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angles_2.html
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<!DOCTYPE html>
<html data-require="math graphie">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Angles 2</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<!-- Used for problems that choose variation of problems that have multiple variations -->
<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>\color{green}{\angle{ABC}} = <var>Tri_Y</var>°</code></li>
<li><code>\color{purple}{\angle{BAC}} = <var>Tri_Z</var>°</code></li>
<li><code>\overline{DE} \parallel \overline{BC}</code></li>
</ul>
<p>What is <code>\color{blue}{\angle{<var>RAND_SWITCH2 === 0 ? "DAF" : "CAE"</var>}} {?}</code></p>
</div>
<div class="problem">
<div class="graphie" id="tri-and-parallel">
init({
range: [[-5, 5], [-3, 5]],
scale: [40, 40]
});
style({
stroke: "#888",
strokeWidth: 2
});
// 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] ]);
style({
fill: "grey"
}, function() {
// Label A, B, C
label([-1.3, 0], "A", "below");
circle([-1.25, 0], 0.05);
label([-4, 3], "B", "above");
circle([-4, 3], 0.05);
label([2.5, 3], "C", "above");
circle([2.5, 3], 0.05);
label([-4, 0], "D", "above");
circle([-4,0], 0.05);
label([3, 0], "E", "above");
circle([3,0], 0.05);
label([-3.75, -2], "F", "above");
circle([-3.75,-2], 0.05);
});
// label angle ABC
arc([-4,3], .75, 312, 360, { stroke: "green" });
label([-3.2, 3], "\\color{green}{" + Tri_Y + "°}",
"below right", {color: "green"});
// label angle BAC
arc([-1.3,0], .75, 38, 125, { stroke: "purple" });
label([-1.3, .7], "\\color{purple}{" + Tri_Z + "°}",
"above", {color: "purple"});
// Label X according to problem variation
if(RAND_SWITCH2 == 0 ) {
//problem variation 1
arc([-1, 0], 1, 180, 210, { stroke: "blue"});
LABEL = label([-3.3, 0], "\\color{blue}{∠DAF} = {?}",
"below", { color: "blue"});
} else {
//problem variation 2
arc([-1, 0], 1, 0, 45, { stroke: "blue"});
LABEL = label([1, 0], "\\color{blue}{∠CAE} = {?}",
"above", { color: "blue"});
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div> <!-- end problem block -->
<div class="solution"><var>Tri_X</var></div>
<div class="hints">
<p>
Remember that the measure of the angles in a triangle sum to 180°.
Solve for <code>\color{orange}{\angle{BCA}}</code> by subtracting the
measures of angles <code>\color{purple}{\angle{BAC}}</code> and
<code>\color{green}{\angle{ABC}}</code> from 180°. We find that
<code>\color{orange}{\angle{BCA}} = <var>Tri_X</var>°</code>.
<span class="graphie" data-update="tri-and-parallel">
// label angle BAC
arc([2.5, 3], .75, 180, 220, { stroke: "orange" });
label([1.8, 3], "\\color{orange}{" + Tri_X + "°}",
"below left", {color: "orange"});
</span>
</p>
<p>
Solve for
<span data-if="RAND_SWITCH2 === 0">
<code>\color{blue}{\angle{DAF}}</code> by using the fact
that it is a corresponding angle to
<span class="graphie" data-update="tri-and-parallel">
LABEL.remove();
LABEL = label([-3.3, 0],
"\\color{blue}{\\angle{DAF}}=" + Tri_X + "°",
"below");
</span>
</span>
<span data-else>
<code>\color{blue}{\angle{CAE}}</code> by using the fact
that it is an alternate interior angle to
<span class="graphie" data-update="tri-and-parallel">
LABEL.remove();
LABEL = label([1, 0],
"\\color{blue}{∠CAE} = " + Tri_X + "°",
"above", { color: "blue"});
</span>
</span>
<code>\color{orange}{\angle{BCA}}.</code> That means those
angles are equal because they are both created by the same
set of parallel lines <code>\overline{BC}</code>
and <code>\overline{DE}</code>, and transversal line
<code>\overline{CF}</code>.
