Two angles are complimentary if they sum to 90 degrees. Two angles are supplementary if their sum is 180 degrees. To find the solution, 3 variables exist.
x: The measure of the angle
y: The measure of the supplement
z: The measure of the complement
x+y=180 (sum of supplemental angles is 180 degrees)
x+z=90 (sum of complementary angles is 90 degrees)
y=6z (The supplementary angle is 6 times the complementary angle)
Using substitution for y, the top equation becomes x+6z=180.
Using elimination with x+6z=180 and subtracting x+z=90, the result is 5z=90 and z=18. So the complementary angle is 18 degrees.
y=6(18)=108 or y=180-72
The solution checks when the measure of the angle is 72, the measure of the supplement is 108, and the measure of the compliment is 18.
Let us assume that the given angle is x
Supplement of x = 180-x
Complement of x = 90 - x
Given supplement of x = 6*complement of x
=> 5x= 360
=> x = 72
Supplement of 72 = 108
Complement of 72 = 18