Saintly

Similar Triangles

24 Problems • 3 sub-topics

Adalynn Le • 5/27/2026

Introduction

Similarity, careful not to be confused with congruency, is a geometric relationship between two figures that are "scale models" of each other. It is probably the single most common topic on the AMC 10 for geometry because of it's adaptability and versatility. Similar triangles are a particularly important facet of similar polygons because they are easy to spot and yet painstakingly nuanced and extensive.

select_check_box What is it?
Similar triangles are triangles that have the same shape but not nescessarily the same size. They are said to be scaled versions of eachother by some constant \(k\). Thus, the corrresponding sides are all scaled by some factor \(k\), but the angle measures remain the same.
select_check_box Why Study?
Every polygon can be created by some combination of triangles, so triangles are really the building blocks of geometry. Similar triangles are perhaps the most common type of similar figures because there are so many and they are so adaptable, yet they are still quintissentially the most simple shape.

Similarity vs Congruence

All congruent polygons are also similar to each other, but not all similar polygons are congruent. Similar polygons need only to have the same shape, not nescessarily the same size. Congruent polygons must have all of the exact same dimensions. This is important to remember on the AMC 10 and in math in general. There is quite a bit of vocabulary associated with these concepts and geometry in general, and to forget or mix them up can cost you points on the AMC 10.

Conditions for Similarity

There are three main conditions for determining if two triangles are similar. A similar triangle will meet all three of these by definition, but to prove similarity you only need to prove one.

AA (Angle-Angle)

Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle. By the Angle Sum Theorem, we know the interior angles of a triangle must sum to \(180^\circ\), so knowing two angles of a triangle can let you find the last one by subtracting both angles from \(180\). If two angles are matching, the remaining angle is nescessarily matching. We know that a similar triangle will have all three angles matching, thus having just two angles that are congruent will dictate similarity. Furthermore, the reason that three congruent angles enforces similarity is that, by trigonometry and law of sines and cosines, it forces the relationships between sides and thus makes it similar.

SSS (Side-Side-Side)

Two triangles are similar if all three sides of one triangle are proportional to the corresponding sides of another triangle. The definition of similarity requires that all sides are proportional, but there is no real proof for this. What is important to realize is that the sides must all be proportional by the same scale value \(k\).

SAS (Side-Angle-Side)

Two triangles are similar if two sides are proportional and the included angle is congruent. When you know the lengths of two sides and their included angle, there is only one way to make a triangle because you just connect the sides. Thus, when the two side lengths are proportional and have the same included angle, you know that the remaining side must be similar and proportional as well.

Interactive Playground: Similar Triangles

Adjust the slider to change the scale factor. Notice how the side lengths change, but the ratios of corresponding sides always remain perfectly equal.

▵ABC (Reference)
A B C c = 4 b = 3 a = 5
▵DEF (Dynamic)
D E F f = 6.0 e = 4.5 d = 7.5
Corresponding Side Proportions:
DE⁄AB = 6.0 / 4 = 1.50
DF⁄AC = 4.5 / 3 = 1.50
EF⁄BC = 7.5 / 5 = 1.50

Conclusion

Similar Triangles are the building blocks of advanced and fundamental geometry. Similar figures in general are just scaled up or down versions of each other. Similar triangles are popular because of their versatility and complexity. They are easy to spot, but they can be used in a variety of ways and can be hidden in plain sight. They are a fundamental topic on the AMC 10 and are worth mastering for any student interested in geometry or competition math.

Question 1:

Loading question...

Welcome Back!

Access your progress across all your devices

Username

Need Some Help?

Contact us through our form

Privacy Policy

Data Collection
  • We use local storage to track and save your ELO on your device
  • We use Supabase backend to track overall acuracy on questions
  • We trackk individual user usage on our website, pages, etc. by tracking time spent and interaction
    • How we use your data
  • To personalize your learning experience
  • We never sell your data to third parties
    • Children's Privacy
  • Saintly is intended for students. We encourage parents to monitor their children's internet usage. We do not knowingly collect personal information from children under 13 without parental consent (per COPPA guidelines).

    • Terms and Conditions

  • By using Saintly you agree to the terms and conditions
  • You may not scrape the site or re-distribute our unique interactive code without permission.
  • Be "Saintly." No harassment, no attempts to hack the site, and no using the platform to cheat on active competitions.
  • Math is hard! While we strive for 100% accuracy, Saintly is provided "as is." We aren't responsible if a specific formula explanation doesn't lead to a perfect AMC score (though we hope it does!).
  • Saintly and its volunteers are not liable for any damages arising from your use of the site.