</p>
</div>
</div>
<!-- Corresponding/Opposite angles of parallel lines -->
<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 AB is parallel to line CD)</li>
<li data-if="RAND_SWITCH3 === 0">
<code>\color{purple}{\angle{EGB}} = <var>X</var>°.</code>
</li><li data-else-if="RAND_SWITCH3 === 1">
<code>\color{purple}{\angle{AGH}} = <var>X</var>°.</code>
</li><li data-else>
<code>\color{purple}{\angle{BGH}} = <var>180 - X</var>°</code>
</li>
</ul>
<p>What is <code>\color{blue}{\angle{<var>RAND_SWITCH2 === 0 ? "GHD" : "CHF"</var>}} {?}</code></p>
</div>
<div class="problem">
<div class="graphie" id="opposite-parallel">
init({
range: [[-7, 6], [-5, 5.2]],
scale: [40, 40]
});
style({
stroke: "#888",
strokeWidth: 2
});
style({
fill: "grey"
}, function() {
// Draw 2 parallel horizontal lines
path([ [-5, 2], [5, 2] ]);
label([-5, 2], "A", "below");
circle([-5,2], 0.05);
label([5, 2], "B", "below");
circle([5,2], 0.05);
path([ [-5, -2], [5, -2] ]);
label([-5, -2], "C", "below");
circle([-5,-2], 0.05);
label([5, -2], "D", "below");
circle([5, -2], 0.05);
// Draw a transversal line.
path([ [-5, -4], [4, 5] ]);
label([4, 5], "E", "below");
circle([4, 5], 0.05);
label([-5, -4], "F", "right");
circle([-5, -4], 0.05);
label([1, 2], "G", "below right");
circle([1,2], 0.05);
label([-3, -2], "H", "below right");
circle([-3, -2], 0.05);
});
// label angle X
if(RAND_SWITCH2 == 0) {
arc([-2.9, -2], 1, 0, 50, { stroke: "blue"});
LABEL = label([-2, -2], "\\color{blue}{\\angle{GHD}}= {?}", "above right");
} else {
arc([-2.9, -2], 1, 180, 220, { stroke: "blue"});
LABEL = label([-4, -2.5], "\\color{blue}{\\angle{CHF}}= {?}", "below left");
}
// label angle Y
if(RAND_SWITCH3 == 0) {
arc([1.2, 2], 1.5, 0, 50, { stroke: "purple" });
label([3.1, 2], "\\color{purple}{\\angle{EGB}}=" + X + "°",
"above right");
} else if (RAND_SWITCH3 == 1) {
arc([1.2, 2], 1.5, 180, 220, { stroke: "purple" });
label([-1, 2], "\\color{purple}{\\angle{AGH}}=" + X + "°",
"below left");
} else {
arc([1.2, 2], 1, 220, 0, { stroke: "purple" });
label([1.5, 1.2], "\\color{purple}{\\angle{BGH}}=" + (180 - X) + "°",
"below right");
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div> <!-- end problem block -->
<div class="solution"><var>X</var></div>
<div class="hints">
<p data-if="RAND_COMBO === 0">
<code>\color{blue}{\angle{GHD}} = \color{purple}{\angle{EGB}}</code>.
We know this because they are 2 complementary angles of a set of
parallel lines bisected by a single line.
</p>
<p data-if="RAND_COMBO === 1">
First solve for <code>\color{orange}{\angle{AGH}}</code>. We know that
<code>\color{orange}{\angle{AGH}} = \color{purple}{\angle{EGB}}</code>
because opposite angles are equal.
<span class="graphie" data-update="opposite-parallel">
arc([1,2], .88, 180, 225, {stroke:"orange"});
label([0,2], "\\color{orange}{" + X + "°}", "below left");
</span>
</p>
<p data-else-if="RAND_COMBO === 10">
<code>\color{blue}{\angle{GHD}} = \color{purple}{\angle{AGH}}</code>
We know this because they are 2 alternate interior angles of a set of
parallel lines bisected by a single line.
