What Is AAA in Geometry?

Geometry is a study of points, lines and angles and how they combine to make different shapes. Similarity and congruence between two figures are important concepts in geometry.

Definition

AAA refers to the angle-angle-angle combination. It compares the angles of one triangle with the angles of another to determine whether congruence or similarity exists between the figures.

Theorem

The AAA theorem is a similarity theorem. It states that if any two angles of two triangles have the same shape, then they are similar. They need not have the same size. For example, take two equilateral triangles, one with side length 2 and the other with side length 3. Because they are equilateral triangles, each angle is 60 degrees. The shape of one triangle is similar to the shape of the other, but because the side lengths are different, one triangle is larger than the other.

Congruence and Similarity

Two figures are congruent only if all sides and the included angles are exactly the same. Two figures are similar if they have the same shape. It is not necessary that they also have the same size.

Differences

Congruency has more stringent requirements because both shape and size have to match for two figures to be congruent. Similarity is the weaker concept because it allows difference in size.

AAA in Non-Euclidian Space

In hyperbolic geometry and spherical geometry, AAA is a sufficient condition for congruence on any given curvature. In these geometries, an angle is a function of size.

References

• Mathwords: Similar
• Learning Math: Similarity