Lesson 10: Angle-Angle Triangle Similarity

Property

If two triangles have two pairs of corresponding angles with the same measure, then the triangles are similar. This is known as the Angle-Angle (AA) similarity criterion.

If two triangles $\triangle ABC$ and $\triangle A'B'C'$ have two pairs of equal angles (e.g., $\angle A = \angle A'$ and $\angle B = \angle B'$), they can be mapped onto one another through a sequence of transformations (translation, rotation, reflection) followed by a dilation. This proves similarity.

Examples

• Triangle 1 has angles $45^\circ$ and $80^\circ$. Triangle 2 has angles $45^\circ$ and $55^\circ$. Since the third angle in Triangle 1 is $180-45-80=55^\circ$, both triangles have angles $45^\circ, 80^\circ, 55^\circ$. By AA similarity, they are similar.

Property

If two angles are known to be congruent (e.g., vertical angles or corresponding angles), you can set their algebraic expressions equal to each other. Solving the resulting equation for the variable allows you to find the angle measures needed to prove two triangles are similar by the Angle-Angle (AA) criterion.

Examples