According to the SAS theorem, what must be true for two triangles to be similar?

The SAS theorem, which stands for Side-Angle-Side similarity, states that two triangles are similar if two pairs of corresponding sides are in proportion and the angle between those sides is congruent. This means that if you take two triangles and identify two sides of each that are proportional (meaning their ratios are equal) and the angle formed between those sides is the same in both triangles, then those triangles are similar.

This concept is crucial in geometry because it allows for the comparison of the shapes of triangles, even when their sizes differ. Similar triangles retain the same shape; thus, their angles are congruent, and their corresponding sides are in proportion. This makes option C the correct choice.

Other options do not accurately capture the essence of the SAS theorem as required for similarity. For instance, simply having equal angles (option B) does not ensure similarity unless the pairs of corresponding sides are proportional. Additionally, while all triangles indeed have angle sums of 180 degrees (option D), this fact applies universally to all triangles and does not specifically relate to the SAS theorem's conditions. Lastly, the notion that sides opposite angles must be equal (option A) is not related to similarity but more to congruence among triangles.