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<title>Theorems, Corollaries, Lemmas</title>
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<h1 align="center">Theorems, Corollaries, Lemmas</h1>
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<p> </p>
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<p>What are all those things? They sound so impressive!</p>
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<p>Well, they are basically just <b>facts</b>: some result that has been arrived at.</p>
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<ul>
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<li>A Theorem is a <b>major</b> result<br />
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</li>
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<li>A Corollary is a theorem that <b>follows on</b> from another theorem<br />
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</li>
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<li>A Lemma is a <b>small</b> result (less important than a theorem) </li>
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</ul>
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<h2>Examples</h2>
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<p>Here is an example from Geometry: </p>
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<div class="example">
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<h3>Example: A Theorem and a Corollary</h3>
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<h4>Theorem: </h4>
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<p><a href="../angle180.html">Angles on one side of a straight line always add to 180°</a>. </p>
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<p align="center"><img src="../geometry/images/angle180.svg" alt="angles add to 180 degrees" /></p>
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<p> </p>
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<h4> Corollary: </h4>
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<p>Following on from that theorem we find that where two lines intersect, the angles opposite each other (called <a href="../geometry/vertical-angles.html">Vertical Angles</a>) are <b>equal</b> (a=c and b=d in the diagram). </p>
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<p align="center"><img src="../geometry/images/vertically-opposite-abcd.svg" alt="vertically-opposite-abcd" /><span class="larger"><br>
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Angle a = angle c<br>Angle b = angle d</span> </p>
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<h4>Proof: </h4>
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<p>Angles a and b add to 180° because they are along a line:</p>
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<p class="so">a + b = 180°</p>
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<p class="so"> a = 180° − b</p>
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<p>Likewise for angles b and c</p>
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<p class="so">b + c = 180°</p>
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<p class="so"> c = 180° − b</p>
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<p>And since both a and c equal 180° − b, then</p>
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<p class="so">a = c </p>
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</div>
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<p> </p>
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<p>And a slightly more complicated example from Geometry:</p>
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<div class="example">
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<h3>Example: A Theorem, a Corollary to it, and also a Lemma!</h3>
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<h4>Theorem: </h4>
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<p align="center"><img src="../geometry/images/inscribed-angle-1.gif" width="183" height="183" alt="inscribed angle 2a and a" /><span class="larger"><br>
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An inscribed angle a° is half of the central angle 2a°</span><br />
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Called the <b><a href="../geometry/circle-theorems.html">Angle at the Center Theorem</a></b>.</p>
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<p><b>Proof: Join the center O to A. </b></p>
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<p class="center"><img src="../geometry/images/inscribed-angle-proof.gif" width="186" height="184" alt="inscribed angle proof" /></p>
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<p>Triangle ABO is <a href="../triangle.html">isosceles</a> (two equal sides, two equal angles), so:</p>
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<div class="so"> Angle OBA = Angle BAO = <b>b°</b> </div>
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<p>And, using <a href="../angle180.html">Angles of a Triangle add to 180°</a>:</p>
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<div class="so">Angle AOB = (180 − 2b)°</div>
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<p>Triangle ACO is isosceles, so:</p>
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<div class="so"> Angle OCA = Angle CAO = <b>c°</b> </div>
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<p>And, using <a href="../angle180.html">Angles of a Triangle add to 180°</a>:</p>
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<div class="so">Angle AOC = (180 − 2c)°</div>
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<p>And, using <a href="../angle360.html">Angles around a point add to 360°</a>: </p><div class="tbl">
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<div class="row"><span class="left">Angle BOC </span><span class="right">= 360° − (180 − 2b)° − (180 − 2c)° </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 2b° + 2c°</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 2(b + c)° </span></div>
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</div>
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<p>Replace <b>b + c</b> with <b>a</b>, we get:</p>
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<p class="center larger">Angle BAC = a° and Angle BOC = 2a° </p>
