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<!DOCTYPE html>
<html data-require="math graphie graphie-helpers">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Angles 2</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="RAND_SWITCH2">rand(2)</var>
<var id="RAND_SWITCH3">rand(3)</var>
<var id="RAND_COMBO">RAND_SWITCH2 + 10 * RAND_SWITCH3</var>
</div>
<div class="problems">
<!-- Opposite angle of alternate interior angle and 180 degrees in a triangle. -->
<div id="opp-alternate-interior">
<div class="vars">
<var id="Tri_Y">rand(15) + 40</var>
<var id="Tri_Z">rand(10) + 100</var>
<var id="Tri_X">180 - Tri_Y - Tri_Z</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul>
<li><code>\green{\angle{ABC}} = <var>Tri_Y</var>^\circ</code></li>
<li><code>\purple{\angle{BAC}} = <var>Tri_Z</var>^\circ</code></li>
<li><code>\overline{BC} \parallel \overline{DE}</code> (line <code>BC</code> is parallel to line <code>DE</code>)</li>
<li><code>C</code> and <code>F</code> lie on a straight line.</li>
</ul>
<p data-if="RAND_SWITCH2 === 0">What is <code>\blue{\angle{DAF}}{?}</code></p>
<p data-else="">What is <code>\blue{\angle{CAE}}{?}</code></p>
<div class="render-answer-area-here"></div>
</div>
<div class="problem">
<div class="graphie" id="tri-and-parallel">
init({
range: [[-5, 5], [-3, 5]],
scale: [40, 40]
});
// label angle ABC
arc([-4, 3], 0.75, 312, 360, { stroke: GREEN });
label([-3.2, 3], "\\green{" + Tri_Y + "^\\circ}", "below right");
// label angle BAC
arc([-1.3, 0], 0.75, 38, 125, { stroke: PURPLE });
label([-1.3, .7], "\\purple{" + Tri_Z + "^\\circ}", "above");
// Label X according to problem variation
if (RAND_SWITCH2 == 0 ) {
arc([-1, 0], 1, 180, 208, { stroke: BLUE });
LABEL = label([-3.3, 0], "\\blue{\\angle{DAF}} = {?}", "below");
} else {
arc([-1.3, 0], 0.75, 0, 35, { stroke: BLUE });
LABEL = label([0.75, 0], "\\blue{\\angle{CAE}} = {?}", "above");
}
// Draw a horizontal line and a crossing line
// to form 2 opposing angles.
path([[-5, 0], [5, 0]]);
path([[-5, -3], [5, 5]]);
path([[-5, 3], [5, 3]]);
path([[-1.2, 0], [-4, 3]]);
ParallelLineMarkers(4.5, 0);
ParallelLineMarkers(4.5, 3);
label([-1.3, 0], "A", "below");
label([-4, 3], "B", "above");
label([2.5, 3], "C", "above");
label([-4, 0], "D", "above");
label([3, 0], "E", "above");
label([-3.75, -2], "F", "above");
var dotStyle = { r: 0.1, fill: BLUE, stroke: "none" };
circle([-1.25, 0], dotStyle);
circle([-4, 3], dotStyle);
circle([2.5, 3], dotStyle);
circle([-4,0], dotStyle);
circle([3,0], dotStyle);
circle([-3.75,-2], dotStyle);
</div>
<p style="color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>Tri_X</var></span><code>\Large{{}^\circ}</code>
</div>
<div class="hints">
<p>The measure of the angles in a triangle sum to <code>180^\circ</code>.</p>
<div>
<p><code>\red{\angle{BCA}} = 180^\circ - \green{\angle{ABC}} - \purple{\angle{BAC}}</code></p>
<div class="graphie" data-update="tri-and-parallel">
arc([2.5, 3], 0.75, 180, 220, { stroke: RED });
graph.TriX = label([1.8, 3], "\\red{\\angle{BCA}}", "below left");
</div>
</div>
<div>
<p><code>
\red{\angle{BCA}} = 180^\circ - \green{<var>Tri_Y</var>^\circ} - \purple{<var>Tri_Z</var>^\circ} =
\red{<var>Tri_X</var>^\circ}
</code></p>
<div class="graphie" data-update="tri-and-parallel">
graph.TriX.remove();
label([1.8, 3], "\\red{" + Tri_X + "^\\circ}", "below left");
