# If an angle is 18 degrees less than twice its complement, how do you find the ratio of the angle to its complement?

May 2, 2015

Let's call the angle we seek $\alpha$

Then the complement is defined as ${90}^{o} - \alpha$
And twice the complement would then be:
$2 \cdot \left({90}^{o} - \alpha\right) = {180}^{o} - 2 \alpha$

So the equation turns into:
$\alpha = \left({180}^{o} - 2 \alpha\right) - {18}^{o} = {162}^{o} - 2 \alpha \to$
Add $2 \alpha$ to both sides, later divide by $3$:
$3 \alpha = {162}^{o} \to \alpha = {54}^{o}$

If $\alpha = {54}^{o} \to$complement=${36}^{o}$
Twice complement=${72}^{o}$
Difference=${18}^{o}$ Check!