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Side-Side-Side Similarity

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Side-Side-Side Similarity

Study Guide

Key Definition

Side-Side-Side Similarity states that when two triangles have corresponding sides that are proportional, the triangles are similar. This is expressed as $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$.

Important Notes

The SSS Similarity is different from SSS Congruence.
Proportional sides mean the ratios of corresponding sides are equal.
If $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$, then the triangles are similar.
SSS Similarity can be used to prove triangles similar in real-world applications.
Remember to confirm proportionality, not just equality.

Mathematical Notation

$=$ represents equality

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

$\triangle$ represents a triangle

$\cong$ represents congruence

$\sim$ represents similarity

Remember to use proper notation when solving problems

Why It Works

When the ratios of all three pairs of corresponding sides of two triangles are equal, each pair of corresponding angles is congruent. This can be shown using the Law of Cosines, which proves that equal side ratios preserve the measures of angles, ensuring the triangles are similar.

Remember

Use $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$ to check for similarity.

Quick Reference

SSS Similarity:$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$

Understanding Side-Side-Side Similarity

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

SSS similarity states that two triangles are similar if their corresponding sides are in proportion. For example, if triangle ABC and triangle DEF satisfy $AB/DE = BC/EF = CA/FD$, then the triangles have equal corresponding angles and are similar.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following pairs of triangles are similar by SSS Similarity if $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Consider two triangles ABC and DEF in a park. Triangle ABC has side lengths AB = 3, BC = 4, and CA = 5. Triangle DEF has side lengths DE = 6, EF = 8, and FD = 10. Determine whether these triangles are similar by SSS.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the similarity of two triangles with sides $x, y, z$ and $2x, 2y, 2z$.

Challenge Quiz

Single Choice Quiz

Advanced

Identify the pair of triangles that are similar by SSS with sides in ratio $\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$.

Recap

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Review key concepts and takeaways