Two angles are complementary. The sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees. What are the measures of the small and large angle?

Question

Two angles are complementary.  The sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees.  What are the measures of the small and large angle?

Violet Aldrich · Accepted Answer

Let the angles be #theta and phi#. Complementary angles are those whose sum is #90^@#. It is given that #theta and phi# are complementary .
#implies theta+phi=90^@##...........(i)# The sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees can be written as a equation. #theta+1/4phi=58.5^@# Multiply both sides by #4#. #implies 4theta+phi=234^@# #implies 3theta+theta+phi=234^@# #implies 3theta+90^0=234^@# #implies 3theta=144^@# #implies theta=48^@# Put #theta=48^@# in #(i)# #implies 48^@+phi=90^@# #implies phi=42^@# Therefore, the small angle is #42^@# and larger angle is #48^@#

Parker Allen · Answer

undefined Let x be the measure of the first angle and y be the measure of the second angle.

Given that the angles are complementary, we have:
x + y = 90

Also, the sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees, so we have:
x + (1/4)y = 58.5

From the first equation, we can express x in terms of y:
x = 90 - y

Substitute this expression for x into the second equation:
90 - y + (1/4)y = 58.5

Combine like terms:
90 + (1/4)y - y = 58.5
90 - (3/4)y = 58.5

Subtract 90 from both sides:
-(3/4)y = 58.5 - 90
-(3/4)y = -31.5

Multiply both sides by -4/3 to solve for y:
y = (-31.5) * (-4/3)
y = 42

Now that we have the measure of the second angle, we can find the measure of the first angle using the equation x = 90 - y:
x = 90 - 42
x = 48

So, the measure of the smaller angle is 48 degrees and the measure of the larger angle is 42 degrees.