SOLUTION: THE RATIO OF THE COMPLIMENTS OF TWO ANGLES IS 3 :5. THE RATIO OF THE SUPPLEMENTS OF THE SAME TWO ANGLES IS 6:7. WHAT ARE THE TWO ANGLES

Question 233491: THE RATIO OF THE COMPLIMENTS OF TWO ANGLES IS 3 :5. THE RATIO OF THE SUPPLEMENTS OF THE SAME TWO ANGLES IS 6:7. WHAT ARE THE TWO ANGLES
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
First statement is ratio of complement of one angle to the complement of the other angle is 3/5.

If x = one angle and y = the other angle, this means that:

(90-x) / (90-y) = 3/5

Second statement is ratio of supplement of one angle to the supplement of the other angle is 6/7.

(180-x) / (180-y) = 6/7

Cross multiply and you get:

5 * (90-x) = 3 * (90-y)
7 * (180-x) = 6 * (180-y)

Solve first equation for y and you get:

First equation is:
5 * (90-x) = 3 * (90-y)
Remove parentheses to get:
5*90 - 5*x = 3*90 - 3*y
Add 3*y to both sides and add 5*x to both sides and subtract 450 from both sides to get:
3*y = 5*x + 270 - 450
This becomes:
3*y = 5*x - 180
Divide both sides by 3 to get:
y = (5*x-180)/3

Second equation is:
7 * (180-x) = 6 * (180-y) becomes:
Remove parentheses to get:
7*180 - 7*x = 6*180 - 6*y
add 7*x to both sides of the equation and add 6*y to both sides of the equation and subtract 7*180 from both sides of the equation to get:
6*y = 7*x + 6*180 - 7*180
This becomes:
6*y = 7*x - 180

Replace y with (5*x-180)/3 in the second equation to get:

6 * (5*x-180)/3 = 7*x - 180
This becomes:
2 * (5*x-180) = 7*x - 180
This becomes:
10*x - 360 = 7*x - 180
Subtract 7*x from both sides and add 360 to both sides to get:
3*x = 180
Divide both sides by 3 to get:
x = 60

You now have x = 60 degrees.

Go back to the first ratio to get:

First ratio is:
(90-x)/(90-y) = 3/5
Replace x with 60 to get:
(90-60)/(90-y) = 3/5
This becomes:
30/(90-y) = 3/5
multiply both sides by 5 and multiply both sides by (90-y) to get:
30*5 = 3 * (90-y)
Simplify and remove parentheses to getr:
150 = 270 - 3*y
Add 3*y to both sides and subtract 150 from both sides to get:
3*y = 270-150 = 120
Divide both sides by 3 to get:
y = 40

You have:
x = 60
y = 40

Plug these values into your original equation to prove these answers are good.

(90-x)/(90-y) = 3/5 becomes (90-60)/(90-40) = 3/5 becomes 30/50 = 3/5 becomes 3/5 = 3/5 which is true.

(180-x)/180-y) = 6/7 becomes (180-60) / (180-40) = 3/5 becomes 120 / 140 = 6/7 becomes 12/14 = 6/7 becomes 6/7 = 6/7 which is true.

Both answer are confirmed to be good.

Your answer is:

The two angles are 40 degrees and 60 degrees.

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