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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Congruency and Similarity in Geometry</title>
<style>
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<body>
<header>
<h1>Congruency and Similarity in Geometry</h1>
</header>
<div class="content">
<h2>1. Congruency</h2>
<p>Congruent shapes are exactly the same in terms of size and shape. If two shapes are congruent, one can be transformed into the other by moving it (translation), rotating it, or flipping it (reflection), without altering its size or shape.</p>
<h3>1.1. Definition of Congruent Figures</h3>
<p>Two geometric figures are said to be congruent if they have:</p>
<ul>
<li> The same size.</li>
<li> The same shape.</li>
<li> Their corresponding sides and angles are equal.</li>
</ul>
<p>If two figures A and B are congruent, we write: <code>A ≅ B</code>.</p>
<h3>1.2. Properties of Congruency</h3>
<ul>
<li><strong>Corresponding Sides are Equal:</strong> In two congruent triangles, the corresponding sides have equal lengths.</li>
<li><strong>Corresponding Angles are Equal:</strong> In two congruent triangles, the corresponding angles have equal measures.</li>
</ul>
<h3>1.3. Congruence in Triangles</h3>
<p>The congruence of triangles can be determined using different criteria, such as:</p>
<ul>
<li><strong>Side-Side-Side (SSS):</strong> If three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.</li>
<li><strong>Side-Angle-Side (SAS):</strong> If two sides and the angle between them in one triangle are equal to two sides and the angle between them in another triangle, then the triangles are congruent.</li>
<li><strong>Angle-Side-Angle (ASA):</strong> If two angles and the side between them in one triangle are equal to two angles and the side between them in another triangle, then the triangles are congruent.</li>
<li><strong>Angle-Angle-Side (AAS):</strong> If two angles and a non-included side in one triangle are equal to two angles and the corresponding non-included side in another triangle, then the triangles are congruent.</li>
<li><strong>Right Angle-Hypotenuse-Side (RHS):</strong> In right triangles, if the hypotenuse and one other side are equal in both triangles, the triangles are congruent.</li>
</ul>
<h3>1.4. Congruence in Other Geometric Figures</h3>
<p>Congruence applies not only to triangles but also to other geometric shapes, such as circles, quadrilaterals, or polygons. Two figures are congruent if their corresponding sides and angles are equal.</p>
<h2>2. Similarity</h2>
<p>Two figures are similar if they have the same shape but not necessarily the same size. The corresponding angles of similar figures are equal, and the corresponding sides are proportional.</p>
<h3>2.1. Definition of Similar Figures</h3>
<p>Two geometric figures are similar if:</p>
<ul>
<li> Their corresponding angles are equal.</li>
<li> Their corresponding sides are in proportion (the ratio of the lengths of corresponding sides is constant).</li>
</ul>
<p>If two figures A and B are similar, we write: <code>A ∼ B</code>.</p>
<h3>2.2. Properties of Similarity</h3>
<ul>
<li><strong>Corresponding Angles are Equal:</strong> In two similar triangles, the corresponding angles are equal.</li>
<li><strong>Corresponding Sides are Proportional:</strong> In two similar triangles, the ratio of the lengths of corresponding sides is constant.</li>
</ul>
<h3>2.3. Criteria for Similarity in Triangles</h3>
<ul>
<li><strong>Angle-Angle (AA) Criterion:</strong> If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.</li>
<li><strong>Side-Angle-Side (SAS) Criterion:</strong> If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, the triangles are similar.</li>
<li><strong>Side-Side-Side (SSS) Criterion:</strong> If the corresponding sides of two triangles are proportional, the triangles are similar.</li>
</ul>
<h3>2.4. Similarity in Other Geometric Figures</h3>
<p>Similarity applies to all shapes, not just triangles. Two polygons are similar if:</p>
<ul>
<li> Their corresponding angles are equal.</li>
<li> The lengths of their corresponding sides are proportional.</li>
</ul>
<h2>3. Formulas Related to Congruency and Similarity</h2>
<h3>3.1. Congruency Formulas</h3>
<ul>
<li><strong>Triangle Congruence (SSS):</strong> If the sides of two triangles are equal, the triangles are congruent.</li>
<li><strong>Triangle Congruence (SAS):</strong> If two sides and the included angle are equal, the triangles are congruent.</li>
<li><strong>Right Triangle Congruence (RHS):</strong> If the hypotenuse and one leg are equal, the triangles are congruent.</li>
</ul>
<h3>3.2. Similarity Formulas</h3>
<ul>
<li><strong>Ratio of Corresponding Sides:</strong> In similar triangles, the ratio of the corresponding sides is constant.</li>
<li><strong>Area Ratio of Similar Figures:</strong> The ratio of the areas of two similar figures is the square of the ratio of their corresponding sides.</li>
<li><strong>Perimeter Ratio of Similar Figures:</strong> The ratio of the perimeters of two similar figures is the same as the ratio of their corresponding sides.</li>
</ul>
<h3>3.3. Other Similarity Applications</h3>
<ul>
<li><strong>Similar Rectangles:</strong> The ratio of corresponding sides in two similar rectangles is constant.</li>
</ul>
</div>
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<p>© 2025 Yoseph Feyisa Wegi | All Rights Reserved</p>
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