Problem:
The difference in the measures of two complementary angles is 120. Find the measures of the angles.

Solution:
To solve this problem, we need to understand what complementary angles are. Complementary angles are two angles whose measures add up to 90 degrees. Let's assume one of the angles is x degrees. The other angle will then be (90 - x) degrees.

Step 1: Define the problem:
We are given that the difference in the measures of two complementary angles is 120 degrees. We need to find the measures of the angles.

Step 2: Formulate a plan:
Since the difference between the measures of the angles is given, we can set up an equation to solve for the unknown angle.

Step 3: Solve the problem:
Let's assume one of the angles is x degrees. Therefore, the other angle is (90 - x) degrees. According to the problem, the difference in their measures is 120 degrees.

So, we can set up the equation:
x - (90 - x) = 120

Simplifying the equation:
x - 90 + x = 120
2x - 90 = 120
2x = 120 + 90
2x = 210
x = 210/2
x = 105

Therefore, one angle measures 105 degrees. The other angle can be found by subtracting 105 from 90:
90 - 105 = -15

However, angles cannot have negative measures. So, we need to check our solution.

Step 4: Check the solution:
If we add the measures of the angles, we should get 90 degrees. Let's check:
105 + (-15) = 90

The sum of the measures of the angles is indeed 90 degrees.

Step 5: Write the solution:
The measures of the angles are 105 degrees and -15 degrees. However, since angles cannot have negative measures, the only valid solution is that one angle measures 105 degrees and the other angle measures 90 - 105 = -15 degrees.