Angles | Brilliant Math & Science Wiki

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<b>Ryan Facio</b> </li> <li> <b>Tatiana Georgieva</b> </li> <li> <b>Calvin Lin</b> </li> <li> <b>Jimin Khim</b> </li> <li> <b>Eli Ross</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/angles/" 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><span class="image-caption left"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-35-34.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-35-34.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-35-34.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-35-34.png?width=3600 3x" alt="" /> </span> </p> <p>An <strong>angle</strong> is a geometric shape formed by the intersection of two <a href="/wiki/linear-equations-intersection-of-lines/" class="wiki_link" title="line segments"target="_blank">line segments</a>, lines, or rays. Angles are a measure of rotational distance as contrasted with linear distance. An angle can also be thought of as a fraction of a circle. The angle between the two line segments is the distance (measured in <a href="/wiki/degrees-radian/" class="wiki_link" title="degrees or radians"target="_blank">degrees or radians</a>) that one segment must be rotated around the intersecting point so that the two segments overlap. Angles are important to defining and studying polygons such as <a href="/wiki/triangles/" class="wiki_link" title="triangles"target="_blank">triangles</a> and <a href="/wiki/quadrilateral-classification/" class="wiki_link" title="quadrilaterals"target="_blank">quadrilaterals</a>. They are used in a variety of disciplines, ranging from animation to carpentry to <a href="/wiki/projectile-motion-easy/" class="wiki_link" title="physics"target="_blank">physics</a>.</p> </div> </div> <div class="toc wiki-toc"> <h4>Contents</h4> <ul class="unstyled"> <li> <a href="#classification-of-angles">Types of Angles</a> </li> <li> <a href="#complementary-and-supplementary-angles">Complementary and Supplementary Angles</a> </li> <li> <a href="#vertical-angles">Vertical Angles</a> </li> <li> <a href="#angle-arithmetic">Angle Chasing</a> </li> <li> <a href="#see-also">See Also</a> </li> </ul> </div> <div id="classification-of-angles" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Types of Angles</h2> </header> </div> <div class="section collapsed" id="section-classification-of-angles"><div class="section-container"> <p>Angles can be classified into five groups, based on their measure in degrees.</p> <p><span class="image-caption center"> <img src="http://f.cl.ly/items/1L2U3H1E0H140s271i0P/Untitled.png" srcset="http://f.cl.ly/items/1L2U3H1E0H140s271i0P/Untitled.png 1x" alt="" /> </span> </p> <blockquote> <ul> <li><strong>Acute</strong>: angles with measure \( < 90^\circ \)</li> <li><strong>Right</strong>: angles with measure \( = 90^\circ \)</li> <li><strong>Obtuse</strong>: angles with measure \( > 90^\circ \) and \( < 180 ^ \circ \)</li> <li><strong>Straight</strong>: angles with measure \( = 180^ \circ \)</li> <li><strong>Reflex</strong>: angles with measure \( > 180^ \circ \) and \( < 360 ^ \circ \)</li> </ul> </blockquote> <blockquote class="example"><p> If \( \angle ABC = 99^ \circ \), what type of angle is it?</p> <p><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"> Since \( 90^ \circ < 99 ^ \circ < 180^ \circ \), angle ABC is an <strong>obtuse angle</strong>. \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If angle \(XYZ \) is acute, which of the following is a possible measure of it?</p> <p>\(\quad \text{A)}\) \( 23 ^ \circ \) <br /> \(\quad \text{B)}\) \( 90 ^ \circ \) <br /> \(\quad \text{C)}\) \( 123 ^ \circ \) <br /> \(\quad \text{D)}\) \( 180 ^ \circ \) <br /> \(\quad \text{E)}\) \( 190 ^ \circ \) </p> <p><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"> The answer is \(\text{A}. \) An acute angle is between \(0^ \circ \) and \( 90^ \circ \). The only value that lies within this range is \( 23 ^ \circ .\) \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-16-56-58.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-16-56-58.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-16-56-58.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-16-56-58.png?width=3600 3x" alt="" /> </span> For the graphic above, which of the angles below is an acute angle?</p> <p>\(\quad \text{A) } AOD\) <br /> \(\quad \text{B) } COA\) <br /> \(\quad \text{C) } BOC\) <br /> \(\quad \text{D) } DOB\) </p> <p><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"> The answer is \(\text{C}.\) \( AOD \) is a straight angle. Both \( COA\) and \(DOB\) are obtuse. Only \(BOC\) is less than \( 90^\circ,\) so it is the only acute angle in the list. \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If \(\angle ABC\) is obtuse and \( \angle ABC = 2 \angle DEF \), what type of angle is \(DEF?