Question

Translate to a system of equations and then solve:

The difference of two complementary angles is $80$degrees. Find the measures of the angles.

Short answer

The measure of angles are${85}^{\circ }$ and $5^{\circ }$.

Step-by-step solution

Step 1. Given

We are given that the difference between two complementary angles is ${80}^{\circ }$.

Step 2. Formation of equations

We know that the sum of complementary angles is equal to ${90}^{\circ }$.

Let the one angle be $x$, and the other angle will be $y$, so the equation will be,

localid="1644600128864" $$\begin{gathered} x+y={90}^{\circ } \\ y={90}^{\circ }-x \end{gathered}$$,

The difference between the angles is ${80}^{\circ }$, so the equation will be,

$$\begin{gathered} x-y={80}^{\circ } \\ x-({90}^{\circ }-x)={80}^{\circ } \end{gathered}$$

Step 3. Solving the equation.

Solving $x-({90}^{\circ }-x)={80}^{\circ }$

$$\begin{gathered} x-{90}^{\circ }+x={80}^{\circ } \\ 2x-{90}^{\circ }={80}^{\circ } \\ 2x={80}^{\circ }+{90}^{\circ } \\ 2x={170}^{\circ } \end{gathered}$$

Dividing by $2$, we get

$x={85}^{\circ }$

And the other angle will be${90}^{\circ }-{85}^{\circ }=5^{\circ }$.

Step 4. Checking the solution  

Checking the solution by putting the value of $x,y$in the equations, we get

$$\begin{gathered} x+y={90}^{\circ } \\ {85}^{\circ }+5^{\circ }={90}^{\circ } \\ {90}^{\circ }={90}^{\circ } \\ x-y={80}^{\circ } \\ {85}^{\circ }-5^{\circ }={80}^{\circ } \\ {80}^{\circ }={80}^{\circ } \end{gathered}$$

This is true, hence the solution is correct.