The difference between two supplementary angles is 121.54. What are the measures of the two angles?

The definition of supplementary angles is that the two angles add up to 180. Let's call the angles angle x and angle y, so that x + y = 180 degrees.

We also know that the difference between these two angles is 10 degrees. Therefore we can say that x - y = 10. Now we have a system of equations.

To solve the system of equations, let's first solve for x in the second equation (you can solve for y as well, whichever makes it easier for you). This gives us the equation: x = 10 + y.

Now that we have that, we can plug in "10+y" for the x in the first equation to get:

(10+y) + y = 180.

Now we just have to solve for y:

(10+y) + y = 180  -->  10 + 2y = 180  -->  2y = 180-10  -->  2y = 170  -->  y = 170/2  -->  y = 85

Now that we have solved for y, we can plug in the value of 85 for y into the second equation (x-y = 10) to get (x-85 = 10) We can then solve for x to get x = 95.  

Now that we have solved for the two angles:

x = 95 and y = 85, we can tell that 85 is the acute angle because it is the angle that is less than 90 degrees.