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195 lines
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<title>How To Find if Triangles are Congruent</title>
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<h1 align="center">How To Find if Triangles are Congruent </h1>
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<table border="0">
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<tr>
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<td width="100"> </td>
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<td><p>Two <a href="triangles-congruent.html">triangles are congruent</a> if they have:</p>
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<ul>
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<li>exactly the same three sides and </li>
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<li>exactly the same three angles.</li>
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</ul>
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<p>But we don't have to know all three sides and all three angles ...usually <b>three out of the six</b> is enough. </p> </td>
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</tr>
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</table>
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<p>There are five ways to find if two triangles are congruent: <b>SSS</b>, <b>SAS</b>, <b>ASA</b>, <b>AAS</b> and <b>HL</b>.</p>
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<h2>1. SSS <i>(side, side, side)</i></h2>
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<p><img src="../algebra/images/triangle-sss.svg" alt="SSS Triangle" /></p>
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<p><b>SSS</b> stands for "side, side, side" and means that we have two triangles with all three sides equal.</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>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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</table>
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<p align="center"><i>(See <a href="../algebra/trig-solving-sss-triangles.html">Solving SSS Triangles</a> to find out more) </i></p>
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<div class="def">
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<p>If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. </p>
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</div>
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<h2>2. SAS <i>(side, angle, side)</i></h2>
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<p><img src="../algebra/images/triangle-sas.svg" alt="SAS Triangle" /></p>
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<p><b>SAS</b> stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.</p>
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<p>For example:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent5a.gif" alt="triangle" width="208" height="97" /></td>
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<td>is<b> </b>congruent to:</td>
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<td><img src="images/tri-congruent5b.gif" alt="triangle" width="203" height="111" /></td>
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</tr>
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</table>
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<p align="center"><i>(See <a href="../algebra/trig-solving-sas-triangles.html">Solving SAS Triangles</a> to find out more) </i></p>
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<div class="def">
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<p>If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. </p>
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</div>
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<h2>3. ASA <i>(angle, side, angle)</i></h2>
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<p><img src="../algebra/images/triangle-asa.svg" alt="ASA Triangle" /></p>
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<p><b>ASA</b> stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.</p>
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<p>For example:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent6a.gif" alt="triangle" width="238" height="81" /></td>
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<td>is<b> </b>congruent to:</td>
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<td><img src="images/tri-congruent6b.gif" alt="triangle" width="192" height="144" /></td>
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</tr>
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</table>
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<p align="center"><i>(See <a href="../algebra/trig-solving-asa-triangles.html">Solving ASA Triangles</a> to find out more)</i></p>
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<div class="def">
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<p>If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. </p>
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</div>
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<h2>4. AAS <i>(angle, angle, side)</i></h2>
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<p><img src="../algebra/images/triangle-aas.svg" alt="AAS Triangle" /></p>
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<p><b>AAS</b> stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.</p>
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<p>For example:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent7a.gif" alt="triangle" width="169" height="184" /></td>
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<td>is<b> </b>congruent to:</td>
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<td><img src="images/tri-congruent7b.gif" alt="triangle" width="152" height="184" /></td>
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</tr>
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</table>
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<p align="center"><i>(See <a href="../algebra/trig-solving-aas-triangles.html">Solving AAS Triangles</a> to find out more)</i></p>
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<div class="def">
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<p>If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. </p>
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</div>
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<h2>5. HL <i>(hypotenuse, leg)</i></h2>
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<p>This one applies only to <a href="../right_angle_triangle.html">right angled-triangles</a>!</p>
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<table border="0">
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<tr>
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<td> <img src="images/tri-hl-1.gif" width="186" height="116" alt="triangle HL" /></td>
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<td> </td>
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<td class="large">or</td>
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<td> </td>
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<td><img src="images/tri-hl-2.gif" width="185" height="114" alt="triangle HL" /></td>
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</tr>
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</table>
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<p><b>HL</b> stands for "<b>H</b>ypotenuse, <b>L</b>eg" (t<span class="Larger">he longest side of a right-angled triangle is called the "hypotenuse"</span>, the other two sides are called "legs")</p>
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<p>It means we have two right-angled triangles with</p>
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<ul>
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<li>the <b>same length of hypotenuse</b> and </li>
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<li>the<b> same length for one of the other two legs</b>. </li>
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</ul>
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<p>It doesn't matter which leg since the triangles could be rotated.</p>
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<p>For example:</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/tri-congruent8a.gif" alt="triangle" width="157" height="187" /></td>
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<td>is<b> </b>congruent to:</td>
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<td><img src="images/tri-congruent8b.gif" alt="triangle" width="222" height="131" /></td>
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</tr>
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</table>
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<p align="center"><i>(See <a href="../pythagoras.html">Pythagoras' Theorem</a> to find out more)</i></p>
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<div class="def">
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<p>If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent. </p>
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</div>
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<h2>Caution! Don't Use "AAA"</h2>
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<p><b>AAA</b> means we are given all three angles of a triangle, but no sides.</p>
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<p><img src="../algebra/images/triangle-aaa.svg" alt="AAA Triangle" /></p>
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<p><b>This is not enough information to decide if two triangles are congruent!</b></p>
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<p>Because the triangles can have the same angles but be <b>different sizes</b>:</p>
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<table width="80%" 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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<br />
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<div class="def">
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<p>Without knowing at least one side, we can't be sure if two triangles are congruent.</p>
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</div>
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<p> </p>
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<div class="related">
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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="similar.html">Similar</a>
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<a href="triangles-similar.html">Similar Triangles</a>
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<a href="triangles-similar-finding.html">Finding Similar Triangles</a>
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<a href="../algebra/trigonometry-index.html">Trigonometry Index</a> </div>
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