\/latex] and [latex]\u22201[\/latex] are the two opposite angles formed by two intersecting lines. [latex]m\u22205=m\u22201=157^\\circ[\/latex] by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior. [latex]m\u22208=m\u22205=157^\\circ[\/latex] by vertical angles because [latex]\u22208[\/latex] and [latex]\u22205[\/latex] are the two opposite angles formed by two intersecting lines. [latex]m\u22207=m\u22202=23^\\circ[\/latex] by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines. [latex]m\u22206=m\u22207=23^\\circ[\/latex] by vertical angles because these are the opposite angles formed by two intersecting lines.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">In the provided figure given that the [latex]m\u22202=48^\\circ[\/latex], find [latex]m\u22201[\/latex], and [latex]m\u22205[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3257\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-300x143.png\" alt=\"Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.\" width=\"342\" height=\"163\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-300x143.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-65x31.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-225x108.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830-350x167.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/22190942\/Screenshot-2023-05-22-150830.png 676w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57884\">Show Solution<\/button><\/p>\n<div id=\"q57884\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\u200b\u200b\u200bm\u2220 1 = {132^ \\circ}\u200b\u200b\u200b, \u200b\u200b\u200bm\u2220 5 = {132^ \\circ}\u200b\u200b\u200b[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Contemporary Mathematics\",\"author\":\"Donna Kirk\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/10-1-points-lines-and-planes\",\"project\":\"10.1 Points, Lines, and Planes\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/10-introduction\"},{\"type\":\"copyrighted_video\",\"description\":\"Math Antics - Points, Lines, & Planes\",\"author\":\"Rob and Jeremy (MathAntics.com)\",\"organization\":\"Math Plus Motion LLC\",\"url\":\"https:\/\/youtu.be\/k5etrWdIY6o\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Types of Angles | Obtuse, Acute, Right, & Straight Angles\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/AuJQZ8Pusdg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"How to use a protractor\",\"author\":\" Christian Meglio\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/t4xCOUNEInI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Radians and Degrees (NancyPi)\",\"author\":\"Nancy (NancyPi.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/l6hSY2Pcch0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Complementary and Supplementary Angles | How to Find Missing Angles | Math with Mr. J\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/E_ulN2FjJac\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine the Complement and Supplement of a Given Angle\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ZQ_L3yJOfqM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"What are Vertical Angles? 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