ASA & AAS

On the previous page you locked a triangle with three sides (SSS) and with two sides and the angle between them (SAS). Both shortcuts leaned heavily on sides. But a triangle is made of angles too — and it turns out that two angles and a single side are also enough to pin the whole figure. That is the lesson of this page: the two angle-driven shortcuts, ASA and AAS.

If you know one side of a triangle and the two angles that "stand on" it, how much freedom is left? Set the side down, send a ray out from each end at its fixed angle, and ask: how many ways can the triangle close?

The game: try to build a different one

The inquiry is the same one you played last page. You are handed a set of parts, drawn on a triangle in green, and your job is to build a different triangle from the same parts — grab a corner and hunt for some other triangle that still honours every green mark. If every nudge springs back, those parts determine the triangle, and you have found a real congruence shortcut.

Start with ASA (two angles and the side between them), then switch to AAS (two angles and a side not between them). Push the corner. Watch what happens. Only when you are convinced should you press Name it.

As before, the tool stays non-judgmental while you explore and reads its verdict straight from the geometry — it cannot call a shortcut valid unless the math agrees that the parts leave the triangle no room to move.

ASA — a side and its two angles meet at one point

Put a segment down. From one end, draw a ray at the first angle; from the other end, draw a ray at the second angle. Two non-parallel rays in a plane cross at exactly one point — and that single intersection is the triangle's third vertex. There is no second place for the rays to meet, so the triangle is forced.

$\text{If } \angle A\cong\angle D,\ \overline{AB}\cong\overline{DE},\ \angle B\cong\angle E, \text{ then } \triangle ABC \cong \triangle DEF. \quad (\text{ASA})$

The word included is doing the same work it did for SAS, just with the roles swapped. ASA is Angle–Side–Angle: the side sits between the two angles you know. That is the configuration the tool draws — the green side is flanked by the two green angles, and their rays close to a single apex.

AAS — because two angles fix the third

Switch the tool to AAS and try to flex it: it springs back just like ASA. The reason is the angle-sum argument above. Knowing $\angle A$ and $\angle B$ forces

$\angle C = 180° - \angle A - \angle B,$

so you really know all three angles and one side. A known side wedged between two known angles is exactly ASA — which you already saw is rigid. AAS is not a separate miracle; it is ASA wearing a disguise, and the $180°$ angle sum is what pulls off the mask.

Where ASA and AAS fit in the whole set

ASA and AAS are the third and fourth of the five valid congruence criteria. Here is the running scorecard, with this page's two rows now filled in — five shortcuts that lock a triangle, and two plausible-looking arrangements that do not:

Notice the pattern across the four valid criteria you now hold: each one fixes three parts, and each one includes at least one side. That last point is the seed of the next page — three angles and no side (AAA) fixes only the shape, not the size, so it cannot guarantee congruence.

This page filled in the ASA and AAS rows. Next comes the right-angle special case HL, set alongside the two famous failures — SSA, where a side can swing to two landing spots, and AAA, where the triangle scales freely: HL & why SSA/AAA fail.