Two angles are complementary and are in the ratio 1:9

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