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79 lines
2.8 KiB
HTML
79 lines
2.8 KiB
HTML
<div class="book-prose">
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<p>
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Deductive geometry uses known facts and logical steps to prove new geometric results.
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A valid proof must justify each statement with a theorem, definition, or given condition.
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</p>
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<h2 id="reasoning-language" class="book-prose__heading">Reasoning language</h2>
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<ul>
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<li><strong>Given</strong>: information provided in the question.</li>
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<li><strong>To prove</strong>: target statement.</li>
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<li><strong>Because / therefore</strong>: links each conclusion to a reason.</li>
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</ul>
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<h2 id="line-angle-facts" class="book-prose__heading">Core angle facts</h2>
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<ul>
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<li>Angles on a straight line sum to \(180^\circ\).</li>
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<li>Angles around a point sum to \(360^\circ\).</li>
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<li>Vertically opposite angles are equal.</li>
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<li>In parallel lines:
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<ul>
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<li>alternate interior angles are equal,</li>
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<li>corresponding angles are equal,</li>
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<li>co-interior angles sum to \(180^\circ\).</li>
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</ul>
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</li>
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</ul>
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<h2 id="triangle-facts" class="book-prose__heading">Triangle facts</h2>
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<ul>
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<li>Interior angles sum to \(180^\circ\).</li>
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<li>Exterior angle equals sum of two opposite interior angles.</li>
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<li>Isosceles triangle: equal sides oppose equal angles.</li>
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</ul>
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<h2 id="polygon-facts" class="book-prose__heading">Polygon facts</h2>
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<p>
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Sum of interior angles of \(n\)-gon:
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</p>
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<p class="book-formula">\[
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(n-2)\times180^\circ.
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\]</p>
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<p>
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Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).
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</p>
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<h2 id="proof-structure" class="book-prose__heading">Proof structure</h2>
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<ol>
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<li>Write what is given and what must be shown.</li>
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<li>Mark known equal angles/lengths on diagram.</li>
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<li>Use relevant theorems in short logical chain.</li>
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<li>End with clear conclusion sentence.</li>
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</ol>
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<h2 id="example-proof" class="book-prose__heading">Example proof pattern</h2>
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<p>
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If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles
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is equal.
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</p>
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<p>
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Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore
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required pair is equal.
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</p>
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<h2 id="common-errors" class="book-prose__heading">Common proof errors</h2>
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<ul>
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<li>Stating result without reason.</li>
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<li>Using theorem conditions not satisfied by diagram.</li>
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<li>Assuming diagram scale implies equality.</li>
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<li>Skipping steps between statements.</li>
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</ul>
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<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
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<ul>
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<li>Use short statements with explicit reasons.</li>
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<li>Reference standard theorem names precisely.</li>
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<li>Keep symbols consistent with question labels.</li>
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<li>Do not rely on appearance; rely on given facts and proven steps.</li>
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</ul>
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</div>
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