\)</p> <p><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"> Since \( \angle ABC \) is obtuse, it follows that \( 90^ \circ < \angle ABC < 180^ \circ \). Similarly, if \( \angle DEF = \frac{ \angle ABC } { 2} \), then \( 45^ \circ < \angle DEF < 90 ^ \circ \). Therefore, \( \angle DEF \) is an <strong>acute angle</strong>. \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If \( \angle AOB \) is acute and \( \angle BOC \) is acute, and the points \(A\) and \(C\) lie on opposite sides of the line \(BO\), what is known about the angle \(AOC? \)</p> <p>\(\quad \text{A) }\) \( 0 ^ \circ < \angle AOC < 90 ^ \circ \) <br /> \(\quad \text{B) }\) \( 0 ^ \circ < \angle AOC < 180 ^ \circ \) <br /> \(\quad \text{C) }\) \( 90 ^ \circ < \angle AOC < 180 ^ \circ \) <br /> \(\quad \text{D) }\) \( 90 ^ \circ < \angle AOC < 360 ^ \circ \) <br /> \(\quad \text{E) }\) \( 180 ^ \circ < \angle AOC < 360 ^ \circ \) </p> <p><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"> \( \angle AOB \) and \( \angle BOC \) are acute, which means that \( 0 ^ \circ < \angle AOB < 90 ^ \circ \) and \( 0 ^ \circ < \angle BOC < 90 ^ \circ \). Since points \(A\) and \(C\) lie on opposite sides of the line \(BC\), we have \( \angle AOC = \angle AOB + \angle BOC \). </p> <p>Therefore: \( 0 ^ \circ < \angle AOC < 90^\circ + 90^ \circ = 180^ \circ \) and the answer is \(\text{B}.\) \(_\square\) </blockquote> </p></blockquote> </div> </div> <div id="complementary-and-supplementary-angles" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Complementary and Supplementary Angles</h2> </header> </div> <div class="section collapsed" id="section-complementary-and-supplementary-angles"><div class="section-container"> <p><strong>Complementary angles</strong> are angles that add together to make a right angle. That is, \(\angle X\) and \(\angle Y\) are complementary if \(\angle X + \angle Y = 90^\circ.\) <br /> <strong>Supplementary angles</strong> are angles that add together to make a straight line. That is, \(\angle X\) and \(\angle Y\) are supplementary if \(\angle X + \angle Y = 180^\circ.\) </p> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-14-00.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-14-00.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-14-00.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/Screen-Shot-2016-05-09-17-14-00.png?width=3600 3x" alt="" /> </span></p> <p>In the image above, \( \angle AOB \) and \( \angle BOC \) are complementary, and \( \angle DOB \) and \( \angle BOA \) are supplementary. </p> <blockquote class="example"><p> Angles \( AOB \) and \(BOC\) are complementary. What is the measure of angle \( AOC? \)</p> <p><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> </p><blockquote class="hidden"> <p>Since the two angles are complementary, their sum is \( 90^\circ \). Therefore, \( \angle AOC = \angle AOB + \angle BOC = 90 ^ \circ.\ _\square \) </blockquote> </p></blockquote> <blockquote class="example"><p> Three points \(X, Y, Z\) lie on a straight line in that order. What can we say about angles \( XYO\) and \( OYZ,\) where \(O\) does not lie on the line?</p> <p><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"> Since the points lie on a straight line, \( \angle XYO + \angle OYZ = 180 ^ \circ \), these angles are supplementary. \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If \( \angle P = 23 ^ \circ \), which of the following is complementary to it?</p> <p>\(\quad \text{A) }\) \(23^\circ \) <br /> \(\quad \text{B) }\) \( 67^\circ \) <br /> \(\quad \text{C) }\) \( 77 ^ \circ \) <br /> \(\quad \text{D) }\) \( 113 ^ \circ \) <br /> \(\quad \text{E) }\) \( 157 ^ \circ \) <br /> <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"> The complementary angles would add together to give \( 90 ^ \circ \), so we have \( 90 ^ \circ - 23 ^ \circ = 67 ^ \circ, \) implying that the answer is \(\text{B)}. \) \(_\square\) </blockquote> </p></blockquote> <p><div class="problem-modal-container anchor" id="problem-complementary-angles"> <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>What is the sum of the complementary angles?</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>90</strong> </div> </div> </div> </div> </p> <blockquote class="example"><p> If \(\angle A \) and \(\angle B \) are supplementary, and \( \angle A - \angle B = 40 ^ \circ \), what are the measures of angles \(A\) and \(B?\) <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"> Given that \( \angle A + \angle B = 180 ^ \circ \) and \( \angle A - \angle B = 40 ^ \circ \), solve the system of equations. Adding the two yields \( 2 \angle A = 220 ^ \circ \), or \( \angle A = 110 ^ \circ \). Finally, \( \angle B = 180 ^ \circ - 110 ^ \circ = 70 ^ \circ \). \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If angles \(X\) and \(Y\) are complementary, and angles \(Y\) and \(Z\) are supplementary, what do we know about angles \(X\) and \(Z?