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2026-01-01
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<p>Last updated on<strong>October 6, 2025</strong></p>
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<p>Last updated on<strong>October 6, 2025</strong></p>
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<p>In geometry, similar triangles are triangles that have the same shape but may differ in size. This means they have equal corresponding angles and proportional corresponding side lengths. In this topic, we will learn the formulas related to similar triangles.</p>
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<p>In geometry, similar triangles are triangles that have the same shape but may differ in size. This means they have equal corresponding angles and proportional corresponding side lengths. In this topic, we will learn the formulas related to similar triangles.</p>
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<h2>List of Math Formulas for Similar Triangles</h2>
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<h2>List of Math Formulas for Similar Triangles</h2>
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<p>The properties<a>of</a>similar triangles are crucial in<a>geometry</a>. Let’s learn the<a>formulas</a>to identify and calculate the corresponding sides and angles of similar triangles.</p>
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<p>The properties<a>of</a>similar triangles are crucial in<a>geometry</a>. Let’s learn the<a>formulas</a>to identify and calculate the corresponding sides and angles of similar triangles.</p>
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<h2>Similarity Criteria for Triangles</h2>
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<h2>Similarity Criteria for Triangles</h2>
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<p>Triangles are similar if they satisfy certain criteria:</p>
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<p>Triangles are similar if they satisfy certain criteria:</p>
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<p>1. AA Criterion (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.</p>
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<p>1. AA Criterion (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.</p>
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<p>2. SSS Criterion (Side-Side-Side): If the corresponding sides of two triangles are in<a>proportion</a>, the triangles are similar.</p>
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<p>2. SSS Criterion (Side-Side-Side): If the corresponding sides of two triangles are in<a>proportion</a>, the triangles are similar.</p>
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<p>3. SAS Criterion (Side-Angle-Side): If one angle of a triangle is equal to one angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.</p>
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<p>3. SAS Criterion (Side-Angle-Side): If one angle of a triangle is equal to one angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.</p>
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<h2>Proportionality in Similar Triangles</h2>
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<h2>Proportionality in Similar Triangles</h2>
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<p>In similar triangles, the corresponding sides are proportional. The proportionality can be expressed using the formula: \([ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ] w\)here \((a_1, b_1, c_1) \)are the sides of one triangle and\( (a_2, b_2, c_2) \)are the sides of the other triangle.</p>
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<p>In similar triangles, the corresponding sides are proportional. The proportionality can be expressed using the formula: \([ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ] w\)here \((a_1, b_1, c_1) \)are the sides of one triangle and\( (a_2, b_2, c_2) \)are the sides of the other triangle.</p>
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<h2>Area Ratio of Similar Triangles</h2>
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<h2>Area Ratio of Similar Triangles</h2>
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<p>The area of similar triangles is proportional to the<a>square</a>of the<a>ratio</a>of their corresponding side lengths. The formula for the ratio of the areas is: \([ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{a_1}{a_2}\right)^2 ]\)</p>
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<p>The area of similar triangles is proportional to the<a>square</a>of the<a>ratio</a>of their corresponding side lengths. The formula for the ratio of the areas is: \([ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{a_1}{a_2}\right)^2 ]\)</p>
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<h2>Importance of Similar Triangles in Geometry</h2>
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<h2>Importance of Similar Triangles in Geometry</h2>
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<p>Similar triangles are fundamental in geometry and are used to solve complex problems. Here are some significant uses:</p>
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<p>Similar triangles are fundamental in geometry and are used to solve complex problems. Here are some significant uses:</p>
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<ul><li>They help in determining distances indirectly using<a>proportions</a>. </li>
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<ul><li>They help in determining distances indirectly using<a>proportions</a>. </li>
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</ul><ul><li>They are used in<a>trigonometry</a>to solve problems involving right triangles. </li>
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</ul><ul><li>They are used in<a>trigonometry</a>to solve problems involving right triangles. </li>
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</ul><ul><li>Understanding similarity helps in grasping concepts related to scale models and maps.</li>
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</ul><ul><li>Understanding similarity helps in grasping concepts related to scale models and maps.</li>
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</ul><h2>Tips and Tricks to Identify Similar Triangles</h2>
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</ul><h2>Tips and Tricks to Identify Similar Triangles</h2>
