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70 lines
2.9 KiB
HTML
70 lines
2.9 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Deductive geometry traces to Euclid's <em>Elements</em>, where geometric knowledge was organized
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into axioms, definitions, and logically derived propositions. This method shaped modern proof.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
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<h3 id="why-proof" class="book-prose__heading">Why proof matters</h3>
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<p>
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Measurement drawings can suggest answers, but proofs guarantee truth for all valid cases. Deductive
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reasoning removes uncertainty caused by drawing scale or rounding.
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">
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Two angles on a straight line are \(3x\) and \(x+40\). Find \(x\).
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</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(3x+(x+40)=180\Rightarrow 4x=140\Rightarrow x=35\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">
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In a triangle, angles are \(x,2x,3x\). Find all three angles.
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</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(x+2x+3x=180\Rightarrow x=30\).</p>
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<p>Angles: \(30^\circ,60^\circ,90^\circ\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">
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If corresponding angles are equal for a transversal, what can you conclude?
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</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> The two lines are parallel (converse theorem).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">
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Find interior-angle sum of a decagon.
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</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((10-2)\times180=1440^\circ\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">
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Explain one reason why “it looks equal” is not valid proof.
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</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Diagram scale can be inaccurate; only theorem-based logic is reliable.</p>
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</div>
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</details>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<ul>
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<li><strong>Engineering drawings:</strong> design correctness depends on provable geometric constraints.</li>
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<li><strong>Architecture:</strong> angle and parallel-line reasoning ensures structural fit and symmetry.</li>
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<li><strong>Computer-aided design:</strong> geometric rule engines rely on deductive relationships.</li>
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</ul>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Euclid, <em>Elements</em> (classical deductive geometry foundation).</li>
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<li>Standard high-school geometry syllabi on angle and polygon proofs.</li>
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</ul>
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</div>
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