The measure of one angle is six more than half the measure of another angle. If the angles are complementary, find both measures.

To solve this, let’s define the angles.

Let:
- \( x \) be the measure of the first angle.
- \( y \) be the measure of the second angle.

### Step 1: Set up the equations
We know two things:
1. The angles are complementary, meaning their sum is \( 90^\circ \). This gives us the equation:
   \[
   x + y = 90
   \]
2. The measure of one angle is six more than half the measure of the other angle. So, we also know:
   \[
   x = \frac{1}{2}y + 6
   \]

### Step 2: Solve the system of equations
We have two equations:
1. \( x + y = 90 \)
2. \( x = \frac{1}{2}y + 6 \)

Substitute the second equation into the first:
\[
\left( \frac{1}{2}y + 6 \right) + y = 90
\]
Now, solve for \( y \).

### Step 3: Solve for \( y \)
Combine like terms:
\[
\frac{1}{2}y + y + 6 = 90
\]
\[
\frac{3}{2}y + 6 = 90
\]
Subtract 6 from both sides:
\[
\frac{3}{2}y = 84
\]
Multiply both sides by \( \frac{2}{3} \) to isolate \( y \):
\[
y = 56
\]

### Step 4: Solve for \( x \)
Now that we know \( y = 56 \), substitute this back into the equation for \( x \):
\[
x = \frac{1}{2}(56) + 6
\]
\[
x = 28 + 6 = 34
\]

### Final Answer
The two angles are:
- \( x = 34^\circ \)
- \( y = 56^\circ \)

These angles are complementary since \( 34 + 56 = 90^\circ \).

Would you like more details on the solution?

Here are 5 related questions:
1. How do we solve systems of linear equations in general?
2. What are complementary angles, and why do they sum to 90 degrees?
3. How can you verify a solution to an angle-related problem?
4. What are some real-world applications of complementary angles?
5. How do you approach problems with unknowns expressed in terms of other variables?

**Tip**: When solving systems of equations, substituting one equation into another can help eliminate a variable, making the problem easier to solve.

Learn how to find the measures of two complementary angles when one angle is six more than half the measure of the other. This problem involves algebraic concepts such as setting up and solving linear equations and is suitable for students in Grades 6-8.

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