HL & why SSA/AAA fail

So far every shortcut has worked: SSS and SAS, then ASA and AAS, each pinned a triangle down to a single shape and size. This page is about the ones that don't — and the honest reason a right angle turns one of them back into a guarantee.

There are two famous near-misses. Both look like they ought to work:

• SSA — two sides and an angle, but the angle is not between the two sides.
• AAA — all three angles, and nothing else.

The way to trust a criterion is to try to break it. Here you finally can: these two arrangements break, and seeing how they break is the whole lesson.

SSA — the side that swings like a door

Keep two sides and one angle, but put the angle off to the side instead of between them. Now one of your sides is hinged at a corner with nothing fixing where its far end lands — it can swing like a door. As you lengthen that swinging side, count how many times it reaches the base.

Start on SSA in the tool below. Drag the swinging side BC slider slowly up from the bottom and watch the dashed swing-circle:

You should see three distinct regimes as the swinging side grows:

1. Too short → no triangle. The side can't stretch down to the base at all. The swing-circle floats above it and never touches.
2. Just long enough → exactly one triangle. The circle becomes tangent to the base — it kisses it at a single point. Notice the little square that appears: the side has met the base at a right angle. Hold onto that.
3. Longer still → two triangles. The circle now cuts the base in two places, so the same three measurements describe two genuinely different triangles. Switch between Solution 1 and Solution 2 to compare them — same angle at $A$, same side $AB$, same swinging side $BC$, but a different third side and a different triangle. Press Name it here and the tool reports the verdict it reads from the geometry: SSA is not a valid congruence criterion.

That middle case — exactly one triangle — only happens at one special length, the length where the swing meets the base perpendicularly.

HL — the right angle that kills the second solution

Look again at regime 2. The ambiguity disappears at exactly the moment the swinging side stands perpendicular to the base. A right angle sits precisely on the boundary between "two triangles" and "no triangle," so it leaves room for only one.

That is the whole idea behind HL (Hypotenuse–Leg). HL is SSA with the angle forced to be a right angle:

$\text{In right triangles, if the hypotenuses are equal and one pair of legs is equal, the triangles are congruent.} \quad (\text{HL})$

Switch the tool to HL and try to nudge the corner: it won't budge. With the right angle fixed, the hypotenuse and one leg, Pythagoras forces the other leg to one exact length — there is no second triangle to find. HL is the one place where "two sides and a non-included angle" becomes trustworthy, and it is trustworthy because the right angle lands exactly on SSA's boundary.

AAA — same shape, any size

The other near-miss keeps all three angles and no sides at all. Switch the tool to AAA and drag the Triangle size slider.

Every triangle in the family has the identical three angles — the identical shape — but the size grows and shrinks freely. There is no length anywhere to anchor the triangle, so the angles alone can never tell you how big it is. A triangle and its enlargement on a photocopier have equal angles and are plainly not congruent. Press Name it and the tool reports: AAA is not a congruence criterion — it gives similarity (same shape), not congruence (same shape and size).

The whole scorecard, complete

With this page the table you have been building across the unit is finished — five arrangements that lock a triangle, and two that only look like they should:

Next, on the CPCTC & first proofs page, you finally use a congruence criterion inside a proof — once two triangles are congruent, every pair of corresponding parts must be equal too.