Question

The difference in the measures of the two complementary angles is $12°$. Find the measure of the angles.

Answer 1

Step 1. Determine the equations.

Assume that the angles are $x\mathrm{and}y$.

The sum of complementary angles is $180°$.

Determine the equation of the sum of angles:

$x+y=180°...\left(i\right)$

The difference of the angles is $12°$.

Determine the equation of the difference of angles:

$x-y=12°...\left(ii\right)$

Step 2. Solve by obtained equation to find the measures of angle.

Add the two obtained equations $\left(i\right)\mathrm{and}\left(ii\right)$ and simplify:

$\begin{array}{rcl}x+y+x-y& =& 180°+12°\\ & \Rightarrow & 2x=192°\\ & \Rightarrow & x=96°\end{array}$

Substitute $x=96°$into $x-y=12°$ and solve:

$\begin{array}{rcl}96°-y& =& 12°\\ & \Rightarrow & y=96°-12°\\ & \Rightarrow & y=84°\end{array}$

So, the measure of the angles are $84°\mathrm{and}96°$