{"id":2960,"date":"2023-05-17T17:46:49","date_gmt":"2023-05-17T17:46:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2960"},"modified":"2024-10-18T20:51:35","modified_gmt":"2024-10-18T20:51:35","slug":"triangles-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/triangles-fresh-take\/","title":{"raw":"Triangles: Fresh Take","rendered":"Triangles: Fresh Take"},"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<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Triangles are one of the fundamental shapes in geometry. They can be classified based on their angle measures and side lengths. These two components help us categorize and understand different types of triangles.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Classification by Side Lengths:\r\n\r\n\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 60 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\r\n<ul>\r\n\t<li><strong>Right Triangle<\/strong>: A right triangle has one angle measuring exactly 90 degrees.<\/li>\r\n\t<li><strong>Acute Triangle<\/strong>: An acute triangle has all three angles measuring less than 90 degrees.<\/li>\r\n\t<li><strong>Obtuse Triangle<\/strong>: An obtuse triangle has one angle measuring greater than 90 degrees.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/1k0G-Y41jRA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\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+the+Different+Types+of+Triangles_+_+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are the Different Types of Triangles? | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Determine the type of triangle based on the angle measures provided. Triangle [latex]XYZ[\/latex] has angle measures [latex]\u2220X = 110\u00b0[\/latex], [latex]\u2220Y = 30\u00b0[\/latex], and [latex]\u2220Z = 40\u00b0[\/latex].[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57883\"]<br \/>\r\nIn triangle [latex]XYZ[\/latex], one angle ([latex]\u2220X[\/latex]) is greater than [latex]90\u00b0[\/latex], while the other two angles ([latex]\u2220Y[\/latex] and [latex]\u2220Z[\/latex]) are less than [latex]90\u00b0[\/latex]. Thus, triangle [latex]XYZ[\/latex] is classified as an obtuse triangle.[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Determine the type of triangle based on the given side lengths. Triangle [latex]PQR[\/latex] has side lengths [latex]PQ = 7[\/latex] cm, [latex]QR = 9[\/latex] cm, and [latex]PR = 8[\/latex] cm.[reveal-answer q=\"57884\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57884\"]<br \/>\r\nIn triangle [latex]PQR[\/latex], all three sides have different lengths. Since no sides are equal, triangle [latex]PQR[\/latex] is classified as a scalene triangle.[\/hidden-answer]<\/section>\r\n<h2>Finding the Measure of the Third Angle in a Triangle<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Finding the measure of the third angle in a triangle involves determining the value of the remaining angle when the measures of two angles are known. Remember all angles in a triangle add up to equal [latex]180^{\\circ}[\/latex].<\/p>\r\n<p>To find the measure of the third angle in a triangle:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Add the measures of the two known angles together.<\/li>\r\n\t<li>Subtract the sum from [latex]180[\/latex] degrees.<\/li>\r\n\t<li>The resulting value is the measure of the third angle.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350396&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=bp5UxYKPie8&amp;video_target=tpm-plugin-cpgzypz4-bp5UxYKPie8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Missing+Angles+in+Triangles+%7C+How+to+Find+the+Missing+Angle+of+a+Triangle+Step+by+Step.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMissing Angles in Triangles | How to Find the Missing Angle of a Triangle Step by Step\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">In triangle [latex]ABC[\/latex], angle [latex]A[\/latex] measures [latex]45^{\\circ}[\/latex] and angle [latex]B[\/latex] measures [latex]60^{\\circ}[\/latex]. Find the measure of angle [latex]C[\/latex].<br \/>\r\n[reveal-answer q=\"57889\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57889\"]<br \/>\r\nStep 1: Add the given angle measures: [latex]45^{\\circ} + 60^{\\circ} = 105^{\\circ}[\/latex]<br \/>\r\nStep 2: Subtract the sum from 180 degrees to find the measure of angle [latex]C[\/latex]: [latex]180^{\\circ} - 105^{\\circ} = 75^{\\circ}[\/latex]<br \/>\r\n<br \/>\r\nTherefore, the measure of angle [latex]C[\/latex] in triangle [latex]ABC[\/latex] is [latex]75^{\\circ}[\/latex].