The measure of Angle MIJ is 125.
From the image, we can see that angle JIK and angle JIM are adjacent angles, which means they share a side.
Since adjacent angles add up to 180 degrees, we can write the following equation:
JIK + JIM = 180 degrees
We are given that JIK = 55 degrees, so we can substitute this value into the equation to solve for JIM:
55 degrees + JIM = 180 degrees
JIM = 180 degrees - 55 degrees
JIM = 125 degrees
Therefore, the measure of angle MIJ is 125 degrees.