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<p align="center"><img src="../geometry/images/inscribed-angle-1.gif" width="183" height="183" alt="inscribed angle 2a and a" /><span class="larger"><br>
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</span></p>
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<p>And we have proved the theorem.</p>
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<p>(That was a "major" result, so is a Theorem.)</p>
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<p> </p>
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<h4>Corollary</h4>
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<p>(This is called the <i>"Angles Subtended by the Same Arc Theorem</i>", but it’s really just a <b>Corollary</b> of the <i>"Angle at the Center Theorem"</i>) </p>
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<p class="larger">Keeping the endpoints fixed ... ... the angle a° is always the same, no matter where it is on the circumference:</p>
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<p class="center"><img src="../geometry/images/inscribed-angle-2.gif" width="181" height="179" alt="inscribed angle a and a" /></p>
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<p>So, Angles Subtended by the Same Arc are equal.</p>
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<p> </p>
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<h4>Lemma</h4>
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<p>(This is sometimes called the <i>"Angle in the Semicircle Theorem"</i>, but it’s really just a <b>Lemma</b> to the <i>"Angle at the Center Theorem"</i>) </p>
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<p class="center"><img src="../geometry/images/inscribed-angle-1.gif" width="183" height="183" alt="inscribed angle 2a and a" /><img src="../geometry/images/angle-semicircle-1a.gif" width="182" height="180" alt="angle semicircle 180 and 90" /></p>
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<p>In the special case where the central angle forms a diameter of the circle:</p>
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<p class="center larger"> 2a° = 180° , so a° = 90° </p>
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<p>So an angle inscribed in a semicircle is always a right angle. </p>
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<p>(That was a "small" result, so it is a Lemma.) </p>
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</div>
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<p> </p>
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<p>Another example, related to <a href="../pythagoras.html">Pythagoras' Theorem</a>:</p>
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<div class="example">
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<h3>Example: </h3>
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<h4>Theorem</h4>
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<p> If m and n are any two whole numbers and </p>
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<ul>
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<li>a = m<sup>2</sup> − n<sup>2</sup></li>
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<li>b = 2mn</li>
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<li>c = m<sup>2</sup> + n<sup>2</sup></li>
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</ul>
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<p>then a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup><br />
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</p>
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<p><b>Proof</b>:</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>a<sup>2</sup> + b<sup>2</sup></b> </span><span class="right">= (m<sup>2</sup> − n<sup>2</sup>)<sup>2</sup> + (2mn)<sup>2</sup> </span></div>
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<div class="row"><span class="left"> </span><span class="right">= m<sup>4</sup> − 2m<sup>2</sup>n<sup>2</sup> + n<sup>4</sup> + 4m<sup>2</sup>n<sup>2</sup> </span></div>
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<div class="row"><span class="left"> </span><span class="right">= m<sup>4</sup> + 2m<sup>2</sup>n<sup>2</sup> + n<sup>4</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= (m<sup>2</sup> + n<sup>2</sup>)<sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right"><b>= c<sup>2</sup></b> </span></div>
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</div>
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<p>(That was a "major" result.)</p>
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<p> </p>
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<h4>Corollary</h4>
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<p> a, b and c, as defined above, are a <a href="../numbers/pythagorean-triples.html">Pythagorean Triple</a></p>
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<p> <b>Proof</b>:</p>
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<p align="center"> From the Theorem <b>a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></b>, <br>
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so a, b and c are a Pythagorean Triple</p>
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<p>(That result "followed on" from the previous Theorem.)</p>
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<p> </p>
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<h4>Lemma</h4>
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<p> If m = 2 and n = 1, then we get the Pythagorean triple 3, 4 and 5</p>
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<p><b>Proof</b>:</p>
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<p>If m = 2 and n = 1, then </p>
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<ul>
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<li>a = 2<sup>2</sup> − 1<sup>2</sup> = 4 − 1 = <b>3</b></li>
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<li>b = 2 × 2 × 1 = <b>4</b></li>
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<li>c = 2<sup>2</sup> + 1<sup>2</sup> = 4 + 1 = <b>5</b></li>
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</ul>
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<p>(That was a "small" result.) </p>
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</div><p> </p>
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<div class="related"><a href="index.html">Algebra Index</a></div>
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