<br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Consider the right triangle [latex]PQR[\/latex]. Angle [latex]Q[\/latex] measures [latex]35^{\\circ}[\/latex]. Find the measure of angle [latex]R[\/latex].<br \/>\r\n[reveal-answer q=\"57888\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57888\"]<br \/>\r\nStep 1: Add the given angle measures: [latex]90^{\\circ} + 35^{\\circ} = 125^{\\circ}[\/latex]<br \/>\r\nStep 2: Subtract the sum from [latex]180[\/latex] degrees to find the measure of angle [latex]R[\/latex]: [latex]180^{\\circ} - 125^{\\circ} = 55^{\\circ}[\/latex]<br \/>\r\n<br \/>\r\nTherefore, the measure of angle [latex]R[\/latex] in triangle [latex]PQR[\/latex] is [latex]55[\/latex] degrees.<br \/>\r\n[\/hidden-answer]<\/section>\r\n<h2>Similar Triangles<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Finding <strong>similar triangles<\/strong> involves comparing the corresponding angles and side lengths of two or more triangles to determine if they have proportional relationships.<\/p>\r\n<p>To determine if triangles are similar:<\/p>\r\n<ul>\r\n\t<li>Compare the measures of their corresponding angles. If all angles have the same measures, the triangles are similar.<\/li>\r\n\t<li>Compare the lengths of their corresponding sides. If the ratios of the corresponding side lengths are equal, the triangles are similar.<\/li>\r\n<\/ul>\r\n<p>To solve for an unknown side of a triangle given two similar triangles we use the following steps:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Identify the corresponding sides in the two similar triangles.<\/li>\r\n\t<li>Write the ratio of the lengths of the corresponding sides.<\/li>\r\n\t<li>Set up a proportion to solve for the unknown.<\/li>\r\n\t<li>Solve the proportion by cross multiplying and dividing to find the unknown.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350397&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8h-BeLqfa3E&amp;video_target=tpm-plugin-2af4npii-8h-BeLqfa3E\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/When+are+Two+Triangles+Similar%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhen are Two Triangles Similar? | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Solve for the unknown side [latex]x[\/latex] given the similar triangles below.<br \/>\r\n<center><img class=\"aligncenter size-medium wp-image-3015\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-300x260.jpg\" alt=\"Two similar triangles are given. Triangle ABC has sides AB 16, BC x, CA 8. Triangle CDF has sides CD 21, DE y, EC 12. Angles B and D are the same.\" width=\"300\" height=\"260\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n[reveal-answer q=\"57887\"]Show Solution[\/reveal-answer] <br \/>\r\n[hidden-answer a=\"57887\"]<br \/>\r\nStep 1: To find the length of [latex]BC[\/latex], we set up a proportion based on the corresponding sides: [latex]\\frac{BC}{CD} = \\frac{AC}{EC}[\/latex]<br \/>\r\nStep 2: Substituting the given values: [latex]\\frac{x}{21} = \\frac{8}{12}[\/latex]<br \/>\r\nStep 3: Cross-multiplying [latex]x*12 = 21*8[\/latex][latex]x*12 = 168[\/latex]<br \/>\r\n<br \/>\r\nStep 4: Divide to solve for the unknown.\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]x = \\frac{168}{12}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x = 14[\/latex]<\/p>\r\n<p>Therefore, the length of side [latex]CB[\/latex] in triangle [latex]ABC[\/latex] is [latex]14[\/latex].<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the video below we show an example of how to find the missing sides of two triangles that are similar.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/FbtCUXgVA3A\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Find+the+Length+of+a+Side+of+a+Triangle+Using+Similar+Triangles.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Length of a Side of a Triangle Using Similar Triangles\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Using the Pythagorean Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>The Pythagorean Theorem<\/strong> is a fundamental concept in geometry that relates to the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.<\/p>\r\n<p>In equation form, the Pythagorean Theorem can be written as: [latex]c^2 = a^2 + b^2[\/latex]<\/p>\r\n<p>Where [latex]c[\/latex] represents the length of the <strong>hypotenuse<\/strong>, and [latex]a[\/latex] and [latex]b[\/latex] represent the lengths of the other two sides (called the legs) of the right triangle.