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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 1</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<!-- Various random angle types -->
<var id="ACCUTEANGLE">rand(25) + 10</var>
<!-- Used for Triangle problems -->
<var id="Tri_Y">70 - rand( 15 )</var>
<var id="Tri_Z">70 - rand( 15 )</var>
<var id="Tri_X">180 - Tri_Y - Tri_Z</var>
<!-- Used for problem variations -->
<var id="RAND2">rand( 2 )</var>
<var id="RAND3">rand( 3 )</var>
</div>
<div class="problems">
<div id="complementary-and-opposite-angles">
<div class="problem">
<div class="question">
Given the following angles:
<ul>
<li>
<code>\overline{AB} \perp \overline{CD}</code>,
lines AB and CD are perpendicular.
</li>
<li>
<span data-if="RAND3===0">
<code>\color{green}{\angle{CGE}} = <var>ACCUTEANGLE</var>&deg;</code>
</span>
<span data-else-if="RAND3===1">
<code>\color{green}{\angle{AGF}} = <var>90 - ACCUTEANGLE</var>&deg;</code>
</span>
<span data-else>
<code>\color{green}{\angle{DGF}} = <var>ACCUTEANGLE</var>&deg;</code>
</span>
</li>
</ul>
What is
<span data-if="RAND3 === 0"><code>\color{blue}{\angle{AGF}} = {?}</code></span>
<span data-else-if="RAND3 === 1"><code>\color{blue}{\angle{CGE}} = {?}</code></span>
<span data-else><code>\color{blue}{\angle{BGE}} = {?}</code></span>
</div>
<div class="graphie" id="complementary-and-opposite">
init({
range: [[-6, 6], [-6, 6]],
scale: [40, 40]
});
style({
stroke: "#888",
strokeWidth: 2
});
// Draw perpendicular lines and 3rd line.
path( [ [-5, 0], [5, 0] ] );
path( [ [0, -5], [0, 5] ] );
path( [ [-2, -5], [2, 5] ] );
//draw points
style({
fill: "grey"
}, function() {
circle( [5, 0], 0.05 );
circle( [-5, 0], 0.05 );
circle( [0, 5], 0.05 );
circle( [0, -5], 0.05 );
circle( [2, 5], 0.05 );
circle( [-2, -5], 0.05 );
});
// Label the sides
label( [-5, 0], "A", "above" );
label( [5, 0], "B", "above" );
label( [0, 5], "C", "left" );
label( [0, -5], "D", "right" );
label( [2, 5], "E", "right" );
label( [-2, -5], "F", "left" );
label( [0, 0], "G", "below right" );
// Label the given angles
if( RAND3 == 0 ) {
label( [0.5, 1.8], "\\color{green}{" + ACCUTEANGLE + "&deg;}",
"above" );
arc( [0, 0], 1.2, 70, 90, { stroke: "green" } );
ORIGINAL_LABEL = label( [-1.2, -0.75],
"\\color{blue}{\\angle{AGF}}= {?}", "below left" );
arc( [0, 0], 1.2, 180, 248, { stroke: "blue" } );
} else if ( RAND3 == 1 ) {
label( [-1.2, -0.75], "\\color{green}{" + (90-ACCUTEANGLE) + "&deg;}",
"below left" );
arc( [0, 0], 1.2, 180, 248, { stroke: "green" } );
ORIGINAL_LABEL = label( [0.5, 1.8],
"\\color{blue}{\\angle{CGE}} = {?}", "above" );
arc( [0, 0], 1.2, 70, 90, { stroke: "blue" } );
} else {
label( [0, -2], "\\color{green}{" + ACCUTEANGLE + "&deg;}",
"below left" );
arc( [0, 0], 1.2, 248, 270, { stroke: "green" } );
ORIGINAL_LABEL = label( [1.5, 0],
"\\color{blue}{\\angle{BGE}} = {?}", "above right" );
arc( [0, 0], 1.2, 0, 70, { stroke: "blue" } );
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong>
Angles not necessarily drawn to scale.</p>
</div>
<div class="solution">
<span data-if="RAND3 === 0 || RAND3 === 2"><var>90-ACCUTEANGLE</var></span>
<span data-else><var>ACCUTEANGLE</var></span>
</div>
<div class="hints">
<p data-if="RAND3 === 0">
Because we know <code>\overline{AB} \perp \overline{CD}</code>, we know
<code>\color{purple}{\angle{CGB}} = 90&deg;</code>
<span class="graphie" data-update="complementary-and-opposite">
label( [2.2, 1.7], "\\color{purple}{90&deg;}",
"above right" );
arc( [0, 0], 3, 0, 90, { stroke: "purple" } );
</span>
</p>
<p data-else-if="RAND3 === 1">
<code>\color{orange}{\angle{EGB}} = \color{green}{\angle{AGF}}
= <var>90 - ACCUTEANGLE</var>&deg;</code>,
because they are opposite angles from each other. Opposite angles
are congruent (equal).
