{"id":219887,"date":"2025-05-27T06:38:20","date_gmt":"2025-05-27T06:38:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219887"},"modified":"2025-05-27T06:38:22","modified_gmt":"2025-05-27T06:38:22","slug":"find-the-following-are-measures","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/27\/find-the-following-are-measures\/","title":{"rendered":"Find the following are measures."},"content":{"rendered":"\n<p>Name: Date: Unit 10: Circles Homework 2: Central Angles &amp; Arc Measures Per: ** This is a 2-page document!<\/p>\n\n\n\n<p>Directions: Find the following are measures.<\/p>\n\n\n\n<p>127<br>164<br>mJL = mBC mABC BC<br>D TO MOR C U 104<br>44 DE mFE- DEF- MCFD = mDFE G E<br>MSOR mROT &#8211; 5.<br>108<br>MYU = 67<br>Y XW 55 M LON MOM- MKNZ mNL &#8211; Z o MXVW VW = mYWU = N<br>Directions: Find the value of x.<\/p>\n\n\n\n<p>(16x-13 31 P S T 83&#8243;<br>(9x + 23) @ R X<br>Given: CH bisects ZDHG. MGF &#8211; (2x + 6). and mDC &#8211; (x-1 (212-9 D c X G Gre Whion gebuc).<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Part 1: Find the following arc or angle measures<\/strong><\/h3>\n\n\n\n<p>The text seems a bit jumbled, so I\u2019ll correct and organize it into readable problems based on typical circle geometry questions. Then I\u2019ll solve them and explain the concepts.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Problem Set 1 \u2013 Arc and Central Angle Measures:<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li>m\u2220JL=mBC\u2322m\\angle J L = m\\overset{\\frown}{BC}, m\u2220ABC=BC\u2322m\\angle ABC = \\overset{\\frown}{BC}<\/li>\n\n\n\n<li>m\u2220D=104\u2218m\\angle D = 104^\\circ<\/li>\n\n\n\n<li>m\u2220DE=44\u2218m\\angle DE = 44^\\circ, find mFE\u2322m\\overset{\\frown}{FE}, m\u2220DEFm\\angle DEF, mCFD\u2322m\\overset{\\frown}{CFD}, and m\u2220DFEm\\angle DFE<\/li>\n\n\n\n<li>m\\angle SOR = \\_\\_\\_), \\( m\\angle ROT = 5^\\circ, m\u2220YU=67\u2218m\\angle YU = 67^\\circ, m\u2220XW=55\u2218m\\angle XW = 55^\\circ<\/li>\n\n\n\n<li>( m\\angle LON = ___), ( m\\angle MOM = ___), ( m\\angle KNZ = ___), ( m\\angle NLZ = ___), ( m\\angle XVW = ___), ( m\\angle VWU = ___), ( m\\angle YWU = ___)<\/li>\n<\/ol>\n\n\n\n<p>You might want to upload or describe the associated diagram so I can solve each item with precision. Circle geometry questions like these depend heavily on the figure. That said, I\u2019ll explain the typical way to approach them.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Concepts for Central Angles &amp; Arc Measures<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Central Angle<\/strong>: An angle whose vertex is the center of a circle and whose sides are radii.<\/li>\n\n\n\n<li><strong>Arc Measure<\/strong>: Equal to the measure of the central angle that intercepts the arc.<\/li>\n\n\n\n<li>If a central angle is \u03b8\u2218\\theta^\\circ, the arc it intercepts is also \u03b8\u2218\\theta^\\circ.<\/li>\n\n\n\n<li>The <strong>sum of all central angles in a circle is 360\u00b0<\/strong>.<\/li>\n\n\n\n<li>If you&#8217;re given expressions like (16x\u221213)\u2218(16x &#8211; 13)^\\circ and another angle 83\u221883^\\circ, and they\u2019re part of a semicircle or a full circle, set up the equation accordingly.