What congruence means
The first lesson of Phase 3 asks a question rather than proving a claim — how much information does it take to pin a shape down?
Every lesson so far in this course has proved something. This one asks a question instead, and spends the whole lesson getting a feel for it, because the next lesson needs the question to already make sense: how much information does it actually take to know a triangle’s exact shape?
A triangle, A, B and C. A and B are fixed. Only C is draggable.
A triangle, A, B and C, where A and B are fixed and only C is draggable. AB's length never changes, but AC and BC change freely as C is dragged, into countless different triangles, all still sharing that one same side length. This needs JavaScript switched on.
All of those different triangles share that one side, AB, and nothing else — proof that a single length leaves a triangle’s shape wide open.
Two shapes that are exactly the same size and the same shape, so that one could be picked up and placed exactly on top of the other, are called congruent. Knowing a shape’s measurements “pins it down” when every triangle that satisfies those measurements is congruent to every other one — no two of them can look different from each other, even though many different triangles could still occupy many different places or point in many different directions.
How much is enough
One length is clearly not enough: this widget just showed a single length, AB, shared by an entire family of triangles that plainly do not all look alike. But some information is enough. If a triangle’s three side lengths, all of them, were fixed instead of just one, dragging C would have nowhere left to go: swap AB for all three lengths, and every triangle satisfying them turns out to be congruent to every other one, wherever it happens to sit or however it happens to be turned. The question this lesson raises — how much information fixes a shape — turns out to have a short list of answers, and finding that list is the next lesson’s job.
“Enough information” is not a vague idea to wave at — it is a question with a precise, checkable answer, and it is worth sitting with the question itself before being handed the answer. Naming the answer before the question has been felt would let “SSS, SAS, ASA” become three memorised words instead of the answer to something the student already wondered. The next lesson names exactly which combinations are enough; this one makes sure the question is felt first, not just stated.
Congruence and the wobble test are the same idea, arrived at from two different directions. The wobble test asks “does this fact keep holding as the givens move”; congruence asks “does this shape stay identical, however it is built, as long as these measurements are fixed.” Both are questions about what varies and what does not, under the same kind of dragging this whole course has used from its very first widget.
Two triangles are congruent. What is guaranteed to be true about their corresponding angles?
They are guaranteed to be equal, one for one — not just the sides. Congruence means the whole shape matches, angles included, not only the lengths that happen to have been listed as given.
A triangle has one side of length 5. Is that enough to know its area?
No. This lesson’s widget showed exactly why: a single fixed length still allows the other two sides, and the angle between them, to vary freely, and a triangle’s area depends on more than one length alone.
Could two triangles share all three angles without being congruent?
Yes — this is worth sitting with before the next lesson answers it properly. Two triangles can have identical angles while being completely different sizes, one a scaled-up copy of the other. Matching angles alone is a different, weaker claim than congruence, and the next lesson is precise about which combinations of measurements avoid that trap.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.