<span class="graphie" data-update="complementary-and-opposite">
label( [1.2, 0], "\\color{orange}{" + (90 - ACCUTEANGLE) + "&deg;}",
"above right" );
arc( [0, 0], 1.2, 0, 68, { stroke: "orange" } );
</span>
</p>
<p data-else>
Because we know <code>\overline{AB} \perp \overline{CD}</code>, we know
<code>\color{purple}{\angle{AGD}} = 90&deg;</code>
<span class="graphie" data-update="complementary-and-opposite">
label( [-2.2, -1.7], "\\color{purple}{90&deg;}",
"below left" );
arc( [0, 0], 3, 180, 270, { stroke: "purple" } );
</span>
</p>
<p data-if="RAND3===0">
<code>\color{orange}{\angle{EGB}} = \color{purple}{90&deg;}
- \color{green}{\angle{CGE}} = <var>90 - ACCUTEANGLE</var>&deg;</code>
<span class="graphie" data-update="complementary-and-opposite">
label( [1.2, 0], "\\color{orange}{" + (90 - ACCUTEANGLE) + "&deg;}",
"above right" );
arc( [0, 0], 1.2, 0, 68, { stroke: "orange" } );
</span>
</p>
<p data-else-if="RAND3===1">
Because we know <code>\overline{AB} \perp \overline{CD}</code>, we know
<code>\color{purple}{\angle{CGB}} = 90&deg;</code>
<span class="graphie" data-update="complementary-and-opposite">
label( [2.2, 1.7], "\\color{purple}{90&deg;}",
"above right" );
arc( [0, 0], 3, 0, 90, { stroke: "purple" } );
</span>
</p>
<p data-else>
<code>\color{orange}{\angle{AGF}} =
\color{purple}{90&deg;} - \color{green}{\angle{DGF}} =
<var>90 - ACCUTEANGLE</var>&deg;</code>
<span class="graphie" data-update="complementary-and-opposite">
label( [-1.2, 0], "\\color{orange}{" + 90 - ACCUTEANGLE + "&deg;}",
"below left" );
arc( [0, 0], 1.2, 180, 248, { stroke: "orange" } );
</span>
</p>
<p data-if="RAND3===0">
<code>\color{blue}{\angle{AGF}} = \color{orange}{\angle{EGB}} =
<var>90 - ACCUTEANGLE</var>&deg;</code>,
because they are opposite from each other. Opposite angles are
congruent (equal).
<span class="graphie" data-update="complementary-and-opposite">
ORIGINAL_LABEL.remove();
label( [-1.2, -0.75],
"\\color{blue}{\\angle{AGF}}=" + (90 - ACCUTEANGLE) + "&deg;",
"below left" );
</span>
</p>
<p data-else-if="RAND3===1">
<code>\color{blue}{\angle{CGE}} =
\color{purple}{90&deg;} - \color{orange}{\angle{EGB}} =
<var>ACCUTEANGLE</var>&deg;
</code>
<span class="graphie" data-update="complementary-and-opposite">
ORIGINAL_LABEL.remove();
label( [0.5, 1.8],
"\\color{blue}{\\angle{CGE}} = " + ACCUTEANGLE + "&deg;",
"above" )
</span>
</p>
<p data-else>
<code>\color{blue}{\angle{BGE}} = \color{orange}{\angle{AGF}} =
<var>90 - ACCUTEANGLE</var>&deg;</code>,
because they are opposite from each other. Opposite angles are
congruent (equal).