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Part 2: Solve for x<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Problem:<\/h4>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>16x\u221213=\u2220PST16x &#8211; 13 = \\angle PST<br>83\u221883^\\circ is the measure of the remaining angle<br>Probably, both are parts of a circle totaling 180\u2218180^\\circ or 360\u2218360^\\circ<\/p>\n<\/blockquote>\n\n\n\n<p>Let\u2019s assume these angles are central angles that sum to 180\u00b0: (16x\u221213)+83=18016x+70=18016x=110x=11016=6.875(16x &#8211; 13) + 83 = 180 \\\\ 16x + 70 = 180 \\\\ 16x = 110 \\\\ x = \\frac{110}{16} = 6.875<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Part 3: Geometry Proof\/Analysis<\/strong><\/h3>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>Given: CH bisects \u2220DHG, m\u2220MGF = (2x + 6), and m\u2220DC = (x \u2013 1)<\/p>\n<\/blockquote>\n\n\n\n<p>Assume you&#8217;re asked to find x or prove congruent angles based on a bisected angle:<\/p>\n\n\n\n<p>If <strong>CH bisects \u2220DHG<\/strong>, then: \u2220DHC=\u2220CHG\\angle DHC = \\angle CHG<\/p>\n\n\n\n<p>If angle measures are expressed as algebraic expressions, set them equal to each other and solve for x.<\/p>\n\n\n\n<p>Let\u2019s assume \u2220DHC=2x+6\\angle DHC = 2x + 6, and \u2220CHG=x\u22121\\angle CHG = x &#8211; 1 2x+6=x\u221212x\u2212x=\u22121\u22126x=\u221272x + 6 = x &#8211; 1 \\\\ 2x &#8211; x = -1 &#8211; 6 \\\\ x = -7<\/p>\n\n\n\n<p>Plug in to verify: \u2220DHC=2(\u22127)+6=\u221214+6=\u22128\u2218(which&nbsp;is&nbsp;invalid)\\angle DHC = 2(-7) + 6 = -14 + 6 = -8^\\circ \\quad \\text{(which is invalid)}<\/p>\n\n\n\n<p>So it\u2019s likely something\u2019s off here\u2014either the problem is miswritten or the expressions belong to different angles.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong> Explanation (Sample)<\/strong><\/h3>\n\n\n\n<p>Understanding central angles and arc measures is essential in circle geometry. A <strong>central angle<\/strong> has its vertex at the center of a circle and its sides are radii, which directly determines the <strong>arc<\/strong> it intercepts. The arc measure is <strong>equal to the central angle<\/strong>. For example, if a central angle measures 60\u00b0, the arc it intercepts also measures 60\u00b0. This is a foundational concept used in many geometry problems involving circles.<\/p>\n\n\n\n<p>In problems involving algebraic expressions like 16x\u22121316x &#8211; 13 and 83\u221883^\\circ, we often set up equations that total 180\u00b0 or 360\u00b0, depending on whether they form a semicircle or full circle. By solving these equations, we find the value of xx, which then allows us to compute unknown angle or arc measures.<\/p>\n\n\n\n<p>In more advanced problems, like those involving <strong>angle bisectors<\/strong>, understanding that the bisector divides the angle into two equal parts is crucial. If an angle is bisected, the two resulting angles are congruent. For instance, if a bisector divides angle \u2220DHG and one part is expressed as 2x+62x + 6 while the other is x\u22121x &#8211; 1, setting these equal and solving gives the correct value of xx.<\/p>\n\n\n\n<p>Circle geometry blends algebra and visual reasoning. With clear diagrams and proper setup, even complex-looking problems can be simplified into solvable equations. It\u2019s always essential to verify answers by substituting back and ensuring the measures make logical sense (i.e., angles should not be negative).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-34.jpeg\" alt=\"\" class=\"wp-image-219888\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Name: Date: Unit 10: Circles Homework 2: Central Angles &amp; Arc Measures Per: ** This is a 2-page document! Directions: Find the following are measures. 127164mJL = mBC mABC BCD TO MOR C U 10444 DE mFE- DEF- MCFD = mDFE G EMSOR mROT &#8211; 5.108MYU = 67Y XW 55 M LON MOM- MKNZ mNL [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-219887","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219887","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=219887"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219887\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}