Question

How does dilation affect angle measures?

Answer 1

Dilation (scaling) does not affect angle measure. It remains the same.
That is, an image of an angle transformed by scaling is an angle of the same measure as an original.

Explanation:

Dilation (scaling) is the main basis for similarity in geometry.
It transforms any line into the one parallel to it.
Therefore, an image of an angle scaled to another location is another angle with correspondingly parallel sides.

As we know, angles with correspondingly parallel sides are congruent and are measured the same.

For details and proof of the above statements please refer to Unizor followed by menu items Geometry - Similarity - Angles.