Triangles | Brilliant Math & Science Wiki

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</div> <div class="alternative"> <p> Already have an account? <a href="https://brilliant.org/account/login/?next=/wiki/triangles/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link"> Log in here. </a> </p> </div> </div> </div> <div class="wiki-top-editors" id="cmp_wiki_top_editors_id"> <b>Aditya Virani</b>, <b>Satvik Golechha</b>, <b>Aditya Raut</b>, and <div class="dropdown tipsy"> <button class="btn-link dropdown-toggle" data-toggle="dropdown"> 16 others </button> <ul class="dropdown-menu"> <li> <b>Christopher Williams</b> </li> <li> <b>Sharky Kesa</b> </li> <li> <b>Arron Kau</b> </li> <li> <b>Niranjan Khanderia</b> </li> <li> <b>Jordan Calmes</b> </li> <li> <b>Ansh Bhatt</b> </li> <li> <b>Anuj Shikarkhane</b> </li> <li> <b>Pi Han Goh</b> </li> <li> <b>Gene Keun Chung</b> </li> <li> <b>Mahindra Jain</b> </li> <li> <b>Kelly Tran</b> </li> <li> <b>Andrew Ellinor</b> </li> <li> <b>A Former Brilliant Member</b> </li> <li> <b>Eli Ross</b> </li> <li> <b>Jimin Khim</b> </li> <li> <b>Calvin Lin</b> </li> </ul> </div> contributed </div> <div id="wiki-main" data-controller="app/newsfeed:feed"> <div class="summary-container" id="cmp_wiki_canonical_page_id"> <div class="summary wiki-content" data-controller="app/wiki:summary,app/zoomable:images" data-cmp-url="/wiki/triangles/" data-page-key="wiki_canonical_page" data-cmp-key="wiki_canonical_page"> <div class="section collapsed" id="section-pre-header-section"><div class="section-container"><p><strong>Triangles</strong> are <a href="/wiki/shifty-shapes/" class="wiki_link" title="polygons"target="_blank">polygons</a> (shapes) with three sides and three <a href="/wiki/angles/" class="wiki_link" title="angles"target="_blank">angles</a>, which can be formed by connecting any three points in a plane. They are one of the first shapes studied in <a href="/wiki/learn-and-practice-geometry-on-brilliant/" class="wiki_link" title="geometry"target="_blank">geometry</a>. </p> <p>Triangles are particularly important because arbitrary polygons (with 4, 5, 6, or \(n\) sides) can be <a target="_blank" rel="nofollow" href="https://brilliant.org/wiki/regular-polygons/#regular-polygons-angles">decomposed into triangles</a>. Thus, understanding the basic properties of triangles allows for deeper study of these larger polygons as well. Interestingly, the triangle is the only rigid polygon formed out of straight line segments, meaning that if the three lengths of the sides are given, the measurements correspond to a unique triangle. Because of this, it is often possible, given some information about a triangle (e.g. some side lengths and some angles), to determine additional facts about the triangle.</p> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/vaHwdAGvLx.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/vaHwdAGvLx.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/vaHwdAGvLx.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/vaHwdAGvLx.png?width=3600 3x" alt="An equilateral triangle, a right triangle, and a scalene triangle" /> <span class="caption">An equilateral triangle, a right triangle, and a scalene triangle</span></span></p> </div> </div> <div class="toc wiki-toc"> <h4>Contents</h4> <ul class="unstyled"> <li> <a href="#triangles-angle-sum">Sum of Angles in a Triangle</a> </li> <li> <a href="#triangle-inequality">Triangle Inequality</a> </li> <li> <a href="#classifying-triangles">Classifying Triangles</a> </li> <li> <a href="#triangle-area-basic">Area of Triangles</a> </li> <li> <a href="#triangles-exterior-angles">Exterior Angles of Triangles</a> </li> <li> <a href="#triangles-problem-solving-easy">Problem Solving with Triangles</a> </li> <li> <a href="#see-also">See Also</a> </li> </ul> </div> <div id="triangles-angle-sum" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Sum of Angles in a Triangle</h2> </header> </div> <div class="section collapsed" id="section-triangles-angle-sum"><div class="section-container"> <blockquote class="theorem"><p> If \(ABC\) is a triangle, then \(\angle ABC + \angle BCA + \angle CAB = 180^\circ.\) In other words, the sum of the \(3\) internal <a href="/wiki/angles/" class="wiki_link" title="angles"target="_blank">angles</a> in any triangle is \(180^\circ.\) </p></blockquote> <blockquote class="proof"><p> <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/i6DJlM7Y9C-13.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/i6DJlM7Y9C-13.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/i6DJlM7Y9C-13.