\)</p> <p><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"> We have \( X + Y = 90 ^ \circ \) and \( Y + Z = 180 ^ \circ \).</p> <p>Therefore, \( Z - X = (Z+Y) - (X+Y) = 180 ^ \circ - 90 ^ \circ = 90 ^ \circ \). \(_\square\) </blockquote> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-complementary-angles-2"> <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>The magnitude of the complementary of an angle is equal to the square of the angle's magnitude. </p> <p>Find the angle in degrees.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>9</strong> </div> </div> </div> </div> </p> </div> </div> <div id="vertical-angles" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Vertical Angles</h2> </header> </div> <div class="section collapsed" id="section-vertical-angles"><div class="section-container"> <p>The two opposite angles formed by a pair of intersecting lines are called <strong>vertical angles</strong>. These angles have equal measures.</p> <p>\(\hspace{5cm}\) <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/kC5UJcgSHH-62.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/kC5UJcgSHH-62.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/kC5UJcgSHH-62.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/kC5UJcgSHH-62.png?width=3600 3x" alt="" /> </span></p> <p>In the diagram above \( \angle ABC \) and \( \angle DBE \) are vertical angles and are therefore equal. </p> <blockquote class="example"><p> If lines \( PQ\) and \(XY\) intersect at \(O\), which angle is vertically opposite \( \angle POX?\) </p> <p><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"> The angle that is vertically opposite \( \angle POX\) will be \( \angle QOY\) \((\)which is also \( \angle YOQ). \) \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> In the following diagram, which pair of angles are vertically opposite?</p> <p>\(\quad \text{A) }\) \( ABC, CBI\) <br /> \(\quad \text{B) }\) \( ABH, HBJ \) <br /> \(\quad \text{C) }\) \( HBJ, JBI \) <br /> \(\quad \text{D) }\) \( ABC, HBI \) <br /> \(\quad \text{E) }\) \( ABH, JBI \) </p> <p>\(\hspace{5cm}\) <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=3600 3x" alt="" /> </span></p> <p><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"> Working through these pairs, we see that the only pair of opposite angles are \(ABC\) and \(HBI \). Therefore, the answer is \(\text{D)}\). \(_\square\)</p> <p><strong>Note</strong>: Another pair of vertical angles are \(ABH\) and \(CBI\). </blockquote> </p></blockquote> <blockquote class="example"><p> Given that \(ABE \) and \(CBD\) are straight lines intersecting each other, if \( \angle ABD = 75 ^ \circ \), what is the measure of \( \angle CBE?\)</p> <p><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> </p><blockquote class="hidden"> <p>\(\hspace{5cm}\) <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/lbaOhO2Lt2-64.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/lbaOhO2Lt2-64.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/lbaOhO2Lt2-64.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/lbaOhO2Lt2-64.png?width=3600 3x" alt="" /> </span></p> <p>\( \angle CBE\) and \( \angle ABD\) are vertical angles, and thus \( \angle CBE = \angle ABD = 75 ^ \circ \). \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> If \( \angle AOB \) and \( \angle COD \) are vertically opposite angles and \( \angle AOB + \angle COD = 90 ^ \circ \), what is the measure of \( \angle AOB?\)</p> <p><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"> Since they are vertically opposite angles, \( \angle AOB = \angle COD \). Therefore, \( 2 \angle AOB = 90 ^ \circ \), or \( \angle AOB = \frac{ 90^ \circ } { 2} = 45 ^ \circ \). \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> In the following diagram, given that \( ABC = 80^\circ \) and \( HBJ= 35^\circ \), what is the measure of \( JBI?\)</p> <p>\(\hspace{6cm}\) <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/LkP88Hr40a-65.png?width=3600 3x" alt="" /> </span></p> <p><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"> Since \( ABC \) and \( HBI \) are vertical angles, they are equal. Thus, \( ABC = HBI = 80^\circ \). But \( HBI = HBJ + JBI \). Thus \( 80^\circ = 35^\circ + JBI \). Therefore, \( JBI = 45^\circ \). \(_\square\) </blockquote> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-find-the-angle-3"> <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/JiRbnBYrTW-14969.svg?width=350" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/JiRbnBYrTW-14969.svg?width=350 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/JiRbnBYrTW-14969.svg?width=700 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/JiRbnBYrTW-14969.svg?width=1050 3x" alt="" style="width:350px;max-width:100%;" /> </span></p> <p>In the figure, with \( \angle BAC = 40 ^ \circ,\) what is the measure of \( \angle ABC \) (in degrees)? </p> <p>\(\) <br /> <strong>Clarification:</strong> \(HFAC, DFG, EDCB, IHGJ\) are all straight lines.