Sss Similarity Theorem

SSS Similarity Theorem (Side-Side-Side)
The SSS Similarity Theorem states that if the three sides of one triangle are in proportion to the three sides of another triangle, then the triangles are similar. This means they have the same shape but not necessarily the same size.
What It Means
When two triangles have all corresponding side lengths in the same ratio, their angles will also match, even if one triangle is larger or smaller. The triangles are essentially scaled versions of each other.
Example
Suppose you have triangle ABC and triangle DEF:
AB is 4 units, and DE is 8 units


BC is 5 units, and EF is 10 units


AC is 6 units, and DF is 12 units


All corresponding side ratios are 1:2 (4/8, 5/10, 6/12). Since all three pairs are proportional, triangle ABC is similar to triangle DEF by the SSS Similarity Theorem.
Key Points to Remember
All three sides must be proportional. It’s not enough to compare just two.


No need to check angles. This theorem relies only on side lengths.


Similar ≠ Congruent. Similar triangles have the same shape, but not necessarily the same size. If the corresponding sides are exactly equal (not just proportional), the triangles are congruent, and the SSS Congruence Theorem applies instead.


Why It's Useful
The SSS Similarity Theorem is commonly used in geometry to prove that triangles are similar when angle information is missing. It’s especially helpful in real-world problems like scaling models, analyzing maps, or solving complex geometric proofs.
In Simple Terms
If you can show that each side of one triangle matches the corresponding side of another triangle in the same ratio, then the triangles are similar. That’s the power of the SSS Similarity Theorem.