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/i6DJlM7Y9C-13.png?width=3600 3x" alt="" /> </span></p> <p>Draw line \(DE\) that is parallel to \(AB\) and passes through \(C\) as in the above figure. <br /> Since \( DE \parallel AB \), applying the principle of <a href="/wiki/parallel-lines/" class="wiki_link" title="alternate interior angles"target="_blank">alternate interior angles</a> shows that \( \angle DCA = \angle CAB \) and \( \angle BCE = \angle CBA.\) <br /> Since angles in a line sum up to 180 degrees, \( \angle DCA + \angle ACB + \angle BCE = 180^ \circ \). <br /> Thus, we conclude that \( \angle CAB + \angle ACB + \angle CBA = \angle DCA + \angle ACB + \angle BCE = 180^ \circ \). \(_\square\) </p></blockquote> </div> </div> <div id="triangle-inequality" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Triangle Inequality</h2> </header> </div> <div class="section collapsed" id="section-triangle-inequality"><div class="section-container"> <p>Main article: <a href="/wiki/triangle-inequality/" class="wiki_link" title="Triangle Inequality"target="_blank">Triangle Inequality</a></p> <p>Triangles have the property that the sum of any two sides of the triangle is always strictly greater than the third side. This property, known as the <strong>triangle inequality</strong>, is explored in the wiki linked above.</p> </div> </div> <div id="classifying-triangles" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Classifying Triangles</h2> </header> </div> <div class="section collapsed" id="section-classifying-triangles"><div class="section-container"> <p>Main article: <a href="/wiki/classification-of-triangles/" class="wiki_link" title="Classification of Triangles"target="_blank">Classification of Triangles</a></p> <p>Triangles can be classified into different categories based on their sides and angles. For example, a triangle with one angle of measure \(90^\circ\) is known as a right triangle, while a triangle with sides of all equal length is known as an equilateral triangle. These classifications and many others are explored in the wiki linked above.</p> </div> </div> <div id="triangle-area-basic" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Area of Triangles</h2> </header> </div> <div class="section collapsed" id="section-triangle-area-basic"><div class="section-container"> <p>Main article: <a target="_blank" rel="nofollow" href="https://brilliant.org/wiki/triangles-calculating-area/">Area of a Triangle</a></p> <p>When determining the area of a triangle, note that a triangle can be thought of as half of a <a href="/wiki/parallelogram/#properties-of-parallelograms" class="wiki_link" title="parallelogram"target="_blank">parallelogram</a>. The following picture should make this point clear:</p> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/oC7FL090gg-screen-shot-2016-05-13-at-41344-am.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/oC7FL090gg-screen-shot-2016-05-13-at-41344-am.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/oC7FL090gg-screen-shot-2016-05-13-at-41344-am.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/oC7FL090gg-screen-shot-2016-05-13-at-41344-am.png?width=3600 3x" alt="" /> </span></p> <p>Because the <a href="/wiki/parallelogram/#properties-of-parallelograms" class="wiki_link" title="area of a parallelogram"target="_blank">area of a parallelogram</a> is equal to the product of its base and height, the area of a triangle is simply half of that area. </p> <blockquote class="theorem"><p> The area of a triangle is \(A = \frac{bh}{2}\), where \(b\) is the length of the base and \(h\) is the height. </p></blockquote> <blockquote class="example"><p> What is the area of a triangle with base 10 and height 6?</p> <hr /> <p>The area of this triangle is \(\frac{10 \times 6}{2} = 30.\) \(_\square\) </p></blockquote> <p><div class="problem-modal-container anchor" id="problem-only-euclid-will-know"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/PoQr5YRPD4-step1.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/PoQr5YRPD4-step1.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/PoQr5YRPD4-step1.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/PoQr5YRPD4-step1.png?width=3600 3x" alt="" /> </span></p> <p>Nihar and Andrew are trying to find the area of \(\triangle ABC\) using the formula</p> <p>\[\dfrac{1}{2} \times \text{base} \times \text{height}. \]</p> <p>Nihar mistakenly multiplies base \(AB\) by the height from \(A\) \((\)instead of \(C).\) He gets a value of 14.</p> <p>Andrew mistakenly multiplies base \(BC\) by the height from \(C\) \((\)instead of \(A).\) He gets a value of 56.</p> <p>Find the actual area of triangle \(\triangle ABC\).