Master the art of shapes, angles, and dimensions with visual logic and expert theorems.
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<p>Geometry is not about measuring; it's about <strong class='text-primary-600'>Seeing</strong>. In competitive exams, 70% of questions can be solved by identifying the hidden shape or property without heavy calculation.</p>
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<strong>Weightage:</strong> High in SSC CGL, CHSL, and increasing in Banking Mains.
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<p>When a Transversal intersects two Parallel Lines, look for these <strong class='text-primary-600 font-bold'>3 Letters</strong>:</p>
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<h4 class='font-bold text-lg text-primary-700 mb-2'>1. The 'Z' Shape (Alternate Angles)</h4>
<p>Angles inside the 'Z' are always <span class='text-primary-600 font-bold'>EQUAL</span>.</p>
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<h4 class='font-bold text-lg text-primary-700 mb-2'>2. The 'F' Shape (Corresponding Angles)</h4>
<p>Angles in the same position (like steps of a ladder) are <span class='text-primary-600 font-bold'>EQUAL</span>.</p>
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<h4 class='font-bold text-lg text-red-700 mb-2'>3. The 'C' Shape (Co-interior Angles)</h4>
<p>Angles inside the 'C' sum to <span class='text-red-600 font-bold'>180┬░</span>.</p>
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<h4 class='text-xl font-bold text-gray-800 mb-4'>The 4 Centers (Don't confuse them!)</h4>
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<h5 class='font-bold text-primary-700 mb-2'>Centroid (G)</h5>
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<li>Formed by <strong>Medians</strong>.</li>
<li>Center of Mass.</li>
<li>Divides median in <strong class='text-primary-600'>2:1</strong>.</li>
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<h5 class='font-bold text-primary-700 mb-2'>Incenter (I)</h5>
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<li>Formed by <strong>Angle Bisectors</strong>.</li>
<li>Center of Incircle.</li>
<li>Equidistant from <strong class='text-primary-600'>sides</strong>.</li>
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<h5 class='font-bold text-primary-700 mb-2'>Circumcenter (O)</h5>
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<li>Formed by <strong>Perpendicular Bisectors</strong>.</li>
<li>Center of Circumcircle.</li>
<li>Equidistant from <strong class='text-primary-600'>vertices</strong>.</li>
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<h5 class='font-bold text-red-700 mb-2'>Orthocenter (H)</h5>
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<li>Formed by <strong>Altitudes</strong> (Heights).</li>
<li>Intersection of perpendiculars from vertices.</li>
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<h4 class='text-xl font-bold text-gray-800 mb-4'>Similarity (The Photo-Resize Rule)</h4>
<p class='mb-4 text-gray-600'>If $\Delta ABC \sim \Delta PQR$, then sides are proportional, but Areas follow the <strong class='text-red-600'>Square Rule</strong>:</p>
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$\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta PQR)} = (\frac{AB}{PQ})^2$
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<strong class='text-primary-700 block mb-1'>Tangent-Radius Rule</strong>
Radius is always <strong class='text-red-600'>Perpendicular (90┬░)</strong> to the Tangent at the point of contact. This creates Right Angled Triangles $\to$ Use Pythagoras!
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<strong class='text-pink-700 block mb-1'>The "Ice-Cream" Cone</strong>
Two tangents drawn from an external point are always <strong class='text-primary-600'>EQUAL</strong> in length ($PA = PB$).
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<strong class='text-primary-700 block mb-1'>Alternate Segment Theorem (The Hidden Equal)</strong>
The angle between a tangent and a chord is equal to the angle in the alternate segment. (Look for this in complex figures!).
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<h4 class='font-bold text-gray-800 mb-3'>The Golden Formula (Tangent-Secant)</h4>
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$PT^2 = PA \times PB$
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<p class='text-sm text-gray-500'>(Where $PT$ is tangent, $PAB$ is secant)</p>
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<p>For a regular polygon with <strong>n</strong> sides:</p>
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<th class='py-5 px-6 font-semibold border-b'>Property</th>
<th class='py-5 px-6 font-semibold border-b'>Formula</th>
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<tr><td class='py-5 px-6 font-medium'>Sum of Interior Angles</td><td class='py-5 px-6 text-primary-700 font-bold'>$(n-2) \times 180^\circ$</td></tr>
<tr><td class='py-5 px-6 font-medium'>Each Interior Angle</td><td class='py-5 px-6 text-primary-700 font-bold'>$\frac{(n-2) \times 180}{n}$</td></tr>
<tr><td class='py-5 px-6 font-medium'>Sum of Exterior Angles</td><td class='py-5 px-6 text-primary-700 font-bold'>$360^\circ$ (Always!)</td></tr>
<tr><td class='py-5 px-6 font-medium'>Each Exterior Angle</td><td class='py-5 px-6 text-primary-700 font-bold'>$\frac{360}{n}$</td></tr>
<tr><td class='py-5 px-6 font-medium'>Diagonals</td><td class='py-5 px-6 text-primary-700 font-bold'>$\frac{n(n-3)}{2}$</td></tr>
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