<\/p>\r\n<p>To use the Pythagorean Theorem, you can follow these steps:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Identify the right triangle: Ensure that you have a triangle with a right angle ([latex]90^\\circ[\/latex]).<\/li>\r\n\t<li>Identify the legs and hypotenuse: Label the lengths of the legs as [latex]a[\/latex] and [latex]b[\/latex], and the length of the hypotenuse as [latex]c[\/latex] .<\/li>\r\n\t<li>Apply the Pythagorean Theorem: Square the lengths of the legs ([latex]a^2[\/latex] and [latex]b^2[\/latex]), then add them together. The result should be equal to the square of the length of the hypotenuse ([latex]c^2[\/latex]).<\/li>\r\n\t<li>Solve for the unknown: If you know the lengths of two sides (legs or hypotenuse), you can use the theorem to find the length of the remaining side by rearranging the equation and solving for the unknown value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Consider a right triangle [latex]XYZ[\/latex] with a hypotenuse of [latex]c = 10[\/latex] units and one leg [latex]a = 6[\/latex] units. We need to find the length of the other leg, [latex]b[\/latex].<br \/>\r\n[reveal-answer q=\"57886\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57886\"]<br \/>\r\nApplying the Pythagorean Theorem ([latex]c^2 = a^2 + b^2[\/latex]), we have:\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]10^2 = 6^2 + b^2[\/latex]<\/p>\r\n<p>We must get [latex]b[\/latex] by itself to solve the equation:<\/p>\r\n<p style=\"text-align: center;\">[latex]b^2 = 10^2 - 6^2 [\/latex]<\/p>\r\n<p>Solving for the unknown we find:<\/p>\r\n<p style=\"text-align: center;\">[latex]b^2 = 100 - 36[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]b^2 = 64[\/latex]<\/p>\r\n<p>Taking the square root of both sides:<\/p>\r\n<p style=\"text-align: center;\">[latex]b = \\sqrt{64}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]b = 8[\/latex] units<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose you have a rectangular garden that needs to be fenced. You want to determine the length of the diagonal of the garden to ensure you purchase enough fencing material. The width of the garden is [latex]8[\/latex] meters and the length is [latex]10[\/latex] meters. Solve for the diagonal.<br \/>\r\n[reveal-answer q=\"57885\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57885\"]<br \/>\r\nTo find the length of the diagonal, we can apply the Pythagorean Theorem ([latex]c^2 = a^2 + b^2[\/latex])In this case, the two sides of the rectangle are the width ([latex]8[\/latex] meters) and the length ([latex]10[\/latex] meters). These are the legs of our right triangle. Let's denote the length of the diagonal as [latex]D[\/latex]. Applying the theorem, we have:\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]D^2 = (\\text{width})^2 + (\\text{length})^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]D^2 = 8^2 + 10^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]D^2 = 64 + 100[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]D^2 = 164[\/latex]<\/p>\r\n<p>Taking the square root of both sides to solve for [latex]D[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]D = \\sqrt{164}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]D \u2248 12.806[\/latex] meters<\/p>\r\n<p>Therefore, the length of the diagonal of the rectangular garden is approximately [latex]12.806[\/latex] meters.<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350398&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=nCD-bAEbB3I&amp;video_target=tpm-plugin-kfei7u55-nCD-bAEbB3I\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/An+Introduction+to+the+Pythagorean+Theorem+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAn Introduction to the Pythagorean Theorem | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the following video we show two more examples of how to use the Pythagorean Theorem to solve application problems.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/2P0dJxpwFMY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solve+Applications+Using+the+Pythagorean+Theorem+(c+only).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve Applications Using the Pythagorean Theorem (c only)\u201d here (opens in new window).<\/a><\/p>\r\n<\/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<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Triangles are one of the fundamental shapes in geometry. They can be classified based on their angle measures and side lengths. These two components help us categorize and understand different types of triangles.