</div>
</div>
<div data-if="RAND_SWITCH2 === 0" data-unwrap="">
<p><code>\blue{\angle{DAF}}</code> is a corresponding angle to <code>\red{\angle{BCA}}</code>.</p>
<div>
<p><code>\blue{\angle{DAF}}</code> = <code>\red{\angle{BCA}} = <var>Tri_X</var>^\circ</code></p>
<div class="graphie" data-update="tri-and-parallel">
LABEL.remove();
label([-3.3, 0], "\\blue{\\angle{DAF}}=" + Tri_X + "^\\circ", "below");
</div>
</div>
</div>
<div data-else="" data-unwrap="">
<p><code>\blue{\angle{CAE}}</code> is an alternate interior angle to <code>\red{\angle{BCA}}</code>.</p>
<div>
<p><code>\blue{\angle{CAE}}</code> = <code>\red{\angle{BCA}} = <var>Tri_X</var>^\circ</code></p>
<div class="graphie" data-update="tri-and-parallel">
LABEL.remove();
label([0.9, 0], "\\blue{\\angle{CAE}} = " + Tri_X + "^\\circ", "above");
</div>
</div>
</div>
</div>
</div>
<div id="opp-parallel">
<div class="vars">
<var id="Y">rand(20) + 100</var>
<var id="X">180 - Y</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul>
<li><code>\overline{AB} \parallel \overline{CD}</code> (line <code>AB</code> is parallel to line <code>CD</code>)</li>
<li data-if="RAND_SWITCH3 === 0"><code>\green{\angle{EGB}} = <var>X</var>^\circ</code></li>
<li data-else-if="RAND_SWITCH3 === 1"><code>\purple{\angle{AGH}} = <var>X</var>^\circ</code></li>
<li data-else=""><code>\green{\angle{BGH}} = <var>180 - X</var>^\circ</code></li>
<li><code>E</code> and <code>F</code> lie on a straight line.</li>
</ul>
<p data-if="RAND_SWITCH2 === 0">What is <code>\blue{\angle{GHD}}{?}</code></p>
<p data-else="">What is <code>\blue{\angle{CHF}}{?}</code></p>
<div class="render-answer-area-here"></div>
</div>
<div class="problem">
<div class="graphie" id="opposite-parallel">
init({
range: [[-6.5, 5.8], [-5, 4.2]],
scale: [40, 40]
});
// label angle X
if(RAND_SWITCH2 === 0) {
arc([-2.9, -2], 1, 0, 50, { stroke: BLUE });
LABEL = label([-2, -2], "\\blue{\\angle{GHD}}= {?}", "above right");
} else {
arc([-2.9, -2], 1, 180, 220, { stroke: BLUE });
LABEL = label([-4, -2.5], "\\blue{\\angle{CHF}}= {?}", "below left");
}
// label angle Y
if(RAND_SWITCH3 === 0) {
arc([1.2, 2], 1, 0, 54, { stroke: GREEN });
label([2.2, 2], "\\green{" + X + "^\\circ}", "above right");
} else if (RAND_SWITCH3 === 1) {
arc([1.4, 2], 1, 180, 210, { stroke: PURPLE });
label([0, 2], "\\purple{" + X + "^\\circ}", "below left");
} else {
arc([1.2, 2], 1, 216, 0, { stroke: GREEN });
label([1.5, 1.2], "\\green{" + (180 - X) + "^\\circ}", "below right");
}
path([[-5, 2], [5, 2]]);
path([[-5, -2], [5, -2]]);
path([[-5, -4], [3, 4]]);
ParallelLineMarkers(4.2, 2);
ParallelLineMarkers(4.2, -2);
label([-5, 2], "A", "below");
label([5, 2], "B", "below");
label([-5, -2], "C", "below");
label([5, -2], "D", "below");
label([3, 4], "E", "below");
label([-5, -4], "F", "right");
label([1, 2], "G", "below right");
label([-3, -2], "H", "below right");
var dotStyle = { r: 0.1, fill: BLUE, stroke: "none" };
circle([-5, 2], dotStyle);
circle([5, 2], dotStyle);
circle([-5, -2], dotStyle);
circle([5, -2], dotStyle);
circle([3, 4], dotStyle);
circle([-5, -4], dotStyle);
circle([1,2], dotStyle);
circle([-3, -2], dotStyle);
</div>
<p style="color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>X</var></span><code>\Large{{}^\circ}</code>
</div>
<div class="hints">
<div data-if="RAND_SWITCH3 === 2" data-unwrap="">
<div>
