If two triangles have three corresponding angles and three corresponding sides that are equal in measure, are the two triangles necessarily congruent? why?

Yes, they are.

If the triangles have three corresponding angles equal means that they are either similar or congruent.

The definition of congruent means that the sides have the same length. That is precisely what the statement says.

So, being the lengths of the corresponding angles the same the two triangles are identical, this is they are congruent.