</p>
<p data-if="RAND_COMBO === 11">
<code>\color{blue}{\angle{CHF}} = \color{purple}{\angle{AGH}}</code>.
We know this because they are 2 corresponding angles of a set of
parallel lines bisected by a single line.
</p>
<p data-else-if="RAND_COMBO === 20 || RAND_COMBO === 21">
First solve for <code>\color{orange}{\angle{AGH}}</code>. We know that
<code>
\color{orange}{\angle{AGH}} = 180° - \color{purple}{\angle{BGH}}
</code>,
because angles along a line plane add up to 180°.
<span class="graphie" data-update="opposite-parallel">
arc([1,2], .88, 180, 225, {stroke:"orange"});
label([0,2], "\\color{orange}{" + X + "°}", "below left");
</span>
</p>
<p data-if="RAND_COMBO === 20">
<code>\color{blue}{\angle{GHD}} = \color{orange}{\angle{AGH}}</code>.
We know those 2 angles are equal because they are alternate interior angles
of 2 parallel lines.
</p>
<p data-else-if="RAND_COMBO === 21 || RAND_COMBO === 1">
<code>\color{blue}{\angle{CHF}} = \color{orange}{\angle{AGH}}</code>.
We know those 2 angles are equal because they are corresponding angles
formed by parallel lines, and a single bisecting lines.
</p>
<p>
Therefore,
<span data-if="RAND_SWITCH2 === 0">
<code>\angle{GHD} = <var>X</var>°</code>
<span class="graphie" data-update="opposite-parallel">
LABEL.remove();
label([-2, -2], "\\color{blue}{\\angle{GHD}}=" + X + "°",
"above right");
</span>
</span>
<span data-else>
<code>\angle{CHF} = <var>X</var>°</code>
<span class="graphie" data-update="opposite-parallel">
LABEL.remove();
label([-4, -2.5], "\\color{blue}{\\angle{CHF}}=" + X + "°",
"below left");
</span>
</span>.
</p>
</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="Tri1_Y">rand(10) + 30</var>
<var id="Tri1_Z">rand(10) + 100</var>
<var id="Tri1_X">180 - Tri1_Y - Tri1_Z</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul>
<li><code>\color{green}{\angle{BDC}°} = <var>Tri1_Y</var></code></li>
<li><code>\color{orange}{\angle{DBE}°} = <var>Tri1_X</var></code></li>
</ul>
<p>
What is <code>\color{blue}{\angle{<var>RAND_SWITCH3 === 0 ? "CHE" : ( RAND_SWITCH3 === 1 ? "GHC" : "DHE" )</var>}} {?}</code>
</p>
</div>
<div class="problem">
<div class="graphie" id="star1">
init({
range: [[-10, 10], [-7, 10]],
scale: [30, 30]
});
style({
stroke: "#888",
strokeWidth: 2
});
// Draw A Star
path([ [-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5] ]);
// 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], "\\color{green}{" + Tri1_Y + "°}",
"above right");
arc([-6, -6], 1.3, 40, 71, { stroke: "green" });
label([0, 7.4], "\\color{orange}{" + Tri1_X + "°}",
"below");
arc([0, 9], 1.3, 245, 290, { stroke: "orange" });
// Label X according to variation on the problem
if(RAND_SWITCH3 == 0) {
LABEL = label([4.8, 1.0], "\\color{blue}{\\angle{CHE}}= {?}", "right");
arc([3.2, 1.3], 1.7, 287, 35, { stroke: "blue" });
} else if (RAND_SWITCH3 == 1) {
LABEL = label([4, 2.5], "\\color{blue}{\\angle{GHC}}= {?}", "above");
arc([3.2, 1.3], 1, 35, 118, { stroke: "blue" });
} else {
LABEL = label([2.5, -0.5], "\\color{blue}{\\angle{DHE}}= {?}", "below");
arc([3.2, 1.3], 1.1, 219, 286, { stroke: "blue" });
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div> <!-- end problem block -->
<p class="solution">
<var data-if="RAND_SWITCH3 === 0">Tri1_Z</var>
<var data-else>180-Tri1_Z</var>
</p>
<div class="hints">
<p>
<code>
\color{purple}{\angle{BHD}} =
180° - \color{green}{\angle{BDC}} - \color{orange}{\angle{DBE}} =
<var>Tri1_Z</var>°.