<p><code>\purple{\angle{AGH}}</code> forms a linear pair with <code>\green{\angle{BGH}}</code>.</p>
<div class="graphie" data-update="opposite-parallel">
arc([1,2], 0.88, 180, 225, { stroke: PURPLE });
graph.AGH = label([0, 2], "\\purple{\\angle{AGH}}", "below left");
</div>
</div>
<div>
<p>Therefore <code>\purple{\angle{AGH}} = 180^\circ - \green{<var>Y</var>^\circ} = <var>X</var>^\circ</code>.</p>
<div class="graphie" data-update="opposite-parallel">
graph.AGH.remove();
label([0,2], "\\purple{" + X + "^\\circ}", "below left");
</div>
</div>
</div>
<div data-else-if="RAND_COMBO === 0" data-unwrap="">
<p><code>\blue{\angle{GHD}}</code> and <code>\green{\angle{EGB}}</code> are complementary angles.</p>
<p>Therefore <code>\blue{\angle{GHD}} = \green{\angle{EGB}}</code>.</p>
</div>
<div data-else-if="RAND_COMBO === 1" data-unwrap="">
<div>
<p><code>\purple{\angle{AGH}}</code> and <code>\green{\angle{EGB}}</code> are opposite angles.</p>
<div class="graphie" data-update="opposite-parallel">
arc([1,2], 0.88, 180, 225, { stroke: PURPLE });
graph.AGH = label([0, 2], "\\purple{\\angle{AGH}}", "below left");
</div>
</div>
<div>
<p>Therefore <code>\purple{\angle{AGH}} = \green{\angle{EGB}} = <var>X</var>^\circ</code>.</p>
<div class="graphie" data-update="opposite-parallel">
graph.AGH.remove();
label([0,2], "\\purple{" + X + "^\\circ}", "below left");
</div>
</div>
</div>
<div data-else-if="RAND_COMBO === 10">
<p><code>\blue{\angle{GHD}}</code> and <code>\purple{\angle{AGH}}</code> are alternate interior angles.</p>
<p>Therefore <code>\blue{\angle{GHD}} = \purple{\angle{AGH}}</code>.</p>
</div>
<div data-if="RAND_COMBO === 20" data-unwrap="">
<p><code>\blue{\angle{GHD}}</code> and <code>\purple{\angle{AGH}}</code> are alternate interior angles.</p>
<p>Therefore <code>\blue{\angle{GHD}} = \green{\purple{AGH}}</code>.</p>
</div>
<div data-if="RAND_SWITCH2 === 0">
<p><code>\angle{GHD} = <var>X</var>^\circ</code></p>
<div class="graphie" data-update="opposite-parallel">
LABEL.remove();
label([-2, -2], "\\blue{\\angle{GHD}}=" + X + "^\\circ", "above right");
</div>
</div>
<div data-else="" data-unwrap="">
<p><code>\blue{\angle{CHF}}</code> and <code>\purple{\angle{AGH}}</code> are corresponding angles.</p>
<p>Therefore <code>\blue{\angle{CHF}} = \purple{\angle{AGH}}</code>.</p>
<div>
<p><code>\angle{CHF} = <var>X</var>^\circ</code></p>
<div class="graphie" data-update="opposite-parallel">
LABEL.remove();
label([-4, -2.5], "\\blue{\\angle{CHF}}=" + X + "^\\circ", "below left");
</div>
</div>
</div>
</div>
</div>
<!-- Star Problem 1 :
Adjacent Angles + 180 deg in triangle + opposite angles (variations 0-2)
(and adjacent in variations 1,2) -->
<div id="star1">
<div class="vars">
<var id="ANGLE_2">rand(10) + 30</var>
<var id="ANGLE_3">rand(10) + 100</var>
<var id="ANGLE_1">180 - ANGLE_2 - ANGLE_3</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul>
<li><code>\green{\angle{BDC}} = <var>ANGLE_2</var>^\circ</code></li>
<li><code>\red{\angle{DBE}} = <var>ANGLE_1</var>^\circ</code></li>
</ul>
<p>
What is <code>\blue{\angle{<var>RAND_SWITCH3 === 0 ? "CHE" : ( RAND_SWITCH3 === 1 ? "GHC" : "DHE" )</var>}} {?}</code>
</p>
<div class="render-answer-area-here"></div>
</div>
<div class="problem">
<div class="graphie" id="star1">
init({
range: [[-10, 10], [-7, 10]],
scale: [25, 25]
});
// Label pts on the star.
