Definition: The SSS criterion states that if three sides of one triangle are congruent to three sides of another, then the two triangles are congruent.
What does SSS say?
You do not need to check any angles. If all three pairs of corresponding sides match in length, the triangles are congruent, and all three pairs of angles then match as well.
Why does it work?
Three side lengths determine a triangle’s shape uniquely. You cannot bend it without changing a side, unlike a four-sided frame, which can be pushed into a different shape.
Example. Triangle ABC has sides 7, 10, and 12. Triangle DEF has sides 12, 7, and 10. Pair the sides by size: 7 with 7, 10 with 10, 12 with 12. All three pairs match, so triangle ABC is congruent to triangle DEF by SSS. In coordinates, use the distance formula to find all three sides of each triangle.
What mistakes are frequent?
Matching sides in the wrong order, so the congruence statement lists the vertices incorrectly. The order of letters must match the correspondence. Also, three angles do not work: that is AAA, which gives similar triangles of possibly different sizes.