The larger of two complementary angles is 12 more than 5 times the measure of the other. Find the measures of the two angles.

The two angles are 77&13

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.