<span class="graphie" data-update="complementary-and-opposite">
ORIGINAL_LABEL.remove();
label( [1.5, 0],
"\\color{blue}{\\angle{BGE}} = " + (90 - ACCUTEANGLE) + "&deg;",
"above right" );
</span>
</p>
</div>
</div>
<div id="supplementary-angles-and-180-degrees-in-triangle">
<div class="problem">
<div class="question">
<p>Given the following:</p>
<div data-if="RAND2===0">
<ul>
<li><code>\color{purple}{\angle{ABC}} = <var>Tri_Z</var>&deg;</code></li>
<li><code>\color{green}{\angle{ACB}} = <var>Tri_Y</var>&deg;</code></li>
</ul>
<p>What is <code>\color{blue}{\angle{DAB}}</code>?</p>
</div>
<div data-else>
<ul>
<li><code>\color{purple}{\angle{ABC}} = <var>Tri_Z</var>&deg;</code></li>
<li><code>\color{green}{\angle{DAB}} = <var>180 - Tri_X</var>&deg;</code></li>
</ul>
<p>What is <code>\color{blue}{\angle{ACB}}</code>?</p>
</div>
</div>
<div class="graphie" id="supplementary-and-triangle">
init({
range: [[-9, 6], [-3, 5]],
scale: [40, 40]
});
style({
stroke: "#888",
strokeWidth: 2
});
// Draw TriAngle
path( [ [-8, -2], [4, -2], [0, 3], [-4, -2] ] );
//draw points
style( {
fill: "grey"
}, function() {
circle( [-8, -2], 0.05 );
circle( [4, -2], 0.05 );
circle( [0, 3], 0.05 );
circle( [-4, -2], 0.05 );
});
// Label the sides
label( [-4, -2], "A", "below" );
label( [0, 3], "B", "above" );
label( [4, -2], "C", "below right" );
label( [-8, -2], "D", "below" );
// Label the angles acording to variation
if( RAND2 == 0 ) {
label( [3, -2], "\\color{green}{" + Tri_Y + "&deg;}",
"above left" );
arc( [4, -2], 1.2, 130, 180, { stroke: "green" } );
label( [0, 1.5], "\\color{purple}{" + Tri_Z + "&deg;}",
"below" );
arc( [0, 3], 1.5, 230, 310, { stroke: "purple" } );
ORIGINAL_LABEL = label( [-4.7, -2], "\\color{blue}{\\angle{DAB}}= {?}",
"above left" );
arc( [-4, -2], .75, 50, 180, { stroke: "blue" } );
} else {
label( [-4.7, -2], "\\color{green}{" + (180 - Tri_X) + "&deg;}",
"above left" );
arc( [-4, -2], .75, 50, 180, { stroke: "green" } );
label( [0, 1.5], "\\color{purple}{" + Tri_Z + "&deg;}",
"below" );
arc( [0, 3], 1.5, 230, 310, { stroke: "purple" } );
ORIGINAL_LABEL = label( [2.80, -2], "\\color{blue}{\\angle{ACB}} = {?}",
"above left" );
arc( [4, -2], 1.2, 130, 180, { stroke: "blue" } );
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong>
Angles not necessarily drawn to scale.</p>
</div>
<div class="solution">
<var data-if="RAND2 === 0">Tri_Y + Tri_Z</var>
<var data-else>Tri_Y</var>
</div>
<div class="hints">
<p data-if="RAND2 === 0">
<code>
\color{orange}{\angle{BAC}} =
180&deg; - \color{purple}{\angle{ABC}} - \color{green}{\angle{ACB}} =
<var>180 - Tri_Y - Tri_Z</var>&deg;
</code>,
This is because angles inside a triangle add up to 180 degrees.
