base can be any of the sides of the triangle. It doesn't really matter which one you take. Base is usually the one that has a 90° angle with the known altitude that goes to the vertex that is opposite to the base side. If none of the altitudes are given then it doesn't matter which side you take as the base.

When constructing triangles you usually take the side that has a known angle next to it as a base. In your first case the angle is between the sides so any of them can be taken as the base.. In your second case only mAB has a known angle (m<A) next to it so you take mAB as the base.

At least that's what I think.