label([-8, 5], "A", "left");
label([0, 9], "B", "above");
label([8, 5], "C", "right");
label([-6, -6], "D", "below");
label([6, -6], "E", "below");
label([-1.8, 5], "F", "above left");
label([1.8, 5], "G", "above right");
label([3.2, 1.3], "H", "below right");
label([0, -1.3], "I", "below");
label([-3.2, 1.3], "J", "below left");
// Label the given angles
label([-5.5, -5.2], "\\green{" + ANGLE_2 + "^\\circ}", "above right");
arc([-6, -6], 1.3, 40, 70, { stroke: GREEN });
label([0, 7.4], "\\red{" + ANGLE_1 + "^\\circ}", "below");
arc([0, 9], 1.3, 247, 290, { stroke: RED });
// Label X according to variation on the problem
if(RAND_SWITCH3 == 0) {
LABEL = label([4.7, 1.0], "\\blue{\\angle{CHE}}= {?}", "right");
arc([3.2, 1.3], 1.5, 287, 35, { stroke: BLUE });
} else if (RAND_SWITCH3 == 1) {
LABEL = label([4, 2.5], "\\blue{\\angle{GHC}}= {?}", "above");
arc([3.2, 1.3], 1, 35, 118, { stroke: BLUE });
} else {
LABEL = label([2.5, -0.5], "\\blue{\\angle{DHE}}= {?}", "below");
arc([3.2, 1.3], 1, 219, 286, { stroke: BLUE });
}
// Draw A Star
path([[-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5]]);
</div>
<p style="color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div>
<div class="solution" data-type="multiple">
<span class="sol">
<var data-if="RAND_SWITCH3 === 0">ANGLE_3</var>
<var data-else="">180 - ANGLE_3</var>
</span>
<code>\Large{{}^\circ}</code>
</div>
<div class="hints">
<p>The measure of the angles in a triangle sum to <code>180^\circ</code>.</p>
<div>
<p><code>\purple{\angle{BHD}} = 180^\circ - \green{\angle{BDC}} - \red{\angle{DBE}}</code></p>
<div class="graphie" data-update="star1">
// label angle BHD
arc([3.2, 1.3], 1, 118, 220, { stroke: PURPLE });
graph.angle3 = label([2.4, 2], "\\purple{\\angle{BHD}}", "below left");
</div>
</div>
<div>
<p><code>
\purple{\angle{BHD}} = 180^\circ - \green{<var>ANGLE_2</var>^\circ} -
\red{<var>ANGLE_1</var>^\circ} = \purple{<var>ANGLE_3</var>^\circ}
</code></p>
<div class="graphie" data-update="star1">
graph.angle3.remove();
label([2.4, 2], "\\purple{" + ANGLE_3 + "^\\circ}", "below left");
</div>
</div>
<div data-if="RAND_SWITCH3 === 0" data-unwrap="">
<p><code>\blue{\angle{CHE}}</code> and <code>\purple{\angle{BHD}}</code> are opposite angles are equal.</p>
<div>
<p>Therefore <code>\blue{\angle{CHE}} = \purple{\angle{BHD}} = <var>ANGLE_3</var>^\circ</code>.</p>
<div class="graphie" data-update="star1">
LABEL.remove();
label([4.8, 1.0], "\\blue{\\angle{CHE}}=" + ANGLE_3 + "^\\circ", "right");
</div>
</div>
</div>
<div data-else="" data-unwrap="">
<p>
<span data-if="RAND_SWITCH3 === 1"><code>\blue{\angle{CHG}}</code></span>
<span data-else=""><code>\blue{\angle{DHE}}</code></span>
and <code>\purple{\angle{BHD}}</code> form a linear pair.