<span class="graphie" data-update="supplementary-and-triangle">
label( [-3.3, -2], "\\color{orange}{" + Tri_X + "&deg;}",
"above right" );
arc( [-4, -2], 0.75, 0, 49, {stroke: "orange"} );
</span>
</p>
<p data-else>
<code>
\color{orange}{\angle{BAC}} =
180&deg; - \color{green}{\angle{DAB}} =
<var>180 - Tri_Y - Tri_Z</var>&deg;
</code>,
because supplementary angles along a line add up to
180 degrees.
<span class="graphie" data-update="supplementary-and-triangle">
label( [-3.3, -2], "\\color{orange}{" + Tri_X + "&deg;}",
"above right" );
arc( [-4, -2], 0.75, 0, 49, {stroke: "orange"} );
</span>
</p>
<p data-if="RAND2 === 0">
<code>
\color{blue}{\angle{DAB}} =
180&deg; - \color{orange}{\angle{BAC}} =
<var>Tri_Y + Tri_Z</var>&deg;
</code>,
because supplementary angles along a line add up to 180&deg;
<span class="graphie" data-update="supplementary-and-triangle">
ORIGINAL_LABEL.remove();
label( [-4.7, -2],
"\\color{blue}{\\angle{DAB}} = " + (Tri_Y + Tri_Z) + "&deg;",
"above left" );
</span>
</p>
<p data-else>
<code>
\color{blue}{\angle{ACB}} =
180&deg; - \color{orange}{\angle{BAC}} - \color{purple}{\angle{ABC}} =
<var>Tri_Y</var>&deg;
</code>,
because angles inside a triangle add up to 180&deg;.
<span class="graphie" data-update="supplementary-and-triangle">
ORIGINAL_LABEL.remove();
label( [2.80, -2],
"\\color{blue}{\\angle{ACB}} = " + Tri_Y + "&deg;",
"above left" );
</span>
</p>
</div>
</div>
<div id="corresponding-and-180-in-triangle">
<div class="problem">
<div class="question">
Given the following:
<ul>
<li><code>\overline{HI} \parallel \overline{JK}</code>,
lines HI and JK are parallel.</li>
<li data-if="RAND3 === 0 || RAND3 === 1">
<code>\color{purple}{\angle{BAC}} = <var>Tri_X</var>&deg;</code>
</li>
<li data-else>
<code>\color{purple}{\angle{AKJ}} = <var>Tri_Y</var>&deg;</code>
</li>
<li data-if="RAND3 === 0">
<code>\color{green}{\angle{AJK}} = <var>Tri_Z</var>&deg;</code>
</li>
<li data-else>
<code>\color{green}{\angle{AHI}} = <var>Tri_Z</var>&deg;</code>
</li>
</ul>
What is
<code data-if="RAND3 === 0">\color{blue}{\angle{AIH}} = {?}</code>
<code data-else-if="RAND3 === 1">\color{blue}{\angle{AKJ}} = {?}</code>
<code data-else>\color{blue}{\angle{BAC}} = {?}</code>
</div>
<div class="graphie" id="parallel-and-triangle">
init( {
range: [[-9, 9.5], [-5.7, 8]],
scale: [30, 30]
});
style( {
stroke: "#888",
strokeWidth: 2
} );
// Draw Parallel lines
path( [ [-7.32, 7.5], [8, -4] ] );
path( [ [-8, 4], [4.03, -5] ] );
// Draw 2 more lines to form a triangle
path( [ [-6, -4], [-4.24, 7.5] ] );
path( [ [-6, -4], [9, -2] ] );
//draw points
style( {
fill: "grey"
}, function() {
circle( [-7.32, 7.5], 0.05 );
circle( [8, -4], 0.05 );
circle( [-8, 4], 0.05 );
circle( [4.03, -5], 0.05 );
label( [-6, -4], "A", "below" );
circle( [-6, -4], 0.05 );
label( [-4.24, 7.5], "B", "right" );
circle( [-4.24, 7.5], 0.05 );
