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?

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Answer 1 · 2016-01-12T20:14:36

Let the angles be $θ and ϕ$ $theta and phi$ .

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

It is given that $θ and ϕ$ $theta and phi$ are complementary .
$\Rightarrow θ + ϕ = 90^{\circ}$ $implies theta+phi=90^@$ $...........(i)$

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.

$theta+1/4phi=58.5^@$

Multiply both sides by $4$ .

$implies 4theta+phi=234^@$

$implies 3theta+theta+phi=234^@$

$implies 3theta+90^0=234^@$

$implies 3theta=144^@$

$implies theta=48^@$

Put $theta=48^@$ in $(i)$

$implies 48^@+phi=90^@$

$implies phi=42^@$

Therefore, the small angle is $42^@$ and larger angle is $48^@$