</p>
<p>Therefore <code>
<span data-if="RAND_SWITCH3 === 1">\blue{\angle{CHG}}</span>
<span data-else="">\blue{\angle{DHE}}</span>
= 180 ^\circ - \purple{<var>ANGLE_3</var>^\circ}</code>.
</p>
<p>
<span data-if="RAND_SWITCH3 === 1">
<code>\blue{\angle{GHC}}</code>
<span class="graphie" data-update="star1">
LABEL.remove();
label([4, 2.5], "\\blue{\\angle{GHC}}=" + (180 - ANGLE_3) + "^\\circ", "above");
</span>
</span>
<span data-else="">
<code>\blue{\angle{DHE}}</code>
<span class="graphie" data-update="star1">
LABEL.remove();
label([2.5, -0.5], "\\blue{\\angle{DHE}}=" + (180 - ANGLE_3) + "^\\circ", "below");
</span>
</span>
<code> = <var>180 - ANGLE_3</var>^\circ</code>
</p>
</div>
</div>
</div>
<!-- Star Problem 2 : 180 deg in triangle + opposite angles -->
<div id="star2">
<div class="vars">
<var id="ANGLE_1">rand(10) + 30</var>
<var id="ANGLE_2">rand(10) + 70</var>
<var id="ANGLE_3">180 - ANGLE_1 - ANGLE_2</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul data-if="RAND_SWITCH2 === 0">
<li><code>\green{\angle{BDC}} = <var>ANGLE_1</var>^\circ</code></li>
<li><code>\red{\angle{AIC}} = <var>180 - ANGLE_2</var>^\circ</code></li>
</ul><ul data-else="">
<li><code>\green{\angle{GCH}} = <var>ANGLE_1</var>^\circ</code></li>
<li><code>\red{\angle{FGH}} = <var>180 - ANGLE_2</var>^\circ</code></li>
</ul>
<p data-if="RAND_SWITCH2 === 0">What is <code>\blue{\angle{AJF}} {?}</code></p><p data-else="">What is <code>\blue{\angle{IHE}} {?}</code></p>
<div class="render-answer-area-here"></div>
</div>
<div class="problem">
<div class="graphie" id="star2">
init({
range: [[-10, 10], [-7, 10]],
scale: [25, 25]
});
// Label pts on the star.
label([-8, 5], "A", "left");
label([0, 9], "B", "above");
label([8, 5], "C", "right");
label([-6, -6], "D", "below");
label([6, -6], "E", "below");
label([-1.8, 5], "F", "above left");
label([1.8, 5], "G", "above right");
label([3.2, 1.4], "H", "below right");
label([0, -1.3], "I", "below");
label([-3.2, 1.4], "J", "below left");
// Label Angles and X according to variation
if( RAND_SWITCH2 == 0) {
// Label the given angles
label([-5.5, -5.2], "\\green{" + ANGLE_1 + "^\\circ}", "above right");
arc([-6, -6], 1.3, 39, 69, { stroke: GREEN });
label([0, -0.4], "\\red{" + (180 - ANGLE_2) + "^\\circ}", "above");
arc([0, -1.2], 0.75, 32, 145, { stroke: RED });
// Label X
LABEL = label([-3.7, 2.5], "\\blue{\\angle{AJF}}= {?}", "above");
arc([-3.2, 1.3], 1, 65, 142, { stroke: BLUE });
} else {
// Label the given angles
label([6.5, 5], "\\green{" + ANGLE_1 + "^\\circ}", "below left");
arc([8, 5], 1.3, 180, 220, { stroke: GREEN });
label([1.3, 4.5], "\\red{" + (180 - ANGLE_2) + "^\\circ}", "below left");
arc([1.7, 5], 1, 180, 289, { stroke: RED });
// Label X
LABEL = label([4.0, -0.3], "\\blue{\\angle{IHE}}= {?}", "below left");
arc([3.1, 1.2], 1, 220, 290, { stroke: BLUE });
}
// Draw A Star
path([[-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5]]);
</div>
<p style="color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>ANGLE_3</var></span>
<code>\Large{{}^\circ}</code>
</div>