label( [9, -2], "C", "above" );
circle( [9, -2], 0.05 );
label( [-5.07, 1.75], "H", "below left" );
label( [-4.47, 5.25], "J", "above right" );
label( [1.25, -3], "I", "below" );
label( [5.7, -2.3], "K", "above" );
} );
//draw given angles
if( RAND3 == 0 ) {
ORIGINAL_LABEL = label( [0, -2.50],
"\\color{blue}{\\angle{AIH}} = {?}", "left" );
arc( [1.25, -3], .75, 135, 190, {stroke: "blue"} );
label( [-4.2, 4.25], "\\color{green}{" + Tri_Z + "&deg;}", "below" );
arc( [-4.47, 5.25], 1, 255, 330, {stroke: "green"} );
label( [-5.5, -3.5], "\\color{purple}{" + Tri_X + "&deg;}",
"above right" );
arc( [-6, -4], 1, 10, 80, {stroke: "purple"} );
} else if( RAND3 == 1 ) {
ORIGINAL_LABEL = label( [3.5, -2.6],
"\\color{blue}{\\angle{AKJ}} = {?}", "above" );
arc( [5.7, -2.3], 0.75, 139, 194, {stroke: "blue"} );
label( [-4.4, 0.65], "\\color{green}{" + Tri_Z + "&deg;}", "below" );
arc( [-5.07, 1.75], 1, 257, 326, {stroke: "green"} );
label( [-5.5, -3.5], "\\color{purple}{" + Tri_X + "&deg;}",
"above right" );
arc( [-6, -4], 1, 10, 80, {stroke: "purple"} );
} else {
ORIGINAL_LABEL = label( [-5.5, -3.5],
"\\color{blue}{\\angle{BAC}} = {?}", "above right" );
arc( [-6, -4], 1, 10, 80, {stroke: "blue"} );
label( [4.1, -2.6], "\\color{purple}{" + Tri_Y + "&deg;}", "above" );
arc( [5.7, -2.3], 0.75, 139, 194, {stroke: "purple"} );
label( [-4.4, 0.65], "\\color{green}{" + Tri_Z + "&deg;}", "below" );
arc( [-5.07, 1.75], 1, 257, 326, {stroke: "green"} );
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong>
Angles not necessarily drawn to scale.</p>
</div>
<div class="solution">
<var data-if="RAND3 === 0 || RAND3 === 1">Tri_Y</var>
<var data-else>Tri_X</var>
</div>
<div class="hints">
<p data-if="RAND3 === 0">
<code>\color{orange}{\angle{AHI}} = \color{green}{\angle{AJK}}</code>,
because they are corresponding angles formed by 2 parallel lines and
a transversal line. Corresponding angles are congruent (equal).
<span class="graphie" data-update="parallel-and-triangle">
label( [-4.60, 0.75], "\\color{orange}{" + Tri_Z + "&deg;}",
"below" );
arc( [-5.07, 1.75], 1, 260, 325, {stroke: "orange"} );
</span>
</p>
<p data-else>
<code>\color{orange}{\angle{AJK}} = \color{green}{\angle{AHI}}</code>,
because they are corresponding angles formed by 2 parallel lines and
a transversal line. Corresponding angles are congruent (equal).
<span class="graphie" data-update="parallel-and-triangle">
label( [-4.00, 4.25], "\\color{orange}{" + Tri_Z + "&deg;}",
"below" );
arc( [-4.47, 5.25], 1, 257, 325, {stroke: "orange"} );
</span>
</p>
<p data-if="RAND3 === 0">
<code>
\color{blue}{\angle{AIH}} =
180&deg; - \color{orange}{\angle{AHI}} - \color{purple}{\angle{BAC}} =
<var>180 - Tri_X - Tri_Z</var>&deg;
</code>,
because the 3 angles are contained in <code>\triangle{AHI}</code>.
Angles inside a triangle add up to 180&deg;.
