To find the measure of angle ABC, we first need to determine the measure of angle AMK.

Since AK = 1/3 AB, we can see that triangle AKM is similar to triangle ABC (by AA similarity). This means that the ratio of corresponding sides is the same in both triangles.

Let x be the length of AB, then AK = x/3. Therefore, AM = 2x/3.

Now since angle AKM = 5, we can set up the following equation:

tan(5) = opposite/adjacent
tan(5) = AM / AK
tan(5) = (2x/3) / (x/3)
tan(5) = 2

Using an inverse tangent function, we find that angle AMK = 63.43 degrees.

Since triangle ABC and triangle AKM are similar, we know that angle ABC = 3 angle AMK. Therefore, angle ABC = 3 63.43 = 190.29 degrees.

Therefore, the measure of angle ABC is 190.29 degrees.