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Mathematics

Triangle Similarity

AA, SAS, SSS Criteria, Proportions, and the Geometric Mean — A TLDR Primer

Similar triangles show up on every geometry test, the SAT, the ACT, and standardized state exams — and they trip students up not because the ideas are hard, but because nobody slowed down to explain them clearly. If you (or your student) can set up the proportion but keep getting the wrong side, or you mix up AA with SAS similarity, or the altitude-to-the-hypotenuse diagram looks like three triangles stacked inside each other and that's somehow useful, this guide is for you.

**TLDR: Triangle Similarity** is a focused, short-by-design guide that covers everything a high school or early college student needs: what similarity actually means (and how it differs from congruence), the three similarity criteria — AA, SAS, and SSS — each with a worked proof, and a systematic method for setting up proportions to solve for missing sides. It continues with the Side-Splitter Theorem for parallel lines, the geometric mean relationships that come from right triangle similarity, and a practical chapter on indirect measurement, shadows, mirrors, and map scaling — exactly the real-world problems that appear on tests and in life.

This is not a textbook. There are no padded chapters, no review questions you'll never use, no fluff. Every section leads with the one thing you need to know, then backs it up with worked examples and common-mistake warnings. If you need a high school geometry test prep book that respects your time, this is it.

Pick it up, work through the examples, and walk into your next exam ready.

What you'll learn
  • Define similarity and distinguish it from congruence.
  • Apply the AA, SAS, and SSS similarity criteria to prove two triangles are similar.
  • Use similarity ratios to find missing side lengths and scale figures.
  • Apply triangle similarity theorems (Side-Splitter, Triangle Proportionality, geometric mean in right triangles).
  • Use similar triangles to solve real-world problems involving indirect measurement.
What's inside
  1. 1. What Similarity Means
    Defines similar triangles, the role of corresponding parts, the scale factor, and how similarity differs from congruence.
  2. 2. The Three Similarity Criteria: AA, SAS, SSS
    Presents and contrasts the three shortcuts for proving two triangles similar, with worked proofs for each.
  3. 3. Solving for Missing Sides with Proportions
    Shows how to set up and solve proportions from similar triangles, including overlapping and shared-vertex configurations.
  4. 4. Side-Splitter and Parallel Lines
    Develops the Triangle Proportionality (Side-Splitter) Theorem and its converse, plus parallel-line transversal results.
  5. 5. Right Triangle Similarity and the Geometric Mean
    Explores how the altitude to the hypotenuse creates three similar triangles and yields the geometric mean relationships.
  6. 6. Using Similarity in the Real World
    Applies similar triangles to indirect measurement, shadows, mirrors, maps, and scaling problems students encounter on tests and in life.
Published by Solid State Press
Triangle Similarity cover
TLDR STUDY GUIDES

Triangle Similarity

AA, SAS, SSS Criteria, Proportions, and the Geometric Mean — A TLDR Primer
Solid State Press

Triangle Similarity

AA, SAS, SSS Criteria, Proportions, and the Geometric Mean — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Similarity Means
  2. 2 The Three Similarity Criteria: AA, SAS, SSS
  3. 3 Solving for Missing Sides with Proportions
  4. 4 Side-Splitter and Parallel Lines
  5. 5 Right Triangle Similarity and the Geometric Mean
  6. 6 Using Similarity in the Real World
Chapter 1

What Similarity Means

Two triangles are similar when they have the same shape but not necessarily the same size. Every angle in one triangle matches an angle in the other, and the sides, while possibly different in length, all shrink or grow by the same factor.

That last part is the core idea. Similarity is a statement about proportion, not about measurement.

Corresponding Parts

When you say two triangles are similar, you have to say which parts match up. A correspondence is just a pairing — vertex to vertex, angle to angle, side to side. The parts that are paired under a given correspondence are called corresponding parts.

Take triangle $ABC$ and triangle $DEF$. If we write the similarity as

$\triangle ABC \sim \triangle DEF$

the symbol $\sim$ means "is similar to." The order of the letters is not decorative — it encodes the correspondence. $A$ corresponds to $D$, $B$ corresponds to $E$, $C$ corresponds to $F$. That means:

  • Corresponding angles are equal: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.
  • Corresponding sides are the sides between those matched vertices: $AB$ corresponds to $DE$, $BC$ corresponds to $EF$, $AC$ corresponds to $DF$.

A common mistake is to match up the triangles however looks convenient and then write a similarity statement that doesn't reflect that pairing. The letters in $\triangle ABC \sim \triangle DEF$ must respect the actual angle matches — if you swap two letters, the ratios you write later will be wrong.

The Scale Factor

Equal angles tell you the shapes match. The sides go further and tell you how much the triangle grew or shrank.

The scale factor (sometimes called the ratio of similarity) is the single number $k$ such that every side of one triangle is $k$ times the corresponding side of the other. If $\triangle ABC \sim \triangle DEF$, then:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$

About This Book

If you're staring down a unit test on congruence and proportions, prepping for the SAT or ACT geometry sections, or trying to make sense of what your teacher said about AA, SAS, and SSS proofs, this book is for you. It also works as a high school geometry test prep book for students who need a clean, fast review before finals or a standardized exam.

This triangle similarity geometry study guide covers every core idea: recognizing similar triangles, writing similarity statements, how to solve proportions with similar triangles, the side splitter theorem and parallel lines geometry problems, and geometric mean altitude hypotenuse explained step by step. A concise overview with no filler.

Read straight through in one sitting. Work every worked example yourself before reading the solution. At the end, the practice problems — including indirect measurement similar triangles problems — let you test whether you can apply the ideas, not just recognize them.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon