---
tag: 0NZVJYZVTZ
kind: statement
role: theorem
name: Angle-angle similarity
aliases: [aa-similarity, aaa-similarity]
msc: [51M04]
uses: [0YF6QXXWB5, 0GJAJC6TNZ, 05MDKQBEAF]
sources:
  - url: https://en.wikipedia.org/wiki/Similarity_(geometry)
    locator: Similar triangles
---

## 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|0GJAJC6TNZ]].
If $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$, then $ABC$ and $A'B'C'$ are [[similar|0YF6QXXWB5]].

## Remarks

By the [[angle sum of a triangle|05MDKQBEAF]], 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.