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171 lines
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<title>Congruent Triangles</title>
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<meta name="description" content="Triangles are congruent when they have exactly the same three sides and exactly the same three angles." />
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<h1 align="center">Congruent Triangles</h1>
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<p class="center">Triangles are congruent when they have <br />
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exactly the <b>same three sides</b> and exactly the <b>same three angles</b>. </p>
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<h2> What is "Congruent" ... ?</h2>
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<p>It means that one shape can become another using Turns, Flips and/or Slides:</p>
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<table border="0" align="center">
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<tr>
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<th><a href="rotation.html">Rotation</a></th>
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<td align="center"><img src="images/rotation-2d.svg" alt="rotation 2d" /></td>
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<th><a href="translation.html">Translation</a></th>
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<th>Slide!</th>
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</tr>
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</table>
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<p align="center"> (see <a href="congruent.html">Congruent</a> for more info)</p>
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<h2>Congruent Triangles</h2>
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<p> When two triangles are congruent they will have exactly <b>the same three sides</b> and exactly <b>the same three angles</b>. </p>
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<p>The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there. </p>
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<h3>Same Sides</h3>
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<p>When the sides are the same then the triangles are congruent.</p>
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<p>For example:</p>
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<table border="0" align="center">
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<tr class="large">
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<td><img src="images/tri-congruent1.gif" alt="triangle" width="143" height="94" /></td>
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<td align="right">is congruent to:</td>
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<td> </td>
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<td><img src="images/tri-congruent2a.gif" alt="triangle" width="144" height="99" /></td>
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</tr>
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<tr class="large">
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<td> </td>
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<td align="right">and also to:</td>
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<td> </td>
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<td><img src="images/tri-congruent2b.gif" alt="triangle" width="121" height="130" /></td>
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</tr>
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</table>
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<p align="center" class="larger">because they all have<b> exactly the same sides</b>.</p>
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<p>But:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent1.gif" alt="triangle" width="143" height="94" /></td>
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<td>is <b>NOT</b> congruent to:</td>
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<td> </td>
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<td><img src="images/tri-congruent2c.gif" alt="triangle" width="160" height="111" /></td>
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</tr>
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</table>
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<p align="center" class="larger">because the two triangles <b>do not have </b>exactly the same sides.</p>
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<p> </p>
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<h3>Same Angles</h3>
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<p>Does this also work with angles? Not always! </p>
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<p>Two triangles with the same angles <b>might be</b> congruent:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent2d.gif" alt="triangle" width="167" height="110" /></td>
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<td><b>is congruent to:</b></td>
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<td><img src="images/tri-congruent2e.gif" alt="triangle" width="167" height="124" /></td>
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</tr>
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</table>
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<p align="center" class="larger">only because they are the <b>same size</b></p>
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<p>But they might <b>NOT</b> be congruent because of <b>different sizes</b>:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent2d.gif" alt="triangle" width="167" height="110" /></td>
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<td>is <b>NOT</b> congruent to:</td>
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<td><img src="images/tri-congruent2f.gif" alt="triangle" width="192" height="130" /></td>
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</tr>
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</table>
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<p align="center" class="larger">because, even though all angles match, <b>one is larger than the other</b>.</p>
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<p>So just having the same angles is no guarantee they are congruent.</p>
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<p> </p>
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<h3>Other Combinations</h3>
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<p class="center">There are other combinations of sides and angles that can work ... <br />
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... read more at <a href="triangles-congruent-finding.html">How To Find if Triangles are Congruent</a></p>
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<h2>Marking</h2>
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<p>When two triangles are congruent we often mark corresponding sides and angles like this:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent3a.gif" alt="triangle" width="199" height="99" /></td>
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<td>is congruent to:</td>
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<td><img src="images/tri-congruent3b.gif" alt="triangle" width="196" height="100" /></td>
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</tr>
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</table>
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</div>
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<p>The sides marked with one line are equal in length. Similarly for the sides marked with two lines. Also for the sides marked with three lines.</p>
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<p>The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs. Also for the angles marked with three arcs.</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(734, 735, 5026, 5027, 1524, 1525, 7492, 7493, 7494, 7495);</script>
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<div class="related"> <a href="congruent.html">Congruent</a> <a href="similar.html">Similar</a> <a href="triangles-congruent-finding.html">Finding Congruent Triangles</a> <a href="triangles-similar.html">Similar Triangles</a> <a href="../algebra/trigonometry-index.html">Trigonometry Index</a> </div>
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