Two angles are complementary. The sum of the measure of the first angle and one-fourth of the second angle is 57.75 degrees. Find the measure of the angles.
To find the measure of the angles, let's assign variables to the angles. Let's call the first angle x and the second angle y.
Given that the two angles are complementary, we know that x + y = 90 degrees since the sum of complementary angles is always 90 degrees.
We also know that the sum of the measure of the first angle (x) and one-fourth of the second angle (1/4 * y) is 57.75 degrees. Mathematically, this can be written as:
x + 1/4 * y = 57.75
Now we can solve the system of equations to find the values of x and y.
First, multiply the second equation by 4 to get rid of the fraction:
4 * (x + 1/4 * y) = 4 * 57.75
4x + y = 231
Now we have a system of equations:
x + y = 90
4x + y = 231
Subtract the first equation from the second equation to eliminate y:
(4x + y) - (x + y) = 231 - 90
3x = 141
Divide both sides by 3 to solve for x:
x = 141 / 3
x = 47
Now we can substitute the value of x back into the first equation to solve for y:
47 + y = 90
y = 90 - 47
y = 43
Therefore, the measure of the first angle is 47 degrees, and the measure of the second angle is 43 degrees.