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<!DOCTYPE html>
<html data-require="math graphie expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Angle addition postulate</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div data-calculator="">
<div class="vars">
<var id="PROBLEM">randRange(0, 2)</var>
<var id="O, A, B, C">randFromArray([
["O", "A", "B", "C"],
["B", "A", "C", "D"],
["O", "L", "M", "N"],
["P", "Q", "R", "S"]
])
</var>
<var id="ANGLE_TOP">"\\angle " + A + O + B</var>
<var id="ANGLE_BOT">"\\angle " + B + O + C</var>
<var id="LARGE_ANGLE">"\\angle " + A + O + C</var>
<var id="ANGLE_ONE, ANGLE_TWO">shuffle([ ANGLE_BOT, ANGLE_TOP ])</var>
<var id="largeAngle">[randRange(30, 170), 180, 90][PROBLEM]</var>
<var id="subAngle1">randRange(15, largeAngle-15)</var>
<var id="subAngle2">largeAngle - subAngle1</var>
<!--Let's add some algebra in here! -->
<var id="X">randRange(1, 20)</var>
<var id="COEF_1">randRange(2, 9)</var>
<var id="CONST_1">subAngle1 - COEF_1 * X</var>
<var id="COEF_2">randRange(2, 9)</var>
<var id="CONST_2">subAngle2 - COEF_2 * X</var>
<var id="ANSWER">randFromArray([[ANGLE_ONE, COEF_1, CONST_1], [ANGLE_TWO, COEF_2, CONST_2]])</var>
</div>
<div class="problem">
<div>
<p>Given</p>
<p>
<span data-if="PROBLEM === 0">
<span id="givenLarge"><code>\qquad m <var>LARGE_ANGLE</var> = <var>largeAngle</var>^\circ</code></span>
</span>
<span data-else-if="PROBLEM === 1">
<span id="givenLarge"><code>\qquad m <var>LARGE_ANGLE</var></code> is a straight angle.</span>
</span>
<span data-else="">
<span id="givenLarge"><code>\qquad \overline{<var>O</var><var>A</var>}\perp\overline{<var>O</var><var>C</var>}</code></span>
</span>
<br>
<span id="given0"><code>\qquad m <var>ANGLE_ONE</var> = <var>COEF_1</var>x + <var>CONST_1</var>^\circ</code></span><br>
<span id="given1"><code>\qquad m <var>ANGLE_TWO</var> = <var>COEF_2</var>x + <var>CONST_2</var>^\circ</code></span>
</p>
<p class="question">Find <code>m<var>ANSWER[0]</var></code>: </p>
<div class="render-answer-area-here"></div>
</div>
<div class="graphie" id="angleGraph">
init({
range: [ [(Math.min( polar(5.5, largeAngle)[0], -.5)-.5), 5.5], [-1, (Math.min( Math.max( polar(5.5,largeAngle)[1], polar(5.5, subAngle2)[1])+.5, 5.5))] ],
scale: 40
});
dotAttr={ r: 0.1, fill: "black", stroke: "none" };
line( [0, 0], [5, 0], { arrows: "-&gt;" } );
line( [0, 0], polar(5, largeAngle), {arrows: "-&gt;"} );
if (ANGLE_ONE === ANGLE_TOP){
line( [0, 0], polar(5, subAngle2), {arrows: "-&gt;"} );
} else {
line( [0, 0], polar(5, subAngle1), {arrows: "-&gt;"} );
}
circle( [0, 0] , dotAttr );
label( [0, 0], O, "below left" );
circle( polar(4.5,largeAngle), dotAttr );
label( polar(4.5,largeAngle), A, (largeAngle &gt; 120) ? "below left" : "left" );
circle( [4.5, 0], dotAttr );
label( [4.5, 0], C, "below" );
if (ANGLE_ONE === ANGLE_TOP){
circle( polar(4.5, subAngle2), dotAttr );
label( polar(4.5, subAngle2), B, (subAngle2 &gt; 45) ? "right" : "above" );
} else {
circle( polar(4.5, subAngle1), dotAttr );
label( polar(4.5, subAngle1), B, (subAngle1 &gt; 45) ? "right" : "above" );
}
</div>
<div style="clear: both;"></div>
</div>
<div class="solution" data-type="multiple">
<span class="sol" data-forms="integer">
<var>ANSWER[1] * X + ANSWER[2]</var>
</span><code>{}^{\circ}</code>
</div>
<div class="hints">
<div>
<p>From the diagram, we see that together <code>\blue{<var>ANGLE_TOP</var>}</code> and <code>\green{<var>ANGLE_BOT</var>}</code> form <code>\purple{<var>LARGE_ANGLE</var>}</code>, so</p>
<p><code>\qquad \blue{m<var>ANGLE_TOP</var>} + \green{m<var>ANGLE_BOT</var>} = \purple{m<var>LARGE_ANGLE</var>}</code>.</p>
<div class="graphie" data-update="angleGraph">
style({ stroke: GREEN, strokeWidth: 3 });
arc([0, 0], 1, 0, ANGLE_ONE === ANGLE_TOP ? subAngle2 : subAngle1).toBack();
style({ stroke: BLUE, strokeWidth: 3 });
arc([0, 0], 1, ANGLE_ONE === ANGLE_TOP ? subAngle2 : subAngle1, largeAngle).toBack();
if (PROBLEM === 0){
style({ stroke: "purple", strokeWidth: 3 });
arc( [ 0, 0 ], 1.2, 0, largeAngle );
$("#givenLarge").addClass("hint_purple");
}
if (ANGLE_ONE === ANGLE_TOP){
$("#given0").addClass("hint_blue");
$("#given1").addClass("hint_green");
} else {
$("#given0").addClass("hint_green");
$("#given1").addClass("hint_blue");
};
</div>
</div>
<div data-if="PROBLEM === 1">
<p>
Since <code><var>LARGE_ANGLE</var></code> is a straight angle,
we know <code>\purple{m<var>LARGE_ANGLE</var> = 180^\circ}</code>.