<span class="graphie" data-update="parallel-and-triangle">
ORIGINAL_LABEL.remove();
label( [0, -2.50],
"\\color{blue}{\\angle{AIH}} = " + (180 - Tri_X - Tri_Z) + "&deg;",
"left" );
</span>
</p>
<p data-else>
<span class="graphie" data-update="parallel-and-triangle">
ORIGINAL_LABEL.remove();
if ( RAND3 === 1 ) {
label( [3.3, -2.6],
"\\color{blue}{\\angle{AKJ}} = " + Tri_Y + "&deg;",
"above" );
} else {
label( [-5.5, -3.5],
"\\color{blue}{\\angle{BAC}} = " + Tri_X + "&deg;",
"above right" );
}
</span>
<code data-if="RAND3 === 1">
\color{blue}{\angle{AKJ}} =
180&deg; - \color{orange}{\angle{AJK}} - \color{purple}{\angle{BAC}} =
<var>Tri_Y</var>&deg;
</code>
<code data-else>
\color{blue}{\angle{BAC}} =
180&deg; - \color{orange}{\angle{AJK}} - \color{purple}{\angle{AKJ}} =
<var>Tri_X</var>&deg;
</code>,
because the 3 angles are contained in <code>\triangle{AJK}</code>.
Angles inside a triangle add up to 180&deg;.
</p>
</div>
</div>
<div id="alternate-exterior-and-complementary-angles">
<div class="problem">
<div class="question">
Given the following:
<ul>
<li><code>\overline{DE} \parallel \overline{FG}</code>,
lines DE and FG are parallel.</li>
<li><code>\overline{KL} \perp \overline{DE}</code>,
Lines KL and DE are perpendicular.</li>
<li data-if="RAND2 === 0">
<code>\color{green}{\angle{GCJ}} = <var>Tri_Y</var>&deg;</code>
</li>
<li data-else>
<code>\color{green}{\angle{IAK}} = <var>Tri_Y</var>&deg;</code>
</li>
</ul>
What is
<span data-if="RAND2 === 0"><code>\color{blue}{\angle{IAK}} = {?}</code></span>
<span data-else><code>\color{blue}{\angle{GCJ}} = {?}</code></span>
</div>
<div class="graphie" id="alternate-exterior-and-complementary">
init( {
range: [[-6, 8], [-5, 5]],
scale: [50, 50]
} );
style( {
stroke: "#888",
strokeWidth: 2
} );
// Draw Parallel lines
path( [ [-5, 2], [7, 2] ] );
path( [ [-5, -2], [7, -2] ] );
// Perpendicular line
path([ [0, 4], [0, -4] ]);
// Transversal line
path( [ [-2, 4], [6, -4] ]);
// Label Points
label( [-5, 2], "D", "above left" );
label( [7, 2], "E", "above right" );
label( [-5, -2], "F", "above left" );
label( [7, -2], "G", "above right" );
label( [-2, 4], "I", "above left" );
label( [6, -4], "J", "above right" );
label( [0, 4], "K", "above right" );
label( [0, -4], "L", "below right" );
label( [0, 2], "A", "below left" );
label( [0, -2], "B", "above left" );
label( [4, -2], "C", "above right" );
// Label given angles
if( RAND2 == 0 ) {
ORIGINAL_LABEL = label( [0, 3.5],
"\\color{blue}{\\angle{IAK}} = {?}", "above left" );
arc( [0, 2], 1, 90, 135, {stroke:"blue"} );
label( [4.75, -2], "\\color{green}{" + Tri_Y + "&deg;}",
"below right" );
arc( [4, -2], .75, 315, 360, {stroke: "green"} );
} else {
label( [0, 3], "\\color{green}{" + Tri_Y + "&deg;}", "above left" );
arc( [0, 2], 1, 90, 135, {stroke:"green"} );
ORIGINAL_LABEL = label( [4.75, -2],
"\\color{blue}{\\angle{GCJ} = {?}}", "below right" );
arc( [4, -2], .75, 315, 360, {stroke: "blue"} )
}
</div>
<p style="font-size:70%; color:grey"><strong>NOTE:</strong>
Angles not necessarily drawn to scale.</p>
</div>
<div class="solution"><var>90 - Tri_Y</var></div>
<div class="hints">
<p data-if="RAND2 === 0">
<code>\color{orange}{\angle{DAI}} = \color{green}{\angle{GCJ}} =
<var>Tri_Y</var>&deg;</code>,
because they are alternate exterior angles, formed by 2 parallel lines
and a transversal line, they are congruent (equal).