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Classification by Side Lengths:\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 60 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<ul>\n<li><strong>Right Triangle<\/strong>: A right triangle has one angle measuring exactly 90 degrees.<\/li>\n<li><strong>Acute Triangle<\/strong>: An acute triangle has all three angles measuring less than 90 degrees.<\/li>\n<li><strong>Obtuse Triangle<\/strong>: An obtuse triangle has one angle measuring greater than 90 degrees.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/1k0G-Y41jRA\" width=\"560\" height=\"315\" 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\/What+are+the+Different+Types+of+Triangles_+_+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are the Different Types of Triangles? | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Determine the type of triangle based on the angle measures provided. Triangle [latex]XYZ[\/latex] has angle measures [latex]\u2220X = 110\u00b0[\/latex], [latex]\u2220Y = 30\u00b0[\/latex], and [latex]\u2220Z = 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\">\nIn triangle [latex]XYZ[\/latex], one angle ([latex]\u2220X[\/latex]) is greater than [latex]90\u00b0[\/latex], while the other two angles ([latex]\u2220Y[\/latex] and [latex]\u2220Z[\/latex]) are less than [latex]90\u00b0[\/latex]. Thus, triangle [latex]XYZ[\/latex] is classified as an obtuse triangle.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Determine the type of triangle based on the given side lengths. Triangle [latex]PQR[\/latex] has side lengths [latex]PQ = 7[\/latex] cm, [latex]QR = 9[\/latex] cm, and [latex]PR = 8[\/latex] cm.<\/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\">\nIn triangle [latex]PQR[\/latex], all three sides have different lengths. Since no sides are equal, triangle [latex]PQR[\/latex] is classified as a scalene triangle.<\/div>\n<\/div>\n<\/section>\n<h2>Finding the Measure of the Third Angle in a Triangle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Finding the measure of the third angle in a triangle involves determining the value of the remaining angle when the measures of two angles are known. Remember all angles in a triangle add up to equal [latex]180^{\\circ}[\/latex].<\/p>\n<p>To find the measure of the third angle in a triangle:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Add the measures of the two known angles together.<\/li>\n<li>Subtract the sum from [latex]180[\/latex] degrees.<\/li>\n<li>The resulting value is the measure of the third angle.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350396&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=bp5UxYKPie8&amp;video_target=tpm-plugin-cpgzypz4-bp5UxYKPie8\" 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\/Missing+Angles+in+Triangles+%7C+How+to+Find+the+Missing+Angle+of+a+Triangle+Step+by+Step.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMissing Angles in Triangles | How to Find the Missing Angle of a Triangle Step by Step\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">In triangle [latex]ABC[\/latex], angle [latex]A[\/latex] measures [latex]45^{\\circ}[\/latex] and angle [latex]B[\/latex] measures [latex]60^{\\circ}[\/latex]. Find the measure of angle [latex]C[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57889\">Show Solution<\/button><\/p>\n<div id=\"q57889\" class=\"hidden-answer\" style=\"display: none\">\nStep 1: Add the given angle measures: [latex]45^{\\circ} + 60^{\\circ} = 105^{\\circ}[\/latex]<br \/>\nStep 2: Subtract the sum from 180 degrees to find the measure of angle [latex]C[\/latex]: [latex]180^{\\circ} - 105^{\\circ} = 75^{\\circ}[\/latex]<\/p>\n<p>Therefore, the measure of angle [latex]C[\/latex] in triangle [latex]ABC[\/latex] is [latex]75^{\\circ}[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Consider the right triangle [latex]PQR[\/latex]. Angle [latex]Q[\/latex] measures [latex]35^{\\circ}[\/latex]. Find the measure of angle [latex]R[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57888\">Show Solution<\/button><\/p>\n<div id=\"q57888\" class=\"hidden-answer\" style=\"display: none\">\nStep 1: Add the given angle measures: [latex]90^{\\circ} + 35^{\\circ} = 125^{\\circ}[\/latex]<br \/>\nStep 2: Subtract the sum from [latex]180[\/latex] degrees to find the measure of angle [latex]R[\/latex]: [latex]180^{\\circ} - 125^{\\circ} = 55^{\\circ}[\/latex]<\/p>\n<p>Therefore, the measure of angle [latex]R[\/latex] in triangle [latex]PQR[\/latex] is [latex]55[\/latex] degrees.