The measure of two complementary angles are in the ratio 1 : 9. What are the degree measures of the two angles?

complementary angles sums up to 90 degrees

1:9 is the division of the total angle that is 90 degrees.

So the sum of ratios = 1+9=10

Dividing total angle by total ratio:

90/10 = 9

Now its clear that each single ratio contains 9 degrees.

so 1:9 means 9 x 1: 9 x 9

which means: 9 : 81