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>19</strong> <div> <div class="solv-nux-solution row"> <p><span class="image-caption"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/image_optimizer/93fd2277fb9b6104dba66b0dd9270ba91614d718.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/image_optimizer/93fd2277fb9b6104dba66b0dd9270ba91614d718.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/image_optimizer/93fd2277fb9b6104dba66b0dd9270ba91614d718.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/image_optimizer/93fd2277fb9b6104dba66b0dd9270ba91614d718.png?width=3600 3x" alt="" /> </span></p> </div> </div> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-finding-the-angles"> <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> 180 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 270 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 360 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 540 </span> </span> </div> <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/qQrngMoByp-17057.svg?width=350" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/qQrngMoByp-17057.svg?width=350 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/qQrngMoByp-17057.svg?width=700 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/qQrngMoByp-17057.svg?width=1050 3x" alt="" style="width:350px;max-width:100%;" /> </span></p> <p>Three line segments \(AD, BE, CF\) intersect at a single point in the diagram at right. </p> <p>Find the sum of the angles \(a+b+c+d+e+f\) in degrees.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>360</strong> </div> </div> </div> </div> </p> </div> </div> <div id="angle-arithmetic" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Angle Chasing</h2> </header> </div> <div class="section collapsed" id="section-angle-arithmetic"><div class="section-container"> <p> <div class="wistia-video center clearfix" data-controller="app/videos:wistiaController" data-width="500" data-video="iodlwrozb1"> <template> <div class="embed-wrapper" style="width:500px;max-width:100%;"> <script src="https://fast.wistia.com/embed/medias/iodlwrozb1.jsonp" async></script> <script src="https://fast.wistia.com/assets/external/E-v1.js" async></script> <div class="wistia_embed wistia_async_iodlwrozb1 videoFoam=true" style="height:100%;width:100%"> </div> </div> </template> </div> </p> <p>It is possible to find a missing angle measure if the values of other angles in the question are known. Start by drawing a diagram and labeling each known angle.</p> <blockquote> <ul> <li>Angles at a point sum to \( 360 ^ \circ \). </li> <li>Angles on a line sum to \( 180 ^ \circ \). </li> <li>Angles in a triangle sum to \( 180 ^ \circ \). </li> <li>Vertical angles are equal. </li> </ul> </blockquote> <blockquote class="example"><p> \(A, B\) and \(C\) are 3 consecutive points on a line. If \( \angle ABX = 35 ^ \circ \), what is the measure of \( \angle CBX?\) <br /> <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"> Angles on a line sum to \( 180 ^ \circ \), so we have \( \angle ABX + \angle CBX = 180 ^ \circ \). Hence, this gives us \( \angle CBX = 180 ^ \circ - \angle ABX = 145 ^ \circ \). \(_\square\) </blockquote> </p></blockquote> <blockquote class="example"><p> Lines \(AB\) and \(CD\) intersect at \( O \). If \( \angle AOD = 70 ^ \circ \), what is the measure of angle \( BOD?\) <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"> Since \(AOD \) and \(BOD\) are angles on the line \(\text{A-O-B}\), they sum up to \( 180 ^ \circ \). Thus </p> <p>\[ \angle BOD = 180^ \circ - 70 ^ \circ = 110^ \circ.\ _\square \] </blockquote> </p></blockquote> <blockquote class="example"><p> In the following image, if \( \angle ADB = 20 ^ \circ \) and \( \angle ADC = 50 ^ \circ \), what is the measure of angle \( BDC?\) <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/adb1d5aed6.f1f836cb4e.kiM4Jr.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/adb1d5aed6.f1f836cb4e.kiM4Jr.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/adb1d5aed6.f1f836cb4e.kiM4Jr.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/adb1d5aed6.f1f836cb4e.kiM4Jr.png?width=3600 3x" alt="image" /> <span class="caption">image</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"> We have \( \angle BDC = \angle ADC - \angle ADB = 50 ^ \circ - 20 ^ \circ = 30 ^ \circ \). \(_\square\) </blockquote> </p></blockquote> <p><div class="problem-modal-container anchor" id="problem-find-the-value-of-x-in-degrees"> <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/dvR40tyyrk-triangle.jpg?