Please see the two attached files.
The student is wrong. We could measure an angle because it doesn't matter how long the two rays are. We could put two points A and B on the two rays, and the angle doesn't change. The angle between the 2 rays is same as the angle AOB, which it could be measured with a protractor.
In geometry, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).
For example, two angles ABC and DFE, say, are "equal" if and only if there are exactly the same angle: that would mean that B and F are just different ways of referring to the same point and that A is on one of the rays FD or FE and C is on the other one.
Two angles are "congruent" if and only if they have the same measure.
Generally speaking, in mathematics, we use the term "equal" to mean "(possibly different) ways of ...
The expert finds the sides of similar triangles. The points when angles two are located.