</code> This is because the interior angles of a triangle add up to 180°.
<span class="graphie" data-update="star1">
// label angle BHD
arc([3.2, 1.3], .75, 118, 220, { stroke: "purple" });
label([2.6, 2], "\\color{purple}{" + Tri1_Z + "^\\circ}",
"below left");
</span>
</p>
<div data-if="RAND_SWITCH3 === 0">
<p><code>\color{blue}{\angle{CHE}} = \color{purple}{\angle{BHD}}</code>.
This is because they are opposite each other, and opposite angles are
equal.</p>
<p><code>\color{blue}{\angle{CHE}} = <var>Tri1_Z</var>°</code></p>
<span class="graphie" data-update="star1">
LABEL.remove();
label([4.8, 1.0],
"\\color{blue}{\\angle{CHE}}=" + Tri1_Z + "^\\circ",
"right");
</span>
</div>
<div data-else>
<p>
<span data-if="RAND_SWITCH3 === 1">
<code>\color{blue}{\angle{CHG}}</code>
</span>
<span data-else>
<code>\color{blue}{\angle{DHE}}</code>
</span>
<code> = 180° - \color{purple}{\angle{BHD}}</code>.
This is because angles along a line add up to
180°.
</p>
<p>
<span data-if="RAND_SWITCH3 === 1">
<code>\color{blue}{\angle{GHC}}</code>
<span class="graphie" data-update="star1">
LABEL.remove();
label([4, 2.5],
"\\color{blue}{\\angle{GHC}}=" + (180 - Tri1_Z) + "^\\circ",
"above");
</span>
</span>
<span data-else>
<code>\color{blue}{\angle{DHE}}</code>
<span class="graphie" data-update="star1">
LABEL.remove();
label([2.5, -0.5],
"\\color{blue}{\\angle{DHE}}=" + (180 - Tri1_Z) + "^\\circ",
"below");
</span>
</span>
<code> = <var>180 - Tri1_Z</var>°</code>
</p>
</div>
</div>
</div>
<!-- Star Problem 2 : 180 deg in triangle + opposite angles -->
<div id="star2">
<div class="vars">
<var id="Tri2_Y">rand(10) + 30</var>
<var id="Tri2_Z">rand(10) + 70</var>
<var id="Tri2_X">180 - Tri2_Y - Tri2_Z</var>
</div>
<div class="question">
<p>Given the following:</p>
<ul data-if="RAND_SWITCH2 === 0">
<li><code>\color{green}{\angle{BDC}°} = <var>Tri2_Y</var>°</code></li>
<li><code>\color{orange}{\angle{AIC}°} = <var>180 - Tri2_Z</var>°</code></li>
</ul><ul data-else>
<li><code>\color{green}{\angle{GCH}°} = <var>Tri2_Y</var>°</code></li>
<li><code>\color{orange}{\angle{FGH}°} = <var>180 - Tri2_Z</var>°</code></li>
</ul>
<p>What is <code>\color{blue}{\angle{<var>RAND_SWITCH2 === 0 ? "AJF" : "IHE"</var>}} {?}</code></p>
</div>
<div class="problem">
<div class="graphie" id="star2">
init({
range: [[-10, 10], [-7, 10]],
scale: [30, 30]
});
style({
stroke: "#888",
strokeWidth: 2
});
// Draw A Star
path([ [-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5] ]);
// 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 Angles and X according to variation
if( RAND_SWITCH2 == 0) {
// Label the given angles
label([-5.5, -5.2], "\\color{green}{" + Tri2_Y + "°}",
"above right");
arc([-6, -6], 1.3, 40, 71, { stroke: "green" });
label([0, -.2], "\\color{orange}{" + (180 - Tri2_Z) + "°}",
"above");
arc([0, -1], 1, 28, 152, { stroke: "orange" });
// Label X
LABEL = label([-3.7, 2.5], "\\color{blue}{\\angle{AJF}}= {?}", "above");
arc([-3.2, 1.3], 1, 65, 142, { stroke: "blue" });
} else {
// Label the given angles
label([6.5, 5], "\\color{green}{" + Tri2_Y + "°}",
"below left");
arc([8, 5], 1.3, 180, 220, { stroke: "green" });
label([1.3, 4.5], "\\color{orange}{" + (180 - Tri2_Z) + "°}",
"below left");
arc([1.6, 5], 0.75, 180, 289, { stroke: "orange" });
// Label X
LABEL = label([4.0, -0.3], "\\color{blue}{\\angle{IHE}}= {?}",
"below left");
arc([3.1, 1.2], .75, 220, 290, { stroke: "blue" });
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong> Angles not drawn to scale.</p>
</div> <!-- end problem block -->
<div class="solution"><var>Tri2_X</var></div>
<div class="hints">
<p data-if="RAND_SWITCH2 === 0">
<code>\color{purple}{\angle{DIJ}} = 180° - \color{orange}{\angle{AIC}}</code>.