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/dvR40tyyrk-triangle.jpg?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/dvR40tyyrk-triangle.jpg?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/dvR40tyyrk-triangle.jpg?width=3600 3x" alt="" /> </span> Find the value of \(x.\)</p> <p><strong>Note:</strong> \(ACD\) is a straight line.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>130</strong> </div> </div> </div> </div> <div class="problem-modal-container anchor" id="problem-question-1-4"> <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 left"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/e4f62701c8.455c1d42ba.C1sIlh.png?width=250" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/e4f62701c8.455c1d42ba.C1sIlh.png?width=250 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/e4f62701c8.455c1d42ba.C1sIlh.png?width=500 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/e4f62701c8.455c1d42ba.C1sIlh.png?width=750 3x" alt="" style="width:250px;max-width:100%;" /> </span> In the figure, what is the measure of \(\angle BKM\) (in degrees)?</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>148</strong> </div> </div> </div> </div> </p> <p>Remembering that the sum of all the interior angles of a triangle is \(180^\circ\) is useful in solving the following examples:</p> <blockquote class="example"><p> In triangle \(ABC\), if \( \angle ABC = 40 ^ \circ \) and \( \angle BCA = 60 ^ \circ \), what is the measure of angle \( CAB?\)</p> <p><span class="image-caption center"> <img src="http://f.cl.ly/items/3q1Q130J3y1p341i2p2e/angle_arithmetic_example.png" srcset="http://f.cl.ly/items/3q1Q130J3y1p341i2p2e/angle_arithmetic_example.png 1x" alt="" /> </span></p> <p><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"> Since angles in a triangle sum to \( 180 ^ \circ \), </p> <p>\[ \angle CAB = 180 ^ \circ - \angle ABC - \angle BCA = 180^ \circ - 40 ^ \circ - 60 ^ \circ = 80 ^ \circ . \ _\square \] </blockquote> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-angles-sum"> <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> 360 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 480 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 540 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 600 </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/uploads/ZKic2yn9Zo-9f6fe9bd911f7094f0bd7fa183667af00b4c1a2879acdefc9131cddfcbd55db5.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/ZKic2yn9Zo-9f6fe9bd911f7094f0bd7fa183667af00b4c1a2879acdefc9131cddfcbd55db5.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/ZKic2yn9Zo-9f6fe9bd911f7094f0bd7fa183667af00b4c1a2879acdefc9131cddfcbd55db5.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/ZKic2yn9Zo-9f6fe9bd911f7094f0bd7fa183667af00b4c1a2879acdefc9131cddfcbd55db5.png?width=3600 3x" alt="" /> </span></p> <p>Find the sum (in degrees) of the four colored angles in the diagram.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>540</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-q-7"> <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/6AiuQ9ZRGt-26821.svg?width=350" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/6AiuQ9ZRGt-26821.svg?width=350 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/6AiuQ9ZRGt-26821.svg?width=700 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/6AiuQ9ZRGt-26821.svg?width=1050 3x" alt="" style="width:350px;max-width:100%;" /> </span></p> <p>In the figure, four line segments \(AB, CD, EF, GH\) intersect at a single point. </p> <p>Find the value (in degrees) of the angle sum \(x + y + w + z.\)</p> <p>\(\) <br /> <strong>Note</strong>: \(\angle EFH = 55 ^ \circ\), \(\angle ACD = 37 ^ \circ\), and \(\angle GHF = 85 ^ \circ\).</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>174</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"> <ul> <li><p><a href="/wiki/amazing-angles-and-shapes-basic/" class="wiki_link" title="Amazing Angles and Shapes - Basic"target="_blank">Amazing Angles and Shapes - Basic</a> </p></li> <li><p><a href="/wiki/amazing-angles-and-shapes/" class="wiki_link" title="Amazing Angles and Shapes - Intermediate"target="_blank">Amazing Angles and Shapes - Intermediate</a> </p></li> <li><p><a href="/wiki/parallel-lines/" class="wiki_link" title="Parallel Lines"target="_blank">Parallel Lines</a> </p></li> <li><p><a href="/wiki/perpendicular-lines/" class="wiki_link" title="Perpendicular Lines"target="_blank">Perpendicular Lines</a></p></li> </ul> </div> </div> </div> </div> </div> <div class="wiki-self-citation" data-controller="app/wiki:getCitationTime"> <strong>Cite as:</strong> Angles. <em>Brilliant.org</em>. 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