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AA, SAS, SSS: Proving Triangles Similar

Similar triangles need equal angles AND proportional sides — but you almost never have to check all of that. Three shortcuts (AA, SAS, SSS) let one or two pieces of evidence force the whole match, and a single missing piece, AAA's cousin SSA, is the cautionary tale.

What we are trying to avoid checking

From the previous guide you already carry the definition: two triangles are similar when their three angles match in pairs and their three sides are in proportion — same shape, possibly different size, related by a single scale factor k. Written out, similarity of triangle ABC and triangle DEF demands six facts at once: three angle equalities and three proportions of sides. That is a lot of evidence to gather. The whole point of this guide is that you almost never need to gather it.

The reason mirrors something you met one rung down, with congruence. Back there you learned that you do not verify all six measurements (three sides, three angles) to prove two triangles identical — three well-chosen pieces, in the right pattern, lock the rest. Similarity has its own version of exactly that miracle. A triangle is a rigid little machine: fix enough of it and the remaining parts have no freedom left. Our task is to find the smallest patterns of evidence that pin a triangle's shape — its angles and side ratios — while leaving its overall size free to float.

AA: two angles are enough

The headline shortcut is AA similarity: if two angles of one triangle equal two angles of another, the triangles are similar. Not three angles — two. The reason is a fact you proved in the triangles rung, the triangle angle sum: a triangle's angles add to 180 degrees. If two of the three angles match across the two triangles, the third is forced, because it is just 180 minus the other two. Two angles secretly hand you the third for free.

But matching all three angles only gives you equal angles — and similarity also demanded proportional sides. Why do equal angles drag the side ratios along? Picture two triangles with identical angles, the smaller laid inside the larger with one vertex shared and a side along a common direction. The third side of the small triangle is parallel to the third side of the big one (equal corresponding angles, from the previous rung's transversal facts). That parallel cut is exactly the side-splitter situation — the parallel line slices the two sides it crosses in the same ratio, and that shared ratio is the scale factor. So equal angles do not just resemble proportional sides; they manufacture them.

SAS and SSS: when sides do the talking

Sometimes you have lengths, not angles. Two more criteria, gathered under SAS and SSS similarity, let sides carry the proof. SSS similarity: if all three pairs of corresponding sides are in the same ratio, the triangles are similar — the matching angles then come along for free. SAS similarity: if two pairs of sides are in the same ratio and the angles squeezed between those two sides are equal, the triangles are similar. Notice the careful word between: the equal angle must be the one nestled between the two proportional sides, the included angle, exactly as in the congruence version one rung down.

triangle ABC  ~  triangle DEF  (proving similar)

SSS:  AB/DE = BC/EF = CA/FD          -> similar, ratio = k

SAS:  AB/DE = AC/DF                  (two sides in ratio)
      and m(angle A) = m(angle D)    (the INCLUDED angle)
                                     -> similar

AA:   m(angle A) = m(angle D)
      and m(angle B) = m(angle E)    -> similar (k from any side pair)
The three similarity tests side by side. The tilde ~ means 'is similar to'; the included angle in SAS sits between the two named sides.

A small worked example makes SAS concrete. Triangle ABC has AB = 4 and AC = 6 with m(angle A) = 50 degrees. Triangle DEF has DE = 6 and DF = 9 with m(angle D) = 50 degrees. Check the ratios: AB/DE = 4/6 = 2/3 and AC/DF = 6/9 = 2/3 — equal. The included angles, angle A and angle D, are both 50 degrees. SAS similarity fires: the triangles are similar with scale factor 2/3, and you now know the third side ratio BC/EF = 2/3 too, plus m(angle B) = m(angle E) and m(angle C) = m(angle F), without measuring any of them.

The traps: SSA, and getting the order wrong

Be honest about what does not work, because the failures are exactly where beginners lose marks. Two non-criteria deserve a warning, both relatives of the genuine tests. First, AAA versus AA: "all three angles equal" sounds stronger than AA, and it is true — but it is not a separate, better tool, because AA already forces the third angle. The real cautionary cousin lives on the congruence side: knowing three equal angles proves similarity but never congruence, since it says nothing about size. AAA is a similarity criterion, not a congruence one — do not promote it past its job.

Second, and sharper, is SSA — two sides and a non-included angle, the angle not squeezed between them. This is the famous broken criterion. With two sides fixed and an angle off to the side rather than between them, the third side can sometimes swing into two different positions, giving two genuinely different triangles. So SSA does not pin down a shape, and there is no SSA similarity rule. The fix is the same discipline as SAS demands: the angle you cite must sit between the two sides whose ratio you used. An angle in the wrong seat proves nothing.

Putting a proof together

Here is how a real similarity proof flows, using the most common setup: two triangles sharing an angle, with a line cutting across. Suppose in triangle ABC a line through points on AB and AC is drawn parallel to BC, meeting AB at P and AC at Q. We claim triangle APQ is similar to triangle ABC. The shared angle at A is one angle equality for free. The parallel line makes angle APQ equal to angle ABC (corresponding angles). That is two angle equalities — AA fires, and the triangles are similar. No lengths needed at all.

  1. Identify the candidate triangles and write the similarity in corresponding order: triangle APQ ~ triangle ABC (so P matches B, Q matches C).
  2. Collect angle equalities: angle A is shared, and angle APQ = angle ABC because PQ || BC cuts the transversal AB into equal corresponding angles.
  3. Two angle pairs match, so by AA similarity the triangles are similar — stop here, no sides required.
  4. Cash in the consequence: corresponding sides are now proportional, AP/AB = AQ/AC = PQ/BC, all equal to one scale factor k.

That last line is the real payoff, and it is what the very next guide grabs hold of. The proportion AP/AB = AQ/AC, born from this similarity, is the side-splitter theorem in disguise, and the same machinery delivers the geometric mean lurking inside a right triangle's altitude. So treat AA, SAS, and SSS not as three facts to memorise but as the three doors into a single, enormously useful room — once two triangles are known similar, every ratio you could want is yours at once.