{"id":2959,"date":"2023-05-17T17:46:53","date_gmt":"2023-05-17T17:46:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2959"},"modified":"2025-08-26T03:22:45","modified_gmt":"2025-08-26T03:22:45","slug":"triangles-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/triangles-learn-it-1\/","title":{"raw":"Triangles: Learn It 1","rendered":"Triangles: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Classify triangles based on their angles and side lengths<\/li>\r\n\t<li>Determine the measure of the third angle in a triangle when the measures of two angles are given<\/li>\r\n\t<li>Apply properties of similar triangles to find unknown side lengths<\/li>\r\n\t<li>Use the Pythagorean theorem to calculate unknown side lengths in triangles<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Types of Triangles<\/h2>\r\n<p>In geometry, triangles are one of the fundamental shapes. They can be classified based on their angle measures and side lengths, which helps us categorize and understand different types of triangles.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>triangle classifications<\/h3>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Classification by Side Lengths:\r\n\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n\t<li><strong>Equilateral Triangle<\/strong>: An equilateral triangle has all three sides of equal length. Consequently, all three angles are also equal, measuring [latex]60[\/latex] degrees each.<\/li>\r\n\t<li><strong>Isosceles Triangle<\/strong>: An isosceles triangle has two sides of equal length. This means two of its angles are also equal.<\/li>\r\n\t<li><strong>Scalene Triangle<\/strong>: A scalene triangle has no sides of equal length. Consequently, all three angles are different from one another.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Classification by Angle Measures:\r\n\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n\t<li><strong>Right Triangle<\/strong>: A right triangle has one angle measuring exactly [latex]90[\/latex] degrees.<\/li>\r\n\t<li><strong>Acute Triangle<\/strong>: An acute triangle has all three angles measuring less than [latex]90[\/latex] degrees.<\/li>\r\n\t<li><strong>Obtuse Triangle<\/strong>: An obtuse triangle has one angle measuring greater than [latex]90[\/latex] degrees.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section>\r\n<center>\r\n[caption id=\"attachment_2976\" align=\"aligncenter\" width=\"684\"]<img class=\"wp-image-2976 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures.png\" alt=\"a diagram of different types of triangles: a equilateral triangle with all equal sides, an isosceles triangle with two equal sides, a scalene triangle with no equal sides, an acute triangle with 3 angles less than 90 degrees, a right triangle with one angle equal to 90 degrees, and an obtuse triangle with one angle greater than 90 degrees\" width=\"684\" height=\"795\" \/> Figure 1. Types of triangles classified by their side lengths and angles[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox example\">Identify the type of triangle based on the given angle measures. Triangle [latex]ABC[\/latex] has angle measures [latex]\u2220A = 60\u00b0[\/latex], [latex]\u2220B = 80\u00b0[\/latex], and [latex]\u2220C = 40\u00b0[\/latex].[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"]In triangle [latex]ABC[\/latex], all three angles ([latex]\u2220A[\/latex], [latex]\u2220B[\/latex], and [latex]\u2220C[\/latex]) are less than [latex]90\u00b0[\/latex]. Therefore, triangle [latex]ABC[\/latex] is classified as an acute triangle.[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Categorize the triangle based on the provided side lengths. Triangle [latex]XYZ[\/latex] has side lengths [latex]XY = 4[\/latex] cm, [latex]YZ = 4[\/latex] cm, and [latex]XZ = 6[\/latex] cm.[reveal-answer q=\"57882\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57882\"]In triangle [latex]XYZ[\/latex], two sides ([latex]XY[\/latex] and [latex]YZ[\/latex]) have the same length, while the third side ([latex]XZ[\/latex]) has a different length. Thus, triangle [latex]XYZ[\/latex] is classified as an isosceles triangle. [\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7022[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Classify triangles based on their angles and side lengths<\/li>\n<li>Determine the measure of the third angle in a triangle when the measures of two angles are given<\/li>\n<li>Apply properties of similar triangles to find unknown side lengths<\/li>\n<li>Use the Pythagorean theorem to calculate unknown side lengths in triangles<\/li>\n<\/ul>\n<\/section>\n<h2>Types of Triangles<\/h2>\n<p>In geometry, triangles are one of the fundamental shapes. They can be classified based on their angle measures and side lengths, which helps us categorize and understand different types of triangles.