Question

Translate to a system of equations and then solve:

The difference of two complementary angles is $20$degrees. Find the measures of the angles.

Short answer

The measure of angles are ${55}^{\circ }$and ${35}^{\circ }$.

Step-by-step solution

Step 1. Given

We are given that the difference between two complementary angles is ${20}^{\circ }$.

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, as per the given statement the equation will be,

$x+y={90}^{\circ }$

$y={90}^{\circ }-x$

localid="1644599820429" $$\begin{gathered} x-y={20}^{\circ } \\ x-({90}^{\circ }-x)={20}^{\circ } \\ x-{90}^{\circ }+x={20}^{\circ } \\ 2x-{90}^{\circ }={20}^{\circ } \\ 2x={20}^{\circ }+{90}^{\circ } \\ 2x={110}^{\circ } \end{gathered}$$

Dividing by $2$, we get

$x={55}^{\circ }$

And the other angle will be${90}^{\circ }-{55}^{\circ }={35}^{\circ }$

Step 4. Checking the solution  

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

$$\begin{gathered} x+y={90}^{\circ } \\ {55}^{\circ }+{35}^{\circ }={90}^{\circ } \\ {90}^{\circ }={90}^{\circ } \\ x-y={20}^{\circ } \\ {55}^{\circ }-{35}^{\circ }={20}^{\circ } \\ {20}^{\circ }={20}^{\circ } \end{gathered}$$

This is true, hence the solution is correct.