What are the Triangle Similarity Theorems?

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Similar triangles are the same shape but not necessarily the same size. When triangles are similar, they have many of the same properties and characteristics. Triangle similarity theorems specify the conditions under which two triangles are similar, and they deal with the sides and angles of each triangle. Once a specific combination of angles and sides satisfy the theorems, you can consider the triangles to be similar.

TL;DR (Too Long; Didn't Read)

There are three triangle similarity theorems that specify under which conditions triangles are similar:

  • If two of the angles are the same, the third angle is the same and the triangles are similar.
  • If the three sides are in the same proportions, the triangles are similar.
  • If two sides are in the same proportions and the included angle is the same, the triangles are similar.

The AA, AAA and Angle-Angle Theorems

If two of the angles of two triangles are the same, the triangles are similar. This becomes clear from the observation that the three angles of a triangle must add up to 180 degrees. If two of the angles are known, the third can be found by subtracting the two known angles from 180. If the three angles of two triangles are the same, the triangles have the same shape and are similar.

The SSS or Side-Side-Side Theorem

If all three sides of two triangles are the same, the triangles are not only similar, they are congruent or identical. For similar triangles, the three sides of two triangles only have to be proportional. For example, if one triangle has sides of 3, 5 and 6 inches and a second triangle has sides of 9, 15 and 18 inches, each of the sides of the larger triangle is three times the length of one of the sides of the smaller triangle. The sides are in proportion to each other, and the triangles are similar.

The SAS or Side-Angle-Side Theorem

Two triangles are similar if two of the sides of two triangles are proportional and the included angle, or the angle between the sides, is the same. For example, if two of the sides of a triangles are 2 and 3 inches and those of another triangle are 4 and 6 inches, the sides are proportional, but the triangles may not be similar because the two third sides could be any length. If the included angle is the same, then all three sides of the triangles are proportional and the triangles are similar.

Other Possible Angle-Side Combinations

If one of the the three triangle similarity theorems is fulfilled for two triangles, the triangles are similar. But there are other possible side-angle combinations that may or may not guarantee similarity.

For the configurations known as angle-angle-side (AAS), angle-side-angle (ASA) or side-angle-angle (SAA), it doesn't matter how big the sides are; the triangles will always be similar. These configurations reduce to the angle-angle AA theorem, which means all three angles are the same and the triangles are similar.

However, the side-side-angle or angle-side-side configurations don't ensure similarity. (Don't confuse side-side-angle with side-angle-side; the "sides" and "angles" in each name refer to the order in which you encounter the sides and angles.) In certain cases, such as for right-angled triangles, if two sides are proportional and angles that are not included are the same, the triangles are similar. In all other cases, the triangles may or may not be similar.

Similar triangles fit into each other, can have parallel sides and scale from one to the other. Determining whether two triangles are similar using the triangle similarity theorems is important when such characteristics are applied to solve geometrical problems.

References

About the Author

Bert Markgraf is a freelance writer with a strong science and engineering background. He has written for scientific publications such as the HVDC Newsletter and the Energy and Automation Journal. Online he has written extensively on science-related topics in math, physics, chemistry and biology and has been published on sites such as Digital Landing and Reference.com He holds a Bachelor of Science degree from McGill University.

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