10.5 Similarity of Triangles
Definition 10.6 Two triangles \(\triangle ABC\) and \(\triangle A'B'C'\) are similar, also called equiangular, if they have congruent angles.
\(\angle ABC\) equals \(\angle A'B'C'\), \(\angle BAC\) equals \(\angle B'A'C'\), and \(\angle BCA\) equals \(\angle B'C'A'\).
When two triangles are similar, we will write \(\triangle ABC \sim \triangle A'B'C'\).
Theorem 10.14 Triangles \(\triangle ABC\) and \(\triangle A'B'C'\) are similar if and only if any two of the corresponding angles are equal.
Theorem 10.15 (SSS Similarity) Triangles \(\triangle ABC\) and \(\triangle A'B'C'\) are similar if and only if the corresponding sides have lengths in the same ratio: \[\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{AC}{A'C'}\]
Theorem 10.16 (SAS Similarity) Triangles \(\triangle ABC\) and \(\triangle A'B'C'\) are similar if and only if two sides have lengths in the same ratio and the angles included between these sides have the same measure.