Two angles are complementary. The sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees. What are the measures of the small and large angle?

1 Answer
Jan 12, 2016

Let the angles be $\theta \mathmr{and} \phi$.

Complementary angles are those whose sum is ${90}^{\circ}$.

It is given that $\theta \mathmr{and} \phi$ are complementary .
$\implies \theta + \phi = {90}^{\circ}$$\ldots \ldots \ldots . . \left(i\right)$

The sum of the measure of the first angle and one-fourth the second angle is 58.5 degrees can be written as a equation.

$\theta + \frac{1}{4} \phi = {58.5}^{\circ}$

Multiply both sides by $4$.

$\implies 4 \theta + \phi = {234}^{\circ}$

$\implies 3 \theta + \theta + \phi = {234}^{\circ}$

$\implies 3 \theta + {90}^{0} = {234}^{\circ}$

$\implies 3 \theta = {144}^{\circ}$

$\implies \theta = {48}^{\circ}$

Put $\theta = {48}^{\circ}$ in $\left(i\right)$

$\implies {48}^{\circ} + \phi = {90}^{\circ}$

$\implies \phi = {42}^{\circ}$

Therefore, the small angle is ${42}^{\circ}$ and larger angle is ${48}^{\circ}$