# The difference of two complementary angles is 40°. What are the angles?

May 24, 2018

$\alpha = {65}^{\circ}$
$\beta = {25}^{\circ}$

#### Explanation:

By definition: Two angles are Complementary when they
add up to ${90}^{\circ}$

so let's call the angles $\alpha$ and $\beta$

$\alpha + \beta = {90}^{\circ}$

$\alpha - \beta = {40}^{\circ}$

$\alpha = {40}^{\circ} + \beta$

substitute:

$\alpha + \beta = {90}^{\circ}$

$\left({40}^{\circ} + \beta\right) + \beta = {90}^{\circ}$

$2 \beta = {50}^{\circ}$

$\beta = {25}^{\circ}$

$\alpha + \beta = {90}^{\circ}$

$\alpha + {25}^{\circ} = {90}^{\circ}$

$\alpha = {65}^{\circ}$

May 24, 2018

One of the angles is 65˚.
The other is 25˚.

#### Explanation:

Complementary angles sum to 90˚.

$\angle 1 + \angle 2 = 90$

For our purposes, we will consider the first angle larger and the second angle smaller.

We are given another piece of information: that the difference of the angles is 40˚. Difference means subtraction, so we can construct the following equation.

$\angle 1 - \angle 2 = 40$

Now, we have two equations. There are many ways to solve this, but we will use the substitution method. Essentially, we will solve for one of the values and then plug it in to the other equation.

$\angle 1 = 40 + \angle 2$

(40+/_2)+/_2)=90

Then, we need to simplify to find $\angle 2$.

$40 + 2 \angle 2 = 90$

$2 \angle 2 = 50$

$\angle 2 = 25$

Finally, we can plug this value back in to the original equation to find the other angle measure.

$\angle 1 + 25 = 90$

$\angle 1 = 65$