Do you have to measure and compare all three sides and all three angles to confirm two triangles overlap perfectly? Actually, no. It turns out that matching just a handful of pieces of information is enough for the rest to automatically match too — these are called congruence conditions. Schools usually teach the three basics, SSS, SAS, and ASA, along with RHS and RHA, which only apply to right triangles.
SSS is when all three side lengths match. Once you fix the lengths of all three sides, the triangle's shape and size are pinned down to exactly one possibility. Build a triangle from three wooden chopsticks, and no matter how you nudge them around by hand, you'll find it can only ever settle into the same shape. SAS is when two side lengths and the angle between them match, and ASA is when one side length and the two angles at its ends match. All three guarantee that "this information alone uniquely determines the triangle."
So why only these particular combinations? Thinking about counterexamples makes it easier to understand. Matching two side lengths and an angle that isn't the one between them (SSA) doesn't guarantee congruence — two different triangle shapes can both satisfy the exact same condition. This is called the "ambiguous case," and if you actually draw it out with a ruler and protractor, you can see for yourself two triangles with different shapes but the same given condition. The same goes for matching all three angles (AAA). In that case, triangles with the same shape but different sizes can come out in endless variety. Two triangles of different sizes can be "similar," but they aren't "congruent."
Congruence conditions aren't just something confined to textbooks. It's exactly this property that's behind why triangular structures (trusses) are used in the framework of roofs and bridges. A triangle is the only polygon whose shape can never be squashed out of shape once its three side lengths are fixed. Push on a rectangular frame and it collapses into a parallelogram-like shape, but a triangle keeps its shape no matter how much force you apply, as long as the side lengths stay the same. That's why triangles show up without fail in structures where sturdiness matters — cranes, power-line towers, bicycle frames. The "triangulation" used in GPS and surveying works on a similar principle: knowing just the angles observed from two points is enough to uniquely pin down a third point's location, which is used to calculate distances.
When studying this with kids, the most effective approach is having them build each condition themselves. Hand them a protractor and ruler and have them draw a triangle from an SAS condition — say, "draw a triangle with sides 5cm and 7cm and a 60-degree angle between them" — then overlap it with a friend's drawing to check whether they're really congruent. Have them try drawing from an SSA condition instead (two sides and an angle not between them), and they'll witness firsthand that the same numbers can produce triangles of different shapes — an experience that drives home "why SSA can't be a congruence condition" far more convincingly than any explanation could.
Once you've learned all the congruence conditions, you'll feel just how efficient it is that judging whether two triangles are congruent only takes checking three pieces of information, instead of all six (3 sides, 3 angles). On our activity page, you can enter your own numbers to match the SSS, SAS, ASA, RHS, and RHA conditions, build a triangle, and overlap two triangles with your own eyes to check whether they're really congruent. Switching between conditions and guessing "will this combination work or not" before checking is also a fun way to test yourself.