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sss Similarity

"SSS" stands for "side-side-side." But what exactly does this mean in the context of triangles? As we''ll soon find out, this represents a special rule we can use to help us determine if two triangles are congruent. But how exactly does this special rule work? What can it teach us about math?

Side-side-side similarity explained

Side-side-side similarity states that:

• When two triangles have corresponding sides of equal length, then we know these triangles are congruent.

Visualizing side-side-side similarity

We can visualize side-side-side congruence more easily by examining the following diagram:

As we can see, these triangles have sides that are shaded in corresponding colors. We know that because all of these sides are shaded with similar colors, they have the same lengths. This means that these two triangles are also congruent.

For triangle similarity, we have a different rule called Side-Side-Side Similarity:

When two triangles have corresponding sides that are proportional then we know these triangles are similar.

In the context of triangles, these ratios are slightly more complex.

For example, we might get two triangles:

One might have side lengths 17, 11.6, and 18.4

The other might have side lengths 8.5, 5.8, and 9.2

These two triangles are similar because their lengths are proportional. Why? Because we can prove that they are proportional with the following formula:

AB/PQ = BC/QR = AC/PR

In this formula, one triangle is ABC while the other is PQR.

Let''s plug in those values:

17/8.5 = 11.6/5.8 = 18.4/9.2

Are they proportional?

Yes, because each fraction can be simplified as 2, and they are all equivalent (or proportional) to one another.

Finding missing values

In some cases, it is possible to find out whether triangles are similar even if you are missing the value of one side. For example, let''s say we have two right triangles with two given sides each. We can use the Pythagorean theorem

(a^2 + b^2 = c^2)
to find the third side and then determine whether the triangles are similar. We also know by the side-angle-side rule that when two sides of a triangle and their included angles are fixed, two proportional sides must also mean that the third side is automatically proportional as well.

Topics related to the sss Similarity

Triangles

Triangle Proportionality Theorem

sss Postulate

Flashcards covering the sss Similarity

Common Core: High School - Geometry Flashcards

Practice tests covering the sss Similarity

Common Core: High School - Geometry Diagnostic Tests