Two equal angles are enough (AA)
You never have to measure a single side to know two triangles are similar.
The last lesson checked three ratios to confirm two triangles were similar. This lesson finds a shortcut that never needs a ruler at all: two triangles, ABC and A′B′C′, all six points draggable and independent of each other, with the angle at A and at A′, and at B and at B′, read live.
Two triangles, ABC and A'B'C', all six points draggable and independent of each other. A live reading gives the angle at A and at A', and at B and at B', says whether AA currently holds, and separately whether the triangles turn out similar — always agreeing with what AA predicts. This needs JavaScript switched on.
Try to break it on purpose — find a drag that keeps the two marked angles matching but stops the triangles from being similar. There isn’t one: two triangles that share two equal angles satisfy the AA condition — short for “angle, angle” — and that turns out to be all it takes.
Why two angles force everything else to match
If two angles of triangle A′B′C′ match two angles of triangle ABC, the third angle matches too: all three angles of a triangle sum to 180 degrees, so once two are pinned down the third has no freedom left at all.
Now imagine scaling triangle A′B′C′ by whatever factor makes A′B′ exactly the same length as AB — scaling changes lengths but never angles, so all three angles still match after the scale. The scaled triangle now has one side equal to AB, with the same two angles either side of it that AB itself has. Two angles and the side between them: ASA, so the scaled triangle is congruent to ABC.
Congruent triangles have every side equal, so the scaled triangle’s other two sides exactly match BC and CA. Since every side of A′B′C′ was scaled by the same one factor to get there, B′C′ and C′A′ must have been that same factor away from BC and CA all along — the same ratio A′B′ was.
This is why the last lesson’s three-ratio check, correct as it is, is more work than similarity actually requires. From here on, two matching angles is the test this course reaches for first.
Angles can be read with a protractor from a distance, or estimated from a photo, in situations where actual side lengths are impossible to measure at all — a shortcut that only works because a triangle’s shape genuinely depends on nothing else.
Testing pairs of angles against the AA condition
Triangle ABC has angles 40° and 70°. Triangle DEF has angles 40° and 70°, matched to the same two vertices. Are they similar?
Yes — AA is satisfied directly, with no side length needed at all.
Triangle ABC has angles 50° and 60°. Triangle DEF has angles 50° and 65°. Are they similar?
No. The first two angles are equal to each other, but the second pair is not — 60° does not equal 65° — so AA fails, and nothing about the sides can rescue it.
Why does checking two angles work, when checking only one clearly would not?
One matching angle leaves the other two free to be anything that sums correctly with it — endlessly many different shapes share one angle. Two matching angles use up all the freedom a triangle’s shape has, which is exactly why the third one is forced to match as well.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.