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Side-Side-Side Similarity

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Side-Side-Side Similarity

Study Guide

Key Definition

Side-Side-Side (SSS) Similarity states that for triangles ABC and XYZ, if their corresponding sides are proportional—where a = AB, b = BC, c = CA and x = XY, y = YZ, z = ZX—such that $\frac{a}{x} = \frac{b}{y} = \frac{c}{z}$, then $\triangle ABC \sim \triangle XYZ$.

Important Notes

SSS Similarity involves proportional sides, not equal sides.
$\triangle$ represents a triangle.
Proportionality is essential for establishing similarity.
Congruence requires exact equality of corresponding sides.
Visual aids can help in understanding proportional relationships.

Mathematical Notation

$\sim$ means similarity

$\cong$ means congruence

$\triangle$ represents a triangle

$\frac{a}{b}$ represents a fraction

$\angle$ represents an angle

Use notation: $\triangle ABC \sim \triangle DEF$ to denote similar triangles

Why It Works

SSS Similarity works because if the ratios of corresponding sides of two triangles are equal, their angles are automatically equal, making the triangles similar.

Remember

SSS Similarity is about proportionality, so always check the ratios of corresponding sides.

Quick Reference

Property:$\frac{a}{x} = \frac{b}{y} = \frac{c}{z} \Rightarrow \triangle ABC \sim \triangle XYZ$

Understanding Side-Side-Side Similarity

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In SSS similarity, we compare the ratios of corresponding sides of two triangles. For example, if triangle ABC has sides 3, 4, 5 and triangle XYZ has sides 6, 8, 10, then comparing side ratios gives 3/6 = 4/8 = 5/10 = 1/2, so the triangles are similar.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following sets of triangles are similar by SSS similarity?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have two model triangles, one with side lengths 17, 11.6, and 18.4, and another with side lengths 8.5, 5.8, and 9.2. Are these triangles similar?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given two triangles, one with side lengths 1, 2, 3 and another with side lengths 2, 4, 6, prove their similarity using SSS.

Challenge Quiz

Single Choice Quiz

Advanced

If $\triangle ABC$ has sides 6, 8, 10 and $\triangle DEF$ has sides 3, 4, 5, are they similar?

Recap

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Review key concepts and takeaways