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>triangle classifications<\/h3>\n<ol style=\"list-style-type: decimal;\">\n<li>Classification by Side Lengths:\n<p>&nbsp;<\/p>\n<ul>\n<li><strong>Equilateral Triangle<\/strong>: An equilateral triangle has all three sides of equal length. Consequently, all three angles are also equal, measuring [latex]60[\/latex] degrees each.<\/li>\n<li><strong>Isosceles Triangle<\/strong>: An isosceles triangle has two sides of equal length. This means two of its angles are also equal.<\/li>\n<li><strong>Scalene Triangle<\/strong>: A scalene triangle has no sides of equal length. Consequently, all three angles are different from one another.<\/li>\n<\/ul>\n<\/li>\n<li>Classification by Angle Measures:\n<p>&nbsp;<\/p>\n<ul>\n<li><strong>Right Triangle<\/strong>: A right triangle has one angle measuring exactly [latex]90[\/latex] degrees.<\/li>\n<li><strong>Acute Triangle<\/strong>: An acute triangle has all three angles measuring less than [latex]90[\/latex] degrees.<\/li>\n<li><strong>Obtuse Triangle<\/strong>: An obtuse triangle has one angle measuring greater than [latex]90[\/latex] degrees.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_2976\" aria-describedby=\"caption-attachment-2976\" style=\"width: 684px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2976 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures.png\" alt=\"a diagram of different types of triangles: a equilateral triangle with all equal sides, an isosceles triangle with two equal sides, a scalene triangle with no equal sides, an acute triangle with 3 angles less than 90 degrees, a right triangle with one angle equal to 90 degrees, and an obtuse triangle with one angle greater than 90 degrees\" width=\"684\" height=\"795\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures.png 684w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures-258x300.png 258w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures-65x76.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures-225x262.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17180039\/Classification-by-Angle-Measures-350x407.png 350w\" sizes=\"(max-width: 684px) 100vw, 684px\" \/><figcaption id=\"caption-attachment-2976\" class=\"wp-caption-text\">Figure 1. Types of triangles classified by their side lengths and angles<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\">Identify the type of triangle based on the given angle measures. Triangle [latex]ABC[\/latex] has angle measures [latex]\u2220A = 60\u00b0[\/latex], [latex]\u2220B = 80\u00b0[\/latex], and [latex]\u2220C = 40\u00b0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57883\">Show Solution<\/button> <\/p>\n<div id=\"q57883\" class=\"hidden-answer\" style=\"display: none\">In triangle [latex]ABC[\/latex], all three angles ([latex]\u2220A[\/latex], [latex]\u2220B[\/latex], and [latex]\u2220C[\/latex]) are less than [latex]90\u00b0[\/latex]. Therefore, triangle [latex]ABC[\/latex] is classified as an acute triangle.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Categorize the triangle based on the provided side lengths. Triangle [latex]XYZ[\/latex] has side lengths [latex]XY = 4[\/latex] cm, [latex]YZ = 4[\/latex] cm, and [latex]XZ = 6[\/latex] cm.<\/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\">In triangle [latex]XYZ[\/latex], two sides ([latex]XY[\/latex] and [latex]YZ[\/latex]) have the same length, while the third side ([latex]XZ[\/latex]) has a different length. Thus, triangle [latex]XYZ[\/latex] is classified as an isosceles triangle. <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7022\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7022&theme=lumen&iframe_resize_id=ohm7022&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2959"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":30,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2959\/revisions"}],"predecessor-version":[{"id":15638,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2959\/revisions\/15638"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2959\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2959"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2959"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2959"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}