If the ratio of two complementary angles is 4 to 5, explain how you would set up an equation to solve for the angle measures.

"A ratio of 4 to 5" means that dividing the two angles (whatever they may be) simplifies to 4/5. One example is 40 degrees and 50 degrees so 40/50 = 4/5. Another example is the pair of angles 20 and 25, so 20/25 = 4/5

For now, we don't know the angles. Let's call them x and y. Based on the info above, we can say

x/y = 4/5

also we know that x and y add to 90 degrees because the angles are complementary. So x+y = 90 which solves to y = 90-x if you isolate y.

Let's cross multiply on the first equation to get ...

x/y = 4/5

5*x = 4*y

Now replace y with 90-x. Solve for x

5x = 4 (90-x)

5x = 360 - 4x

5x+4x = 360

9x = 360

x = 360/9

x = 40

Use this to find y

y = 90 - x = 90 - 40 = 50

So the two angles are 40 degrees and 50 degrees.