Angles, yes.  Sides, no.  "Similar" triangles are those that can be mapped to each other by translation, rotation, inversion, or uniform scaling.  The scaling operation makes the sides longer by the same scale factor.  Since all the sides increase by the same factor, the angles stay the same.

You can prove (and may have proved in your homework) that sides in proportion implies that the angles are the same, or that angles being the same means the sides are in proportion.  So no matter how you define similar triangles, it always means that corresponding angles are congruent but on the ratio of side lengths is fixed.