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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.

The measure of an angle is 1/3 times more than its complement. Find the measure of each angle.

Learn how to find the measure of an angle when its measure is 1/3 times more than its complement. This problem involves algebraic manipulation and understanding the complementary angle theorem. Suitable for students in Grades 7-10.

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