\n<\/div>\n<\/div>\n<\/section>\n<h2>Similar Triangles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Finding <strong>similar triangles<\/strong> involves comparing the corresponding angles and side lengths of two or more triangles to determine if they have proportional relationships.<\/p>\n<p>To determine if triangles are similar:<\/p>\n<ul>\n<li>Compare the measures of their corresponding angles. If all angles have the same measures, the triangles are similar.<\/li>\n<li>Compare the lengths of their corresponding sides. If the ratios of the corresponding side lengths are equal, the triangles are similar.<\/li>\n<\/ul>\n<p>To solve for an unknown side of a triangle given two similar triangles we use the following steps:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify the corresponding sides in the two similar triangles.<\/li>\n<li>Write the ratio of the lengths of the corresponding sides.<\/li>\n<li>Set up a proportion to solve for the unknown.<\/li>\n<li>Solve the proportion by cross multiplying and dividing to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350397&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8h-BeLqfa3E&amp;video_target=tpm-plugin-2af4npii-8h-BeLqfa3E\" 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\/When+are+Two+Triangles+Similar%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhen are Two Triangles Similar? | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Solve for the unknown side [latex]x[\/latex] given the similar triangles below.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-3015\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-300x260.jpg\" alt=\"Two similar triangles are given. Triangle ABC has sides AB 16, BC x, CA 8. Triangle CDF has sides CD 21, DE y, EC 12. Angles B and D are the same.\" width=\"300\" height=\"260\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-300x260.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-1024x886.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-768x665.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-1536x1329.jpg 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-2048x1772.jpg 2048w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-1200x1039.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-65x56.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-225x195.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17191106\/shutterstock_2149588925-350x303.jpg 350w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57887\">Show Solution<\/button> <\/p>\n<div id=\"q57887\" class=\"hidden-answer\" style=\"display: none\">\nStep 1: To find the length of [latex]BC[\/latex], we set up a proportion based on the corresponding sides: [latex]\\frac{BC}{CD} = \\frac{AC}{EC}[\/latex]<br \/>\nStep 2: Substituting the given values: [latex]\\frac{x}{21} = \\frac{8}{12}[\/latex]<br \/>\nStep 3: Cross-multiplying [latex]x*12 = 21*8[\/latex][latex]x*12 = 168[\/latex]<\/p>\n<p>Step 4: Divide to solve for the unknown.<\/p>\n<p style=\"text-align: center;\">[latex]x = \\frac{168}{12}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x = 14[\/latex]<\/p>\n<p>Therefore, the length of side [latex]CB[\/latex] in triangle [latex]ABC[\/latex] is [latex]14[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<p>In the video below we show an example of how to find the missing sides of two triangles that are similar.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/FbtCUXgVA3A\" width=\"560\" height=\"315\" 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\/Ex+1_+Find+the+Length+of+a+Side+of+a+Triangle+Using+Similar+Triangles.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Length of a Side of a Triangle Using Similar Triangles\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using the Pythagorean Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>The Pythagorean Theorem<\/strong> is a fundamental concept in geometry that relates to the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.<\/p>\n<p>In equation form, the Pythagorean Theorem can be written as: [latex]c^2 = a^2 + b^2[\/latex]<\/p>\n<p>Where [latex]c[\/latex] represents the length of the <strong>hypotenuse<\/strong>, and [latex]a[\/latex] and [latex]b[\/latex] represent the lengths of the other two sides (called the legs) of the right triangle.