SSS & SAS

On the last page we defined congruence through motion: two triangles are congruent when a rigid motion carries one exactly onto the other. That is the precise idea — but in practice you almost never get to pick a triangle up and slide it. You get a few measurements: a side here, an angle there. So the real question becomes:

How many parts of a triangle do you have to pin down before the triangle has no choice but to be a single shape and size?

A triangle has six parts — three sides and three angles. It turns out you never need all six. Some three of them are enough to lock the whole triangle. The trick is discovering which threes work, and the honest way to discover that is to try to break them.

The game: try to build a different one

Here is the inquiry. You are handed a set of parts, drawn on a triangle in green. Your job is to build a different triangle from the same parts — to grab a corner and find some other triangle that still honours every green mark. If you can't — if every nudge springs back — then those parts determine the triangle, and you have found a genuine congruence shortcut.

Start with SSS (all three sides), then switch to SAS (two sides and the angle between them). Push the corner. Watch what happens. Only when you are convinced should you press Name it.

Notice that the tool never tells you the answer up front. It lets you prod the figure first, stays non-judgmental while you explore ("it won't budge"), and only names the criterion when you decide you have seen enough. The verdict it gives is read straight from the geometry — the figure cannot claim a shortcut is valid unless the math agrees.

SSS — three sides lock it

Give a triangle its three side lengths and it has nowhere to flex. This is the property that makes triangles the workhorse of bridges, roof trusses, and bicycle frames: a triangle of fixed-length bars is rigid, while a four-bar square sags into a rhombus the moment you lean on it. There is no second triangle hiding in three fixed sides.

$\text{If } \overline{AB}\cong\overline{DE},\ \overline{BC}\cong\overline{EF},\ \overline{CA}\cong\overline{FD}, \text{ then } \triangle ABC \cong \triangle DEF. \quad (\text{SSS})$

SAS — and the angle must be between

Now drop one side and keep two sides with the angle wedged between them. Set the two sides, fix the angle at their shared corner, and the third side is forced to connect their far ends — there is only one way to close it up.

$\text{If } \overline{AB}\cong\overline{DE},\ \angle A\cong\angle D,\ \overline{AC}\cong\overline{FD}, \text{ then } \triangle ABC \cong \triangle DEF. \quad (\text{SAS})$

The word included is doing real work here. SAS is Side–Angle–Side: the angle sits between the two sides you know. Keep that order in mind — when the angle drifts to a position not between the two sides, the guarantee can fail, and that failure (the infamous SSA) is the whole story of a later page.

The whole set, at a glance

SSS and SAS are the first two of five valid congruence criteria. Here is the complete scorecard you are building toward across this unit — five shortcuts that lock a triangle, and two plausible-looking arrangements that do not:

This page nailed down the first two rows. The next pages take on ASA & AAS, then the right-angle special case HL alongside the two famous failures, SSA and AAA.