What is the measure of angle b in the figure below?

Since its a right triangle, we can use the Pythagorean theorem and see what kind of right triangle it is (30-60-90 or 45-45-90):

a^2 + b^2 = c^2
c^2 - a^2 = b^2
(3sqrt2)^2 - 3^2 = b^2
9*2 - 9 = b^2
18 - 9 = b^2
b^2 = 9
b = sqrt9
b = 3

Since a and b are equal (3), we know this is a 45-45-90 triangle (it is characteristic for these triangles to have 2 equal sides).