<\/p>\n<p>To use the Pythagorean Theorem, you can follow these steps:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify the right triangle: Ensure that you have a triangle with a right angle ([latex]90^\\circ[\/latex]).<\/li>\n<li>Identify the legs and hypotenuse: Label the lengths of the legs as [latex]a[\/latex] and [latex]b[\/latex], and the length of the hypotenuse as [latex]c[\/latex] .<\/li>\n<li>Apply the Pythagorean Theorem: Square the lengths of the legs ([latex]a^2[\/latex] and [latex]b^2[\/latex]), then add them together. The result should be equal to the square of the length of the hypotenuse ([latex]c^2[\/latex]).<\/li>\n<li>Solve for the unknown: If you know the lengths of two sides (legs or hypotenuse), you can use the theorem to find the length of the remaining side by rearranging the equation and solving for the unknown value.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Consider a right triangle [latex]XYZ[\/latex] with a hypotenuse of [latex]c = 10[\/latex] units and one leg [latex]a = 6[\/latex] units. We need to find the length of the other leg, [latex]b[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57886\">Show Solution<\/button><\/p>\n<div id=\"q57886\" class=\"hidden-answer\" style=\"display: none\">\nApplying the Pythagorean Theorem ([latex]c^2 = a^2 + b^2[\/latex]), we have:<\/p>\n<p style=\"text-align: center;\">[latex]10^2 = 6^2 + b^2[\/latex]<\/p>\n<p>We must get [latex]b[\/latex] by itself to solve the equation:<\/p>\n<p style=\"text-align: center;\">[latex]b^2 = 10^2 - 6^2[\/latex]<\/p>\n<p>Solving for the unknown we find:<\/p>\n<p style=\"text-align: center;\">[latex]b^2 = 100 - 36[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]b^2 = 64[\/latex]<\/p>\n<p>Taking the square root of both sides:<\/p>\n<p style=\"text-align: center;\">[latex]b = \\sqrt{64}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]b = 8[\/latex] units<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Suppose you have a rectangular garden that needs to be fenced. You want to determine the length of the diagonal of the garden to ensure you purchase enough fencing material. The width of the garden is [latex]8[\/latex] meters and the length is [latex]10[\/latex] meters. Solve for the diagonal.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57885\">Show Solution<\/button><\/p>\n<div id=\"q57885\" class=\"hidden-answer\" style=\"display: none\">\nTo find the length of the diagonal, we can apply the Pythagorean Theorem ([latex]c^2 = a^2 + b^2[\/latex])In this case, the two sides of the rectangle are the width ([latex]8[\/latex] meters) and the length ([latex]10[\/latex] meters). These are the legs of our right triangle. Let&#8217;s denote the length of the diagonal as [latex]D[\/latex]. Applying the theorem, we have:<\/p>\n<p style=\"text-align: center;\">[latex]D^2 = (\\text{width})^2 + (\\text{length})^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]D^2 = 8^2 + 10^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]D^2 = 64 + 100[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]D^2 = 164[\/latex]<\/p>\n<p>Taking the square root of both sides to solve for [latex]D[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]D = \\sqrt{164}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]D \u2248 12.806[\/latex] meters<\/p>\n<p>Therefore, the length of the diagonal of the rectangular garden is approximately [latex]12.806[\/latex] meters.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350398&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=nCD-bAEbB3I&amp;video_target=tpm-plugin-kfei7u55-nCD-bAEbB3I\" 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\/An+Introduction+to+the+Pythagorean+Theorem+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAn Introduction to the Pythagorean Theorem | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the following video we show two more examples of how to use the Pythagorean Theorem to solve application problems.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/2P0dJxpwFMY\" width=\"560\" height=\"315\" 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\/Solve+Applications+Using+the+Pythagorean+Theorem+(c+only).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve Applications Using the Pythagorean Theorem (c only)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Image of two triangles in example\",\"author\":\"MATHEUS 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