Question

$\angle e$ and $\angle f$ are complementary. the measure of $\angle e$ is $54^{circ}$ more than the measure of $\angle f$. find the measure of each angle.
$m\angle f =$
$m\angle e =$
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Explanation:

Step1: Set up equations

Let \(m\angle F = x\). Since \(\angle E\) and \(\angle F\) are complementary, \(m\angle E+m\angle F = 90^{\circ}\). Also, \(m\angle E=x + 54^{\circ}\).
Substitute \(m\angle E\) into the complementary - angle equation: \((x + 54^{\circ})+x=90^{\circ}\).

Step2: Solve for \(x\)

Combine like terms: \(2x+54^{\circ}=90^{\circ}\).
Subtract \(54^{\circ}\) from both sides: \(2x=90^{\circ}- 54^{\circ}\), so \(2x = 36^{\circ}\).
Divide both sides by \(2\): \(x=\frac{36^{\circ}}{2}=18^{\circ}\).

Step3: Find \(m\angle E\)

Since \(m\angle E=x + 54^{\circ}\), substitute \(x = 18^{\circ}\): \(m\angle E=18^{\circ}+54^{\circ}=72^{\circ}\).

Answer:

\(m\angle F = 18^{\circ}\)
\(m\angle E = 72^{\circ}\)