Triangle Similarity – Comprehensive Guide

Triangle similarity is one of the key concepts in geometry. Two triangles are similar if they have the same angles and their corresponding sides are in the same ratio. In this guide, we will explain everything you need to know.

Table of Contents

• What is similarity of geometric figures
• Similarity ratio (scale factor)
• Triangle similarity theorems
• Examples
• Related Articles
• Practice

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What is similarity of geometric figures

Two geometric figures are similar if they have the same shape but can differ in size. This means that:

• All corresponding angles are equal
• All corresponding sides are in the same ratio

For example, any two squares are similar, and any two circles are similar. However, this does not automatically hold for triangles — certain conditions must be met.

👉 Detailed explanation: Similarity of Geometric Figures

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Similarity ratio (scale factor)

If two triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ are similar, there exists a number $k$ that expresses the ratio of corresponding sides:

This number $k$ is called the similarity ratio or scale factor.

• If $k>1$ — triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is larger (enlargement)
• If $k<1$ — triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is smaller (reduction)
• If $k=1$ — the triangles are congruent (the same size)

👉 Detailed explanation with examples: Similarity Ratio (Scale Factor)

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Triangle similarity theorems

To prove that two triangles are similar, we use three fundamental theorems:

SSS Theorem (Side – Side – Side)

If the ratios of all three pairs of corresponding sides are equal, the triangles are similar.

SAS Theorem (Side – Angle – Side)

If the ratios of two pairs of corresponding sides are equal and the included angles are equal, the triangles are similar.

AA Theorem (Angle – Angle)

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. The third angle is automatically equal because the sum of angles in a triangle is $180°$.

👉 Detailed explanation of all three theorems: Triangle Similarity Theorems – SSS, SAS, AA

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Examples

Quick example: Triangle $ABC$ has sides $a=3$ cm, $b=4$ cm, $c=5$ cm. Triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ has sides $a^{\prime }=6$ cm, $b^{\prime }=8$ cm, $c^{\prime }=10$ cm.

Side ratios: $\frac{6}{3}=\frac{8}{4}=\frac{10}{5}=2$

The triangles are similar with a scale factor of $k=2$.

👉 More solved examples: Triangle Similarity – Solved Examples

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Related Articles

• Similarity of figures: Similarity of Geometric Figures
• Similarity ratio: Similarity Ratio (Scale Factor)
• Similarity theorems: Triangle Similarity Theorems – SSS, SAS, AA
• Solved examples: Triangle Similarity – Solved Examples

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Practice

• 🔢 Similarity ratio: Calculate the Similarity Ratio
• 📐 Missing sides: Find the Missing Side
• 🔍 Similarity theorems: Identify the Similarity Theorem
• 🧮 Mixed problems: Mixed Similarity Problems