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<title>Solving AAS Triangles</title>
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<h1 align="center">Solving AAS Triangles</h1>
<p align="center"><i>&quot;AAS&quot; means &quot;Angle, Angle, Side&quot;</i></p>
<table width="85%" border="0" align="center">
<tr>
<td><img src="images/triangle-aas.svg" alt="AAS Triangle" /></td>
<td>
<p>&quot;<i>AAS</i>&quot; is when we know two angles and one side (which is <b>not</b> between the angles).</p>
</td>
</tr>
</table>
<p>&nbsp;</p>
<table width="90%" align="center">
<tr>
<td>
<p>To solve an AAS triangle</p>
<ul>
<li>use <a href="../proof180deg.html">the three angles add to 180&deg;</a> to find the other angle </li>
<li>then <a href="trig-sine-law.html">The Law of Sines</a> to find each of the other two sides. </li>
</ul>
</td>
</tr>
</table>
<p>&nbsp;</p>
<div class="example">
<h3>Example 1</h3>
<p> <img src="images/trig-aasex1.gif" width="161" height="108" alt="trig AAS example" /></p>
<p>In this triangle we know: </p>
<ul>
<li>angle A = 35&deg; </li>
<li>angle C = 62&deg; </li>
<li>and side c = 7 </li>
</ul>
<p>&nbsp;</p>
<p>It's easy to find angle B by using 'angles of a triangle add to 180&deg;':</p>
<p align="center"> B = 180&deg; &minus; 35&deg; &minus; 62&deg; = <b>83&deg;</b></p>
<p>&nbsp;</p>
<p>We can also find side a by using The Law of Sines:</p>
<p align="center">a/sin A = c/sin C</p>
<p align="center"> a/sin(35&deg;) = 7/sin(62&deg;)</p>
<p align="center">a = (7 &times; sin(35&deg;))/sin(62&deg;)
<br> a = <b>4.55</b> to 2 decimal places</p>
<p>&nbsp;</p>
<p>Also we can find b by using The Law of Sines:</p>
<p align="center">b/sin B = c/sin C</p>
<p align="center"> b/sin(83&deg;) = 7/sin(62&deg;)</p>
<p align="center">b = (7 &times; sin(83&deg;))/sin(62&deg;)
<br> b = <b>7.87</b> to 2 decimal places</p>
<p>&nbsp;</p>
<p>Now we have completely solved the triangle!</p>
</div>
<p>Did you notice that we used <b>b/sin B = c/sin C</b> rather than <b>b/sin B = a/sin A</b> for the last calculation?</p>
<p>There's a good reason for that. What if we made a mistake when finding <b>a</b>? Then our answer for <b>b</b> would also be wrong!</p>
<p><b>As a rule, it is always better to use the sides and angles that are given rather than ones we've just worked out.</b></p>
<p>&nbsp;</p>
<div class="example">
<h3>Example 2</h3>
<p> <img src="images/trig-aasex2.gif" width="227" height="126" alt="trig AAS example" /></p>
<p>This is also an AAS triangle.</p>
<p>First find angle A by using 'angles of a triangle add to 180&deg;':</p>
<div class="so">A = 180&deg; &minus; 41&deg; &minus; 105&deg; = <b>34&deg;</b></div>
<p>&nbsp;</p>
<p>Now find side <b>c</b> by using The Law of Sines:</p>
<div class="so">c/sin C = b/sin B</div>
<div class="so">c/sin(41&deg;) = 12.6/sin(105&deg;)</div>
<div class="so">c = (12.6 &times; sin(41&deg;))/sin(105&deg;) </div>
<div class="so">c = <b>8.56</b> to 2 decimal places</div>
<p>&nbsp;</p>
<p>Similarly we can find side <b>a</b> by using The Law of Sines and using the given side b = 12.6 rather than c that we just worked out:</p>
<div class="so">a/sin A = b/sin B</div>
<div class="so">a/sin(34&deg;) = 12.6/sin(105&deg;)</div>
<div class="so">a = (12.6 &times; sin(34&deg;))/sin(105&deg;)</div>
<div class="so">a = <b>7.29</b> to 2 decimal places</div>
<p>Done!</p>
</div>
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