We know that a complementary angle is a part of angles that adds up to a total of 90 degrees. Given there are two angle measures to be deduced here, and that we have two pieces of information, this is a system of equations. We know that the smaller of the angles (angle a) and the larger of the angles (angle b) add up to a total of 90 degrees. Given this information, we can write our first equation.
a+b=90
We also know that angle a, the larger of the two is 12 more than five times the measure of angle b. Given this information we can write our second equation.
a=5b+12
Now we have to isolate for a variable in our first equation.
a+b=90
a=90-b
Now we can substitute 90-b in for a in our second equation
90-b=5b+12
90=6b+12
78=6b
13=b
b=13
Now we can substitute the measure of b back in to our first equation to get our second answer.
a+b=90
a+13=90
a=77
We now know that using our knowledge of systems of equations, the measures of these angles are 13 and 77 degrees.
