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73 lines
2.4 KiB
HTML
73 lines
2.4 KiB
HTML
<div class="book-prose">
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<p>
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Congruence and similarity compare shapes by size and shape. They are central for triangle proofs,
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scale drawings, and indirect measurement.
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</p>
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<h2 id="congruence" class="book-prose__heading">Congruence</h2>
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<p>
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Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal.
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</p>
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<p>
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For triangles, common criteria:
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</p>
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<ul>
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<li>SSS (three sides equal)</li>
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<li>SAS (two sides and included angle equal)</li>
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<li>ASA / AAS (two angles and one corresponding side equal)</li>
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<li>RHS (right triangle: hypotenuse and one side equal)</li>
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</ul>
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<h2 id="similarity" class="book-prose__heading">Similarity</h2>
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<p>
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Similar figures have the same shape but possibly different sizes.
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</p>
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<ul>
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<li>Corresponding angles are equal.</li>
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<li>Corresponding sides are in one constant ratio (scale factor).</li>
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</ul>
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<p>
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Triangle similarity criteria:
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</p>
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<ul>
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<li>AA (two angles equal)</li>
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<li>SAS in proportion</li>
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<li>SSS in proportion</li>
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</ul>
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<h2 id="scale-factor" class="book-prose__heading">Scale factor relationships</h2>
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<p>
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If linear scale factor from shape A to B is \(k\):
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</p>
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<ul>
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<li>Lengths multiply by \(k\).</li>
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<li>Areas multiply by \(k^2\).</li>
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<li>Volumes multiply by \(k^3\).</li>
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</ul>
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<h2 id="using-similarity" class="book-prose__heading">Using similarity to solve problems</h2>
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<ol>
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<li>Identify corresponding vertices in correct order.</li>
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<li>Write ratio equation using matching sides.</li>
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<li>Solve unknown lengths/areas.</li>
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<li>Check ratio consistency across all pairs.</li>
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</ol>
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<h2 id="proofs" class="book-prose__heading">Congruence and similarity in proofs</h2>
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<p>
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Common proof flow:
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</p>
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<ul>
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<li>Show two triangles are congruent/similar by criteria.</li>
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<li>Conclude equal angles/sides or proportional segments.</li>
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<li>Use result to prove target statement.</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>State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).</li>
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<li>Match vertices in correct correspondence order.</li>
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<li>Do not mix congruence and similarity conditions.</li>
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<li>For ratio problems, keep fractions in consistent side order.</li>
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</ul>
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</div>
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