</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>28.0</strong> </div> </div> </div> </div> </p> <p>For more advanced methods of finding the area of a triangle, such as <a href="/wiki/herons-formula/" class="wiki_link" title="Heron's formula"target="_blank">Heron's formula</a>, see the wiki linked at the top of this section.</p> </div> </div> <div id="triangles-exterior-angles" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Exterior Angles of Triangles</h2> </header> </div> <div class="section collapsed" id="section-triangles-exterior-angles"><div class="section-container"> <p>The measure of an <strong>exterior angle</strong> of a triangle is the sum of its two remote interior angles. <strong>Remote interior angles</strong> are the interior angles of a triangle that are opposite to the exterior angle under consideration. </p> <p>For <strong>regular polygons</strong>, the formula to find the exterior angle of a polygon is \(\frac{360^\circ}{n},\) where \(n\) is the number of sides.</p> <blockquote class="example"><p> Prove that \(\angle ABC+\angle CAB=\angle DCA. \)</p> <p><span class="image-caption center"> <img src="https://i.imgur.com/KfmvyIv.jpg" srcset="https://i.imgur.com/KfmvyIv.jpg 1x" alt="ExteriorAngle" /> <span class="caption">ExteriorAngle</span></span> <div class="show-hide-btn-container" data-controller="util/ui:showAndHideContent"> <button class="btn ax-click" data-ax-type="button" data-ax-id="clicked_show_here_button"> ANSWER </button> </div> <blockquote class="hidden"> From the property of <a target="_blank" rel="nofollow" href="https://brilliant.org/wiki/triangles-angle-sum/">triangles-angle sum</a>, the sum of measures of all angles in a triangle is \(180^\circ\). So, in \(\triangle ABC\), it follows that \( \angle ABC + \angle BCA + \angle CAB = 180^\circ \). </p> <p>Since \( \angle BCA \) and \( \angle DCA \) form a straight line, \( \angle BCA + \angle DCA = 180^\circ, \) It follows that both sets of angles are equal to \( 180^\circ: \)</p> <p>\[ \angle ABC + \angle BCA + \angle CAB = \angle BCA + \angle DCA.\]</p> <p>Therefore, </p> <p>\[ \angle ABC + \angle CAB = \angle DCA. \ _\square \] </blockquote> </p></blockquote> </div> </div> <div id="triangles-problem-solving-easy" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Problem Solving with Triangles</h2> </header> </div> <div class="section collapsed" id="section-triangles-problem-solving-easy"><div class="section-container"> <p><div class="problem-modal-container anchor" id="problem-a-symmetric-triangle"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>Triangle \(ABC\) has the property that \(\angle A+\angle B= \angle C\). Find the measure of \(\angle C\) in degrees.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>90</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-geom-1"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[ 90^ \circ \] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[ 180 ^ \circ \] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[ 270 ^ \circ \] </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> \[ 360 ^ \circ \] </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/cb654ac8f3.cb539c60d4.j7otcV.png?width=250" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/cb654ac8f3.cb539c60d4.j7otcV.png?width=250 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/cb654ac8f3.cb539c60d4.j7otcV.png?width=500 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/cb654ac8f3.cb539c60d4.j7otcV.png?width=750 3x" alt="" style="width:250px;max-width:100%;" /> </span> Mary drew a triangle on the board. What is the sum of all interior angles in a triangle?</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>\[ 180 ^ \circ \]</strong> </div> </div> </div> </div> </p> </div> </div> <div id="see-also" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>See Also</h2> </header> </div> <div class="section collapsed" id="section-see-also"><div class="section-container"> <p>Interested in learning more about triangles? Check out these pages:</p> <ul> <li><a href="/wiki/congruent-and-similar-triangles/" class="wiki_link" title="Congruent and Similar Triangles"target="_blank">Congruent and Similar Triangles</a> </li> <li><a href="/wiki/pythagorean-theorem/" class="wiki_link" title="Pythagorean Theorem"target="_blank">Pythagorean Theorem</a> </li> <li><a href="/wiki/triangles-calculating-area/" class="wiki_link" title="Triangles - Calculating Area"target="_blank">Triangles - Calculating Area</a></li> </ul> </div> </div> </div> </div> </div> <div class="wiki-self-citation" data-controller="app/wiki:getCitationTime"> <strong>Cite as:</strong> Triangles. <em>Brilliant.org</em>. 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