two angles are complementary. The sum of the measure of the first and the one-fourth of the second angle is 62.25 degrees find the measures of the angle.

To solve this problem, let's assume that the measures of the two angles are x and y degrees respectively.

Given that the two angles are complementary, we know that the sum of their measures is 90 degrees. Therefore, we can write the following equation:

x + y = 90 --(Equation 1)

Now, according to the given information, the sum of the measure of the first angle (x) and one-fourth of the second angle (1/4 * y) is 62.25 degrees. Therefore, we can write another equation:

x + (1/4)y = 62.25 --(Equation 2)

To find the measures of the angles, we can solve this system of equations simultaneously.

Let's use the method of substitution to solve these equations.

From Equation 1, we can express x in terms of y:
x = 90 - y

Now, substitute the value of x in Equation 2:
90 - y + (1/4)y = 62.25

Simplify the equation:
90 + (1/4)y - y = 62.25
90 - (3/4)y = 62.25

Move the constant term to the other side:
(3/4)y = 90 - 62.25
(3/4)y = 27.75

Now, multiply both sides of the equation by 4 to get rid of the fraction:
(3/4)y * 4 = 27.75 * 4
3y = 111

Divide both sides of the equation by 3 to isolate y:
(3y) / 3 = 111 / 3
y = 37

Now, substitute the value of y back into Equation 1 to find x:
x + 37 = 90
x = 90 - 37
x = 53

Therefore, the measures of the two angles are 53 degrees and 37 degrees, respectively.