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<p>Identifying similar triangles can be tricky. Here are some tips: -</p>
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<p>Identifying similar triangles can be tricky. Here are some tips: -</p>
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<ul><li>Look for equal angles in the triangles. - Check if the sides are in proportion. </li>
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<ul><li>Look for equal angles in the triangles. - Check if the sides are in proportion. </li>
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</ul><ul><li>Use the criteria (AA, SSS, SAS) to confirm similarity. </li>
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</ul><ul><li>Use the criteria (AA, SSS, SAS) to confirm similarity. </li>
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</ul><ul><li>Practice with different orientations and configurations of triangles to get comfortable with identifying similarity.</li>
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</ul><ul><li>Practice with different orientations and configurations of triangles to get comfortable with identifying similarity.</li>
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</ul><h2>Common Mistakes and How to Avoid Them While Using Similar Triangles</h2>
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</ul><h2>Common Mistakes and How to Avoid Them While Using Similar Triangles</h2>
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<p>Students often make errors when working with similar triangles. Here are some mistakes and the ways to avoid them.</p>
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<p>Students often make errors when working with similar triangles. Here are some mistakes and the ways to avoid them.</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>If triangle ABC is similar to triangle DEF and AB = 4 cm, BC = 6 cm, and DE = 8 cm, find EF.</p>
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<p>If triangle ABC is similar to triangle DEF and AB = 4 cm, BC = 6 cm, and DE = 8 cm, find EF.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>EF = 12 cm</p>
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<p>EF = 12 cm</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Since the triangles are similar, the corresponding sides are proportional.</p>
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<p>Since the triangles are similar, the corresponding sides are proportional.</p>
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<p>\([ \frac{AB}{DE} = \frac{BC}{EF} \implies \frac{4}{8} = \frac{6}{EF} \implies EF = \frac{6 \times 8}{4} = 12 \text{ cm} ]\)</p>
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<p>\([ \frac{AB}{DE} = \frac{BC}{EF} \implies \frac{4}{8} = \frac{6}{EF} \implies EF = \frac{6 \times 8}{4} = 12 \text{ cm} ]\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>Two similar triangles have sides in the ratio 3:4. If the area of the smaller triangle is 27 cm², find the area of the larger triangle.</p>
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<p>Two similar triangles have sides in the ratio 3:4. If the area of the smaller triangle is 27 cm², find the area of the larger triangle.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The area of the larger triangle is 48 cm².</p>
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<p>The area of the larger triangle is 48 cm².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The ratio of the areas of similar triangles is the square of the ratio of their sides.</p>
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<p>The ratio of the areas of similar triangles is the square of the ratio of their sides.</p>
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<p>\( [ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{3}{4}\right)^2 \implies \frac{27}{\text{Area}_2} = \frac{9}{16} \implies \text{Area}_2 = \frac{27 \times 16}{9} = 48 \text{ cm}^2 ]\)</p>
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<p>\( [ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{3}{4}\right)^2 \implies \frac{27}{\text{Area}_2} = \frac{9}{16} \implies \text{Area}_2 = \frac{27 \times 16}{9} = 48 \text{ cm}^2 ]\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>In similar triangles, the side lengths are in the ratio 5:7. If the shortest side of the larger triangle is 21 cm, find the length of the corresponding side in the smaller triangle.</p>
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<p>In similar triangles, the side lengths are in the ratio 5:7. If the shortest side of the larger triangle is 21 cm, find the length of the corresponding side in the smaller triangle.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The corresponding side in the smaller triangle is 15 cm.</p>
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<p>The corresponding side in the smaller triangle is 15 cm.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the side ratio: \([ \frac{5}{7} = \frac{x}{21} \implies x = \frac{5 \times 21}{7} = 15 \text{ cm} ]\)</p>
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<p>Using the side ratio: \([ \frac{5}{7} = \frac{x}{21} \implies x = \frac{5 \times 21}{7} = 15 \text{ cm} ]\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Triangle XYZ is similar to triangle UVW. If XY = 9 cm, YZ = 12 cm, and UV = 6 cm, find VW.</p>
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<p>Triangle XYZ is similar to triangle UVW. If XY = 9 cm, YZ = 12 cm, and UV = 6 cm, find VW.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>VW = 8 cm</p>
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<p>VW = 8 cm</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the proportionality of corresponding sides:\([ \frac{XY}{UV} = \frac{YZ}{VW} \implies \frac{9}{6} = \frac{12}{VW} \implies VW = \frac{12 \times 6}{9} = 8 \text{ cm} ]\)</p>
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<p>Using the proportionality of corresponding sides:\([ \frac{XY}{UV} = \frac{YZ}{VW} \implies \frac{9}{6} = \frac{12}{VW} \implies VW = \frac{12 \times 6}{9} = 8 \text{ cm} ]\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>The sides of two similar triangles are in the ratio of 2:3. If the perimeter of the smaller triangle is 30 cm, find the perimeter of the larger triangle.</p>