<span class="graphie" data-update="alternate-exterior-and-complementary">
label( [-.80, 2], "\\color{orange}{" + Tri_Y + "&deg;}",
"above left" );
arc( [0, 2], 1, 135, 180, {stroke: "orange"} );
</span>
Alternatively, you can pair up using opposite angles and alternate interior
angles to achieve the same result (as seen using
<code>\color{pink}{pink}</code>).
<span class="graphie" data-update="alternate-exterior-and-complementary">
label( [1, 2], "\\color{pink}{" + Tri_Y + "&deg;}",
"below right" );
arc( [0, 2], 1, 315, 360, {stroke: "pink"} );
label( [3, -2], "\\color{pink}{" + Tri_Y + "&deg;}",
"above left" );
arc( [4, -2], 1, 135, 180, {stroke: "pink"} );
</span>
</p>
<p>
<code>\color{purple}{\angle{DAK}} = 90&deg;</code>,
because angles formed by perpendicular lines are equal to 90&deg;.
<span class="graphie" data-update="alternate-exterior-and-complementary">
label( [-1.68, 2], "\\color{purple}{90&deg;}", "above left" );
arc( [0, 2], 1.65, 90, 180, {stroke: "purple"} );
</span>
</p>
<p data-if="RAND2 === 0">
<code>\color{blue}{\angle{IAK}} = 90&deg; - \color{orange}{\angle{DAI}} =
<var>90 - Tri_Y</var>&deg;</code>,
because angles <code>\color{blue}{\angle{IAK}}</code>
and <code>\color{orange}{\angle{DAI}}</code> make up angle
<code>\color{purple}{\angle{DAK}}</code>.
<span class="graphie" data-update="alternate-exterior-and-complementary">
ORIGINAL_LABEL.remove();
label( [0, 3.5],
"\\color{blue}{\\angle{IAK}} = " + (90 - Tri_Y) + "&deg;",
"above left" );
</span>
</p>
<p data-if="RAND2 !== 0">
<code>\color{orange}{\angle{IAK}} = 90&deg; - \color{green}{\angle{IAK}} =
<var>90 - Tri_Y</var>&deg;</code>,
because angles <code>\color{green}{\angle{IAK}}</code>
and <code>\color{orange}{\angle{DAI}}</code>, make up angle
<code>\color{purple}{\angle{DAK}}</code>.
<span class="graphie" data-update="alternate-exterior-and-complementary">
label( [-.80, 2], "\\color{orange}{" + (90-Tri_Y) + "&deg;}",
"above left" );
arc( [0, 2], 1, 135, 180, {stroke: "orange"} );
</span>
</p>
<p data-if="RAND2 !== 0">
<code>\color{blue}{\angle{GCJ}} = \color{orange}{\angle{DAI}} =
<var>90 - Tri_Y</var>&deg;</code>,
because they are alternate exterior angles formed by 2 parallel lines
and a transversal line, they are congruent (equal).
Alternatively, you can pair up using opposite angles and alternate interior
angles to achieve the same result (as seen using
<code>\color{pink}{pink}</code>).
<span class="graphie" data-update="alternate-exterior-and-complementary">
label( [1, 2], "\\color{pink}{" + (90-Tri_Y) + "&deg;}",
"below right" );
arc( [0, 2], 1, 315, 360, {stroke: "pink"} );
label( [3, -2], "\\color{pink}{" + (90-Tri_Y) + "&deg;}",
"above left" );
arc( [4, -2], 1, 135, 180, {stroke: "pink"} );
ORIGINAL_LABEL.remove();
label( [4.75, -2],
"\\color{blue}{\\angle{GCJ} = " + (90-Tri_Y) + "&deg;}",
"below right" );
</span>
</p>
</div>
</div>
</div><!-- End of problems -->
</div> <!-- End of exercises -->
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