</p>
<div class="graphie" data-update="angleGraph">
style({ stroke: "purple", strokeWidth: 3 });
arc( [ 0, 0 ], 1.2, 0, 180 );
$("#givenLarge").addClass("hint_purple");
</div>
</div>
<div data-if="PROBLEM === 2">
<p>
Since we are given that <code>\overline{<var>O</var><var>A</var>}\perp\overline{<var>O</var><var>C</var>}</code>,
we know <code>\purple{m<var>LARGE_ANGLE</var> = 90^\circ}</code>.
</p>
<div class="graphie" data-update="angleGraph">
style({ stroke: "purple", strokeWidth: 3 });
path( [[0, .5], [.5, .5], [.5, 0]] );
$("#givenLarge").addClass("hint_purple");
</div>
</div>
<div>
<p>Substitute in the expressions that were given for each measure:</p>
<p data-if="ANGLE_TOP === ANGLE_ONE">
<code>\qquad \blue{<var>COEF_1</var>x + <var>CONST_1</var>^\circ} + \green{<var>COEF_2</var>x + <var>CONST_2</var>^\circ} =
\purple{<var>largeAngle</var>^\circ}</code>.
</p><p data-else="">
<code>\qquad \blue{<var>COEF_2</var>x + <var>CONST_2</var>^\circ} + \green{<var>COEF_1</var>x + <var>CONST_1</var>^\circ} =
\purple{<var>largeAngle</var>^\circ}</code>.
</p>
</div>
<div>
<p>Combine like terms:</p>
<p><code>\qquad<var>COEF_1 + COEF_2</var>x + <var>CONST_1 + CONST_2</var>^\circ = <var>largeAngle</var>^\circ</code>.</p>
</div>
<div>
<p data-if="CONST_1 + CONST_2 > 0">Subtract <code><var>CONST_1 + CONST_2</var>^\circ</code> from both sides:</p>
<p data-else="">Add <code><var>-(CONST_1 + CONST_2)</var>^\circ</code> to both sides:</p>
<p><code>\qquad<var>COEF_1 + COEF_2</var>x = <var>largeAngle - CONST_1 - CONST_2</var>^\circ</code>.</p>
</div>
<div>
<p>Divide both sides by <code><var>COEF_1 + COEF_2</var></code> to find <code>x</code>:</p>
<p><code>\qquad \pink{x = <var>X</var>^\circ}</code>.</p>
</div>
<div data-if="ANSWER[0] === ANGLE_BOT">
<p>
Substitute <code>\pink{<var>X</var>}^\circ</code> for <code>\pink{x}</code> in the expression that was given for
<code>\green{m<var>ANSWER[0]</var>}</code>:
</p>
<p>
<code>\qquad \green{m<var>ANSWER[0]</var> =
<var>ANSWER[1]</var>(}\pink{<var>X</var>^\circ}\green{) + <var>ANSWER[2]</var>^\circ}</code>.
</p>
</div>
<div data-else="">
<p>
Substitute <code>\pink{<var>X</var>^\circ}</code> for <code>\pink{x}</code> in the expression that was given for
<code>\blue{m<var>ANSWER[0]</var>}</code>:
</p>
<p>
<code>\qquad \blue{m<var>ANSWER[0]</var> =
<var>ANSWER[1]</var>(}\pink{<var>X</var>^\circ}\blue{) + <var>ANSWER[2]</var>^\circ}</code>.
</p>
</div>
<div data-if="ANSWER[0]===ANGLE_BOT">
<p>Simplify:</p>
<p><code>\qquad \green{m<var>ANSWER[0]</var>^\circ = <var>ANSWER[1] * X</var>^\circ + <var>ANSWER[2]</var>^\circ}</code>.</p>
</div>
<div data-else="">
<p>Simplify:</p>
<p><code>\qquad \blue{m<var>ANSWER[0]</var>^\circ = <var>ANSWER[1] * X</var>^\circ + <var>ANSWER[2]</var>^\circ}</code>.</p>
</div>
<p data-if="ANSWER[0]===ANGLE_BOT">So <code>\green{m<var>ANSWER[0]</var> = <var>ANSWER[1] * X + ANSWER[2]</var>^\circ}</code>.</p>
<p data-else="">So <code>\blue{m<var>ANSWER[0]</var> = <var>ANSWER[1] * X + ANSWER[2]</var>^\circ}</code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>
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