The problem reads:

"The measure of an angle is $\frac{1}{3}$ times more than its complement. Find the measure of each angle."

Let the measure of the angle be $x$. Its complement would then be $90^\circ - x$, as the sum of complementary angles is always $90^\circ$.

The angle is said to be $\frac{1}{3}$ times more than its complement, so we can express this relationship as:

$x = \frac{1}{3}(90^\circ - x) + (90^\circ - x)$

We will now solve this equation.

Step 1: Simplify the equation

First, combine like terms:

$x = \frac{1}{3}(90^\circ - x) + (90^\circ - x)$

Distribute the $\frac{1}{3}$ across the terms in parentheses:

$x = \frac{90^\circ}{3} - \frac{x}{3} + (90^\circ - x)$

This simplifies to:

$x = 30^\circ - \frac{x}{3} + 90^\circ - x$

Now combine the constants:

$x = 120^\circ - \frac{x}{3} - x$

Step 2: Eliminate fractions

Multiply the entire equation by 3 to get rid of the fraction:

$3x = 360^\circ - x - 3x$

Step 3: Combine like terms

Bring all the $x$-terms to one side:

$3x + x + 3x = 360^\circ$

This simplifies to:

$7x = 360^\circ$

Step 4: Solve for $x$

Divide both sides by 7:

$x = \frac{360^\circ}{7} \approx 51.43^\circ$

So, the angle is approximately $51.43^\circ$.

Step 5: Find the complement

The complement of the angle is:

$90^\circ - 51.43^\circ = 38.57^\circ$

Thus, the angle measures approximately $51.43^\circ$, and its complement is $38.57^\circ$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are complementary angles, and why do their measures add to 90 degrees?
2. How do we solve equations involving fractions step-by-step?
3. How do angles relate to different geometric figures, like triangles?
4. How can you express the relationship between two complementary angles algebraically?
5. What are some real-world applications of complementary angles?

Tip: When solving equations with fractions, it's often helpful to clear the fractions by multiplying through by a common denominator early on to simplify calculations.