Angle-angle similarity

Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the angle at a point. If $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$, then $ABC$ and $A'B'C'$ are similar.

Remarks

By the angle sum of a triangle, the third angles are then equal too. So the criterion is also called angle-angle-angle similarity. Another criterion for similarity is side-angle-side: two equal ratios of sides, and equal angles between those sides. In hyperbolic geometry, triangles with equal angles are congruent, so the criterion has a different meaning there.

Explanations

L1 human

The shape of a triangle depends only on its angles. If two angles match, the third one matches too, since the three always add up to $\pi$. So two triangles with matching angles are scaled copies of each other.