Two angles are called “congruent” if they are equal. Two triangles are called “similar” if all their corresponding angles are congruent. Two triangles are called “congruent” if all their corresponding sides are congruent.
Do you think that if two triangles are similar then they are congruent? Do you think that if two triangles are congruent then they are similar?
There is no tool in GeoGebra for copying an angle, but you can use the compass tool to do it in a couple of steps. Chat about why this works. Watch this YouTube clip: http://www.youtube.com/watch?v=qngWUFgyyHc. It gives a clear view of one way to copy an angle.
When you drag point A, do the sides of triangle DEN stay congruent to the corresponding sides of ABC? Do the vertex angles of triangle DEN stay congruent to the corresponding angles of ABC?