SSS, SAS, ASA, agreed
The last lesson asked how much information fixes a shape. This one names the three answers.
One length was not enough. This lesson names three combinations that are, and shows one of them directly: match a triangle’s three sides, and there is nowhere left for its shape to differ.
Two triangles: ABC, fixed, and DEF, entirely draggable.
Two triangles: ABC, fixed, with sides 3, 4 and 5 units, and DEF, entirely draggable. Drag D, E and F until DEF's three sides match ABC's three sides, in any order: once all three match, the two triangles are guaranteed congruent, however differently they are placed or turned. This needs JavaScript switched on.
DEF does not need to sit in the same place as ABC, or point the same way — only the three lengths matter. Try to make DEF’s shape genuinely different from ABC’s while keeping all three matched. It cannot be done.
Matching all three sides is called SSS — side, side, side. Two more combinations are agreed to be equally enough: SAS, two sides and the angle between them, and ASA, two angles and the side between them. Each is a different minimum amount of information that leaves no freedom for a triangle’s shape to vary.
Why each combination pins the shape down
SSS leaves nothing free because a triangle’s shape is entirely decided by how its three vertices sit relative to each other, and three lengths pin down every one of those relative positions at once: fix AB’s length, then C has to sit exactly on a circle of the right radius around A and exactly on another circle of the right radius around B, and two circles cross in at most two points, which are mirror images of each other — the same triangle, flipped, not a different one.
SAS works by the same kind of argument, one step earlier: fix two sides and the angle trapped between them, and the third side is not free either — it is whatever length happens to close the gap between the two fixed sides at that fixed angle, which is one specific length, not a range of them. Once the third side is forced, the triangle is an SSS case in disguise.
ASA works by fixing the shape from the other end: two angles and the side between them decide where the third vertex has to be, because two lines leaving the ends of a fixed side, at two fixed angles, only ever cross in one place. There is no freedom left for a third vertex to wander to.
“The minimum enough” is worth knowing exactly, not approximately. This course could stop at “more information generally pins a shape down more” and leave students guessing which combinations actually work. Geometry does not need to be guessed at here: SSS, SAS and ASA are a short, complete, checkable list, and knowing the list means never wondering whether a particular set of measurements was enough — it either matches one of these three shapes, or it was not.
Three matching angles, with no side given at all, is deliberately absent from this list — the last lesson asked you to sit with exactly that gap. The next two lessons, on isosceles triangles, are where SAS gets its first real job: proving a claim about angles by constructing a matching pair of triangles, rather than only checking whether two given triangles already match.
A triangle has sides 5, 7 and 9. Another triangle also has sides 5, 7 and 9, listed in a different order around its vertices. Are they congruent?
Yes. SSS does not care about vertex order or which side is called which — three matching lengths, any correspondence, forces the same shape.
Two triangles share two equal sides and one equal angle, but the equal angle is not the one between the two equal sides. Are they guaranteed congruent?
Not by SAS — that specific arrangement (two sides and a non-included angle) is not on the agreed list, which is exactly why the list has to be learned precisely rather than approximately. It can fail to pin down a unique triangle in a way SAS’s included-angle version cannot.
Why is AAA (three matching angles, no sides at all) not on this list?
Because three matching angles fix a triangle’s shape but not its size: a small triangle and a much larger one can share every angle while being completely different sizes. AAA guarantees the same shape scaled, not the same triangle — a different, weaker claim than congruence.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.