This is because angles along a line total 180°.
<span class="graphie" data-update="star2">
// label angle JID
arc([0, -1.2], .75, 143, 220, { stroke: "purple" });
label([-.75, -1.2], "\\color{purple}{" + Tri2_Z + "°}",
"left");
</span>
</p>
<p data-else>
<code>\color{purple}{\angle{HGC}} = 180° - \color{orange}{\angle{FGH}}</code>.
This is because angles along a line or flat plane total 180°.
<span class="graphie" data-update="star2">
// label angle HGC
arc([1.8, 5], 1, 280, 0, { stroke: "purple" });
label([2.5, 4.3], "\\color{purple}{" + Tri2_Z + "°}",
"below right");
</span>
</p>
<p data-if="RAND_SWITCH2 === 0">
<code>\color{teal}{\angle{DJI}} = 180° - \color{green}{\angle{BDC}} -
\color{purple}{\angle{DIJ}}</code>.
We know this because the sum of angles inside a triangle add up to 180°.
<span class="graphie" data-update="star2">
// label angle JID
arc([-3.2, 1.3], .75, 260, 320, { stroke: "teal" });
label([-3.2, 0.50], "\\color{teal}{" + Tri2_X + "°}",
"below right");
</span>
</p>
<p data-else>
<code>\color{teal}{\angle{CHG}} = 180° - \color{green}{\angle{ACD}} -
\color{purple}{\angle{HGC}}</code>.
We know this, because the sum of angles inside a triangle add up to 180°.
<span class="graphie" data-update="star2">
// label angle CHG
arc([3.2, 1.3], .75, 38, 120, { stroke: "teal" });
label([3.4, 1.78], "\\color{teal}{" + Tri2_X + "°}",
"above");
</span>
</p>
<p data-if="RAND_SWITCH2 === 0">
<code>\color{blue}{\angle{AJF}} = \color{teal}{\angle{DJI}}</code>.
We know they are equal because they are opposite angles.
</p>
<p data-else>
<code>\color{blue}{\angle{IHE}} = \color{teal}{\angle{CHG}}</code>.
We know they are equal because they are opposite angles.
</p>
<p>
Therefore,
<span data-if="RAND_SWITCH2 === 0">
<code>\angle{AJF} = <var>Tri2_X</var>°</code>
<span class="graphie" data-update="star2">
LABEL.remove();
label([-3.7, 2.5],
"\\color{blue}{\\angle{AJF}}=" + Tri2_X + "°", "above");
</span>
</span>
<span data-else>
<code>\angle{IHE} = <var>Tri2_X</var>°</code>
<span class="graphie" data-update="star2">
LABEL.remove();
label([4.0, -0.3],
"\\color{blue}{\\angle{IHE}}=" + Tri2_X + "°",
"below left");
</span>
</span>.
</p>
</div>
</div>
</div>
</div>
</body>
</html>