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<p>The sides of two similar triangles are in the ratio of 2:3. If the perimeter of the smaller triangle is 30 cm, find the perimeter of the larger triangle.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The perimeter of the larger triangle is 45 cm.</p>
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<p>The perimeter of the larger triangle is 45 cm.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.</p>
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<p>The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.</p>
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<p>\([ \frac{\text{Perimeter}_1}{\text{Perimeter}_2} = \frac{2}{3} \implies \frac{30}{\text{Perimeter}_2} = \frac{2}{3} \implies \text{Perimeter}_2 = \frac{30 \times 3}{2} = 45 \text{ cm} ]\)</p>
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<p>\([ \frac{\text{Perimeter}_1}{\text{Perimeter}_2} = \frac{2}{3} \implies \frac{30}{\text{Perimeter}_2} = \frac{2}{3} \implies \text{Perimeter}_2 = \frac{30 \times 3}{2} = 45 \text{ cm} ]\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Similar Triangles Math Formulas</h2>
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<h2>FAQs on Similar Triangles Math Formulas</h2>
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<h3>1.What is the AA criterion for similar triangles?</h3>
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<h3>1.What is the AA criterion for similar triangles?</h3>
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<p>The AA criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.</p>
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<p>The AA criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.</p>
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<h3>2.How do you use the SSS criterion for similarity?</h3>
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<h3>2.How do you use the SSS criterion for similarity?</h3>
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<p>The SSS criterion states that if the corresponding sides of two triangles are in proportion, the triangles are similar.</p>
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<p>The SSS criterion states that if the corresponding sides of two triangles are in proportion, the triangles are similar.</p>
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<h3>3.What is the ratio of areas of similar triangles?</h3>
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<h3>3.What is the ratio of areas of similar triangles?</h3>
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<p>The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths.</p>
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<p>The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths.</p>
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<h3>4.How do you find a missing side in similar triangles?</h3>
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<h3>4.How do you find a missing side in similar triangles?</h3>
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<p>To find a missing side in similar triangles, use the proportionality of corresponding sides. Set up a proportion<a>equation</a>and solve for the missing length.</p>
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<p>To find a missing side in similar triangles, use the proportionality of corresponding sides. Set up a proportion<a>equation</a>and solve for the missing length.</p>
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<h3>5.What is the SAS criterion for similar triangles?</h3>
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<h3>5.What is the SAS criterion for similar triangles?</h3>
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<p>The SAS criterion states that if one angle of a triangle is equal to one angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.</p>
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<p>The SAS criterion states that if one angle of a triangle is equal to one angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.</p>
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<h2>Glossary for Similar Triangles Math Formulas</h2>
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<h2>Glossary for Similar Triangles Math Formulas</h2>
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<ul><li><strong>Similar Triangles:</strong>Triangles that have the same shape but different sizes, with equal corresponding angles and proportional sides.</li>
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<ul><li><strong>Similar Triangles:</strong>Triangles that have the same shape but different sizes, with equal corresponding angles and proportional sides.</li>
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</ul><ul><li><strong>AA Criterion:</strong>A condition for similarity where two angles of one triangle are equal to two angles of another triangle.</li>
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</ul><ul><li><strong>AA Criterion:</strong>A condition for similarity where two angles of one triangle are equal to two angles of another triangle.</li>
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</ul><ul><li><strong>SSS Criterion:</strong>A condition for similarity where the corresponding sides of two triangles are in proportion.</li>
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</ul><ul><li><strong>SSS Criterion:</strong>A condition for similarity where the corresponding sides of two triangles are in proportion.</li>
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</ul><ul><li><strong>SAS Criterion:</strong>A condition for similarity where one angle is equal in two triangles, and the including sides are proportional.</li>
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</ul><ul><li><strong>SAS Criterion:</strong>A condition for similarity where one angle is equal in two triangles, and the including sides are proportional.</li>
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</ul><ul><li><strong>Proportionality:</strong>In similar triangles, the corresponding side lengths are proportional.</li>
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</ul><ul><li><strong>Proportionality:</strong>In similar triangles, the corresponding side lengths are proportional.</li>
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</ul><h2>Jaskaran Singh Saluja</h2>
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</ul><h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>