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198 lines
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198 lines
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<title>How To Find if Triangles are Similar</title>
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<h1 align="center">How to Find if Triangles are Similar</h1>
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<p>Two <a href="triangles-similar.html">triangles are similar</a> if they have:</p>
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<ul>
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<li>all their angles equal</li>
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<li>corresponding sides are in the same ratio</li>
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</ul>
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<p>But we don't need to know all three sides and all three angles ...<b>two or three out of the six</b> is usually enough. </p>
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<p>There are three ways to find if two triangles are similar: <b>AA</b>, <b>SAS</b> and <b>SSS</b>:</p>
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<h2>AA</h2>
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<p>
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<b>AA</b> stands for "angle, angle" and means that the triangles have two of their angles equal. </p>
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<div class="def">
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<p>If two triangles have two of their angles equal, the triangles are similar. </p>
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</div>
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<div class="example">
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<h3>Example: these two triangles are similar:</h3>
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<p align="center"><img src="images/tri-similar3.gif" width="404" height="149" alt="triangles similar both have angles 72 and 35" /></p>
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<p>If two of their angles are equal, then the third angle must also be equal, because <a href="../proof180deg.html">angles of a triangle always add to make 180°</a>. </p>
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<p>In this case the missing angle is 180° − (72° + 35°) = 73°</p>
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</div>
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<p>So AA could also be called AAA (because when two angles are equal, all three angles must be equal).</p>
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<h2>SAS</h2>
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<p>
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SAS stands for "side, angle, side" and means that we have two triangles where:</p>
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<ul>
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<li>the ratio between two sides is the same as the ratio between another two sides</li>
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<li>and we we also know the included angles are equal.</li>
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</ul>
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<div class="def">
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<p>If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. </p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p align="center"><img src="images/tri-similar4.gif" alt="triangles similar both have angle 75 but sides (15,21,a) and (10,14,x)" /></p>
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<p>In this example we can see that:</p>
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<ul>
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<li>one pair of sides is in the ratio of 21 : 14 = <b>3 : 2</b></li>
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<li>another pair of sides is in the ratio of 15 : 10 = <b>3 : 2</b></li>
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<li>there is a matching angle of 75° in between them</li>
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</ul>
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<p>So there is enough information to tell us that the <b>two triangles are similar</b>.</p>
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</div>
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<h3>Using Trigonometry</h3>
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<p>We could also use <a href="../algebra/trigonometry-index.html">Trigonometry</a> to calculate the other two sides using the <a href="../algebra/trig-cosine-law.html">Law of Cosines</a>:</p>
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<div class="example">
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<h3>Example Continued</h3>
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<p>In Triangle ABC:</p>
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<ul>
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<li>a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2bc cos A</li>
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<li>a<sup>2</sup> = 21<sup>2</sup> + 15<sup>2</sup> - 2 × 21 × 15 × Cos75°</li>
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<li>a<sup>2</sup> = 441 + 225 - 630 × 0.2588...</li>
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<li>a<sup>2</sup> = 666 - 163.055...</li>
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<li>a<sup>2</sup> = 502.944...</li>
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<li>So a = √502.94 = <b>22.426...</b></li>
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</ul>
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<p>In Triangle XYZ:</p>
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<ul>
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<li>x<sup>2</sup> = y<sup>2</sup> + z<sup>2</sup> - 2yz cos X</li>
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<li>x<sup>2</sup> = 14<sup>2</sup> + 10<sup>2</sup> - 2 × 14 × 10 × Cos75°</li>
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<li>x<sup>2</sup> = 196 + 100 - 280 × 0.2588...</li>
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<li>x<sup>2</sup> = 296 - 72.469...</li>
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<li>x<sup>2</sup> = 223.530...</li>
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<li>So x = √223.530... = <b>14.950...</b></li>
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</ul>
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<p>Now let us check the ratio of those two sides:</p>
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<p align="center" class="larger">a : x = 22.426... : 14.950... = <b>3 : 2 </b></p>
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<p align="center" class="larger">the same ratio as before!</p>
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</div>
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<p>Note: we can also use the <a href="../algebra/trig-sine-law.html">Law of Sines</a> to show that the other two angles are equal.</p>
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<h2>SSS</h2><p>
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SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.</p>
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<div class="def">
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<p>If two triangles have three pairs of sides in the same ratio, then the triangles are similar.</p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p align="center"><img src="images/tri-similar5.gif" alt="triangles (4,6,8) and (5,7.5,10)" /></p>
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<p>In this example, the ratios of sides are:</p>
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<ul>
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<li>a : x = 6 : 7.5 = 12 : 15 = <b>4 : 5</b></li>
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<li>b : y = 8 : 10 = <b>4 : 5 </b></li>
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<li>c : z = <b>4 : 5</b></li>
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</ul>
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<p>These ratios are all equal, so the two triangles are similar.</p>
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</div>
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<h3>Using Trigonometry</h3>
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<p>Using <a href="../algebra/trigonometry-index.html">Trigonometry</a> we can show that the two triangles have equal angles by using the <a href="../algebra/trig-cosine-law.html">Law of Cosines</a> in each triangle:</p>
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<div class="example">
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<p>In Triangle ABC:</p>
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<ul>
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<li>cos A = (b<sup>2</sup> + c<sup>2</sup> - a<sup>2</sup>)/2bc</li>
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<li>cos A = (8<sup>2</sup> + 4<sup>2</sup> - 6<sup>2</sup>)/(2× 8 × 4)</li>
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<li>cos A = (64 + 16 - 36)/64</li>
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<li>cos A = 44/64</li>
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<li>cos A = 0.6875</li>
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<li>So Angle A = <b>46.6°</b></li>
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</ul>
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<p>In Triangle XYZ:</p>
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<ul>
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<li>cos X = (y<sup>2</sup> + z<sup>2</sup> - x<sup>2</sup>)/2yz</li>
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<li>cos X = (10<sup>2</sup> + 5<sup>2</sup> - 7.5<sup>2</sup>)/(2× 10 × 5)</li>
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<li>cos X = (100 + 25 - 56.25)/100</li>
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<li>cos X = 68.75/100</li>
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<li>cos X = 0.6875</li>
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<li>So Angle X = <b>46.6°</b></li>
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</ul>
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<p align="center" class="larger">So angles A and X are equal!</p>
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</div>
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<p>Similarly we can show that angles B and Y are equal, and angles C and Z are equal.
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</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(7861, 7862, 7863, 7864, 7865, 7866, 7867, 7868, 7869, 7870, 7871, 7872);</script>
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</div>
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<div class="related">
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<a href="similar.html">Similar</a>
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<a href="triangles-similar.html">Similar Triangles</a>
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<a href="triangles-similar-theorems.html">Similar Triangle Theorems</a>
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<a href="congruent.html">Congruent</a>
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<a href="triangles-congruent.html">Congruent Triangles</a>
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<a href="triangles-congruent-finding.html">Finding Congruent Triangles</a>
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<a href="../algebra/trigonometry-index.html">Trigonometry Index</a> </div>
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