<div class="hints">
<div data-if="RAND_SWITCH2 === 0" data-unwrap="">
<p><code>\purple{\angle{DIJ}}</code> forms a linear pair with <code>\red{\angle{AIC}}</code>.</p>
<div>
<p>Therefore, <code>\purple{\angle{DIJ}} = 180^\circ - \red{<var>180 - ANGLE_2</var>^\circ}</code>.</p>
<div class="graphie" data-update="star2">
// label angle JID
arc([0, -1.2], 0.75, 143, 220, { stroke: PURPLE });
label([-.75, -1.2], "\\purple{" + ANGLE_2 + "^\\circ}", "left");
</div>
</div>
<p>The measure of the angles in a triangle sum to <code>180^\circ</code>.</p>
<div>
<p>Therefore, <code>\pink{\angle{DJI}} = 180^\circ - \green{\angle{BDC}} - \purple{\angle{DIJ}}</code>.</p>
<div class="graphie" data-update="star2">
// label angle JID
arc([-3.2, 1.3], 1, 256 , 318, { stroke: PINK });
graph.DJI = label([-3.2, 0.50], "\\pink{\\angle{DJI}}", "below right");
</div>
</div>
<div>
<p><code>
\pink{\angle{DJI}} = 180^\circ - \green{<var>ANGLE_1</var>^\circ} - \purple{<var>ANGLE_2</var>^\circ}
= \pink{<var>ANGLE_3</var>^\circ}
</code></p>
<div class="graphie" data-update="star2">
graph.DJI.remove();
label([-3.2, 0.50], "\\pink{" + ANGLE_3 + "^\\circ}", "below right");
</div>
</div>
<p><code>\blue{\angle{AJF}}</code> and <code>\pink{\angle{DJI}}</code> are opposite angles.</p>
<div>
<p>Therefore, <code>\angle{AJF} = \angle{DJI} = <var>ANGLE_3</var>^\circ</code>.</p>
<div class="graphie" data-update="star2">
LABEL.remove();
label([-3.7, 2.5], "\\blue{\\angle{AJF}}=" + ANGLE_3 + "^\\circ", "above");
</div>
</div>
</div>
<div data-else="" data-unwrap="">
<p><code>\purple{\angle{HGC}}</code> forms a linear pair with <code>\red{\angle{FGH}}</code>.</p>
<div>
<p>Therefore, <code>\purple{\angle{HGC}} = 180^\circ - \red{<var>180 - ANGLE_2</var>^\circ}</code>.</p>
<div class="graphie" data-update="star2">
// label angle HGC
arc([1.7, 5], 1, 285, 0, { stroke: PURPLE });
label([2.4, 4.6], "\\purple{" + ANGLE_2 + "^\\circ}", "below right");
</div>
</div>
<p>The measure of the angles in a triangle sum to <code>180^\circ</code>.</p>
<div>
<p>Therefore, <code>\pink{\angle{CHG}} = 180^\circ - \green{\angle{ACD}} - \purple{\angle{HGC}}</code>.</p>
<div class="graphie" data-update="star2">
// label angle CHG
arc([3.2, 1.3], 0.75, 35, 118, {stroke: PINK });
graph.CHG = label([3.5, 1.8], "\\pink{\\angle{CHG}}", "above");
</div>
</div>
<div>
<p><code>
\pink{\angle{CHG}} = 180^\circ - \green{<var>ANGLE_1</var>^\circ} - \purple{<var>ANGLE_2</var>^\circ}
= \pink{<var>ANGLE_3</var>^\circ}
</code></p>
<div class="graphie" data-update="star2">
graph.CHG.remove();
label([3.5, 1.8], "\\pink{" + ANGLE_3 + "^\\circ}", "above");
</div>
</div>
<p><code>\blue{\angle{IHE}}</code> and <code>\pink{\angle{CHG}}</code> are opposite angles.</p>
<div>
<p>Therefore, <code>\angle{IHE} = \pink{\angle{CHG}} = <var>ANGLE_3</var>^\circ</code>.</p>
<div class="graphie" data-update="star2">
LABEL.remove();
label([4.0, -0.3], "\\blue{\\angle{IHE}}=" + ANGLE_3 + "^\\circ", "below left");
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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