an someone help me a Geometry question?

Question

the question is:
2 angles r such that the ratio of the measures of their complements is 3:2, while the ratio of the measures of their supplements is 9:8 find the measure of each of the original angles

Please help me & tell me how you got the answer

thanks

Answer 1

Complementary angles add up to 90, and supplementary angles add up to 180. So let's set up some variables:

x = first angle
90 - x = complement of first angle
180 - x = supplement of first angle

y = second angle
90 - y = complement of second angle
180 - y = supplement of second angle

The ratios help set up the equations:
3/2 = (90 - x)/(90 - y)
3(90 - y) = 2(90 - x)
270 - 3y = 180 - 2x
2x - 3y = -90

9/8 = (180 - x)/(180 - y)
9(180 - y) = 8(180 - x)
1620 - 9y = 1440 - 8x
8x - 9y = -180

Now we have two equations and two unknowns, and we can solve that thorugh some manipulation. If we multiply the first equation by -3, and then add the two equations, we can eliminate the y terms and solve for x:

-3(2x - 3y = -90) = -6x + 9y = 270
(-6x + 9y = 270) + (8x - 9y = -180) = (2x = 90)
x = 45

Now we can solve for the other angle y:
2x - 3y = -90
2(45) - 3y = -90
90 - 3y = -90
-3y = -180
y = 60

So let's check these out. The complement of 45 is 45, and the complement of 60 is 30, and the ratio is 45/30 = 3/2. The supplement of 45 is 135, and the supplement of 60 is 120, and the ratio is 135/120 = 